Anti-inflammatory activity of flavonols via inhibiting MAPK and NF-κB signaling pathways in RAW264.7 macrophages

Fisetin (Fis), quercetin (Que), and myricetin (Myr) are flavonols with similar structure but different number of hydroxyl groups. The present research focused on the anti-inflammatory effect of these three flavonols in lipopolysaccharide-stimulated RAW264.7 cells. The number and site of hydroxyl group in flavonols obviously affected their anti-inflammation activity. These flavonols suppressed the overproduction of nitric oxide. Fis showed the best activity with an inhibition rate of 52% at 20 μM. Moreover, the flavonols reduced the levels of ROS, TNF-α, and IL-6. The mechanistic study showed that they inhibited the activation of NF-κB and MAPK pathways by suppressing the phosphorylation of IκBα, p65, JNK, ERK, p38, MEK, and reducing the nuclear translocation of NF-κB p65. In addition, the metabolism of the flavonols was examined. The results indicated that Fis was both methylated and glucuronidated. Que and Myr were mainly transformed into methylated products. This study highlights the anti-inflammatory activity of flavonols, particularly Fis, which has the potential for the prevention or treatment of inflammation as an adjuvant medicine or food additive.

Introduction
Inflammation is a kind of physiological responses in immune system that protects human bodies against tissue injury or infection (Chen et al., 2020b;Liu et al., 2021). Inflammatory response under normal condition has protective effects on body functions, but it would cause host injury when lacking control (Ayala et al., 2019). For instance, chronic inflammatory diseases have been reported to associated with rheumatoid arthritis , cardiovascular disease (Furman et al., 2019), and Alzheimer's disease (Qian et al., 2021). Macrophage is an important immune cell and plays a pivotal role during inflammation in host defenses against pathogens infection. Lipopolysaccharide (LPS)-induced inflammation in RAW264.7 murine macrophagies is a widely used inflammation model. The stimulated cells exhibit typical inflammatory responses by producing nitric oxide (NO) and reactive oxygen species (ROS) levels, and generating pro-inflammatory mediators such as interleukin-6 (IL-6), tumor necrosis factor-alpha (TNF-α), and inducible NO synthase (iNOS) (Lim et al., 2021;Tian et al., 2021). Several signaling pathways were demonstrated to be closely associate with inflammation in LPS-induced RAW264.7 cells. The transcriptional regulator nuclear factor-κB (NF-κB) is vitally involved in the pathogenesis of various inflammatory diseases, which modulates the production of many cytokines and mediators (Chen et al., 2020a,b). Besides, NF-κB could be activated through mitogen activated protein kinase (MAPK), such as c-Jun NH2-terminal kinase (JNK), extracellular signal-regulated kinase (ERK) and p38, which are regulated by mitogen-activated protein kinase (MEK) (Buchanan et al., 2010;Huang et al., 2012).
Dietary flavonols are typical and promising phytochemicals belonging to the flavonoids. They have been recognized to exert various effects such as anti-oxidantion, anti-microbe, anticancer, antiinflammation, and anti-hypoglycemia (Barreca et al., 2021;Wang et al., 2006;Zhao et al., 2020). Fis (3,3 ′ ,4 ′ ,7-tetrahydroxyflavone), Que (3,3 ′ ,4 ′ ,5,7-pentahydroxyflavone), and Myr (3,3 ′ ,4 ′ ,5,5 ′ ,7-hexahydroxyflavone) are three structural related flavonols, which have the same 3-hydroxyflavone backbone (Fig. 1A). However, they have minor structural differences. Fis does not have a 5-OH group and Myri has a 3 ′ -OH group when they were compared with Que. These minor differences might attribute to the difference in their bioactivities (Funakoshi-Tago et al., 2011). Fis is widely found in strawberry, onion, and grape (Arai et al., 2000). Que is a major flavonoid of many fruits and vegetables such as mango, apple and tea (Kandemir et al., 2021). Also, Myr is distributed in many foods including berry, onion, and red wine . These three compounds are common flavonols distributing in many foods, which would have more application value for exploring potential adjuvant treatment for inflammation. Different structures of polyphenols were observed to have various pharmacological activities, absorption, metabolism (Hollman, 2004;Tian et al., 2021). However, there is no comparative and systematic research on the anti-inflammatory effect and metabolism of Fis, Que, and Myr in vitro. Therefore, in this present work, we selected the compounds of Fis, Que, and Myr to examine their anti-inflammatory potential and the underlying mechanisms and metabolism in LPS-induced RAW264.7 macrophage model.

Cell culture and flavonols treatment
A mouse macrophage cell line RAW264.7 was purchased from the American Type Culture Collection (ATCC, VA, USA). The RAW264.7 cells were cultured in DMEM/High Glucose with 1% P/S (v/v) and 10% (v/v) FBS and maintained in the atmosphere of 95% humidity and 5% CO2 at 37 • C. The cells were sub-cultured into a new 50 cm 2 flask (Thermo Fisher Scientific, MA, USA) when they reached 80% confluence. Fis, Que, and Myr were dissolved in DMSO to yield stock solutions of 100 mM and stored at − 20 • C. Then they freshly diluted with DMEM with 10% FBS to make working solutions, which the concentrations of DMSO were less than 0.1%.

Cytotoxicity assay
Cell viability was detected by MTT method with a few modifications (Miao et al., 2019). RAW 264.7 cells (1 × 10 4 cells/well) were seeded into 96-well plates and adhered overnight. Then, the cells were treated with DMEM or increasing concentrations of flavonols (Fis at 5, 10, and 20 μM; Que at 10, 20, and 40 μM; Myr at 20, 40, 80 μM) for 24 h. Then, the supernatant was removed and 100 μL solution of MTT (10 mg/ml) with flesh DMEM (MTT: DMEM = 1: 10) was added to each well, incubating at 37 • C for 3 h. Thereafter, the MTT media were carefully replaced with 150 μL DMSO for dissolving the formazan crystals. After shaking for 30 min, the absorbance at 490 nm was read through a microplate spectrophotometer reader (Molecular Devices, USA). And the cell viability of the control group was regarded as 100%. The experiments were performed in triplicate of each flavonol.

Measurement of NO
RAW264.7 cells were seeded in 24-well plates at a density of 4 × 10 4 cells/well and incubated overnight at 37 • C with 5% CO 2 . After pretreatment with indicated concentrations of three flavonols for 4 h, cells were stimulated with LPS (1 μg/ml) for another 12 h. Then, 100 μL supernatants of were transformed into a new 96-well plate. And the Griess reagent was added and the 96-well plate was shaken for 30 min in dark conditions. The absorbance of the final solution was measured at 540 nm by a microplate reader. The inhibition rates of NO release were shown to express the percentage of NO production in flavonols and only LPS groups. This assay was performed in triplicate.

Enzyme linked immune Sorbent assay (ELISA)
RAW 264.7 cells were pretreated with different concentrations of Fis, Que, and Myr for 4 h and stimulated with LPS (1 μg/mL) for 12 h. The cell supernatant samples were collected to detect the cytokine levels of IL-6 and TNF-α by ELISA kits according to the manufacturers' protocols. The absorbance at 450 nm was read through a microplate spectrophotometer reader.

Detection of intracellular ROS production
RAW 264.7 (2 × 10 5 cells/well) cells were seeded in 6-well plates and cultured for 12 h. Cells were pretreated with Fis at 20 μM, Que at 40 μM, and Myr at 80 μM for 4 h, and exposed to LPS (1 μg/ml) for 12 h. DCFH-DA reagent (10 mM) was diluted to 5 μM by DMEM (1:2000) in dark condition. The cells were discarded and added 1 mL DCFH-DA diluted solution, incubating at 37 • C for 20 min. Then, the solution was discarded and cells were washed three times with DMEM to remove DCFH-DA that existing on the cell surfaces. Finally, cells were collected into 1.5 mL tubes and suspended with 200 μL PBS to determine the intracellular ROS production by flow cytometer (Beckman, USA). The FlowJo software (BD, America) was conducted for analysis.
The immunofluorescence analysis was also done to investigate the ROS production. The preliminary procedure was almost the same as for flow cytometer detection. However, the cells were not need to be collected instead of filling with 300 μL PBS, and they were taken photos using DMI8 inverted fluorescent microscope (Leica, Germany).

Western blot analysis
RAW 264.7 (2 × 10 5 cells/well) cells were seeded in 6-well plates and cultured for 12 h. Cells were stimulated with LPS (1 μg/mL) for 30min or 12 h in the absence or presence of different concentrations of flavonols for 4 h. The supernatants were discarded after centrifugating for 6 min at 2000 rpm. And 100 μL RIPA buffer containing with 1% protease inhibitor cocktail and 1% phenylmethylsulfonyl fluoride (PMSF) (Beyotime, China) was added into harvested cells and incubated for 40 min on ice. Then they were centrifuged (1500 rpm, 15 min) at 4 • C and collected the supernatants to obtain the total protein. Nuclear protein was fractionated with nuclear extraction reagent from Nuclear and Cytoplasmic Protein Extraction kit (Beyotime, China) according to the manufacture's instruction. The concentrations of total protein and nuclear protein were determined by Thermo Pierce BCA protein assay kit (Thermo Scientific, USA). Proteins were dissolved by SDS/PAGE loading buffer and boiled for 10 min at 99 • C. Subsequently, the same amounts of protein (20 μg) were isolated by 8%-10% SDS-PAGE gels and transferred to polyvinylidene fluoride (PVDF) membranes (Bio-Rad, USA). The membranes were blocked by 5% de-fatted dry milk in Tris-Buffered Saline/Tween 20 (TBST) for 1 h and probed with corresponding specific primary antibodies diluted with TBST at 1:1000 overnight at 4 • C (GAPDH, Lamin B, IκBα, p-IκBα, NFκB p65, p-p65, JNK, p-JNK, ERK, p-ERK, p38, p-p38, MEK, p-MEK, and iNOS). The membranes were washed with TBST for 3 times and incubated with the secondary antibody diluted with TBST at 1:2000 for 1 h at room temperature. After washing with TBST for another 3 times, the protein bands were visualized by an enhanced chemiluminescence detection kit and scanned by Bio-Rad ChemiDoc™ MP Imaging System (Bio-Rad, USA). The bands were quantified by Image J densitometry software.

Immunofluorescence assay
RAW 264.7 cells (1 × 10 4 ) were cultured in the confocal dishes (NEST Biotechnology Co., Ltd, China) overnight. Then, cells were pretreated with the highest concentration of selected flavonols for 4 h, followed by inducing LPS (1 μg/mL) for 30 min. After treatment and stimulation, cells were washed with cold PBS for 3 times and immediately fixed in 4% paraformaldehyde for 15 min. Cold PBS was used to washed the cells for another three times and 0.5% Triton X-100 solution was added for cell permeabilization for 20 min at room temperature. Next, cells were blocked with 3% BSA for 1 h in the cell incubator and followed by incubating with primary antibody rabbit NF-κB p65 (1: 500 dilution) for 2 h at room temperature. Cells were washed with ice-cold PBS for 3 times and incubated with secondary antibody Alexa Fluor 568 goat anti-rabbit IgG (H + L) (1:500 dilution) for 1 h at dark at room temperature. Lastly, cells were stained by 4 ′ ,6-diamidino-2-phenylindole (DAPI) for 10 min at dark for labeling cell nucleus and photographed by Leica TCS SP8 Confocal Laser Scanning Microscope System (Leica, USA).

Metabolism of flavonols in RAW264.7 cells
RAW cells were cultured in 6-well plates of DMEM (10% FBS) culture medium without P/S. Different time points (5 min, 30 min, 6 h, 16h) after adding flavonols were set up to collect the cells. The concentration treatments of flavonols and LPS were the same as section 2.3. Besides, LPS did not added into cells at 5 min and 30 min, but added after pretreating flavonols for 4 h. At different time points, 500 μL of cell culture medium supernatant was removed and pipetted into a 1.5 mL centrifuge tube containing 1000 μL of methanol (HPLC grade, − 20 • C). The remaining medium was immediately discarded and cells were washed with 1 mL PBS three times. Cells were then scraped using 100 μL MiliQ water once and 100 μL of methanol (HPLC grade, − 20 • C) twice respectively. The solutions were combined in a 1.5 mL centrifuge tube and sonicated at 0 o C for 30 min. Finally, all tubes were centrifuged at 20,627 g for 15 min at 0 • C and the supernatants were used to analyze the metabolism of flavonols in RAW264.7 cells by UPLC-LTQ Orbitrap XL hybrid FTMS system in positive ion (ESI) mode.
UPLC-MS/MS analysis was performed on a Thermo UltiMate 3000 UHPLC system and a Thermo Scientific LTQ Orbitrap XL hybrid FT Mass Spectrometer (San Jose, CA, USA). Chromatographic separation on the system was achieved on a Waters ACQUITY UPLC HSS T3 1.8 μm, 2.1 × 150 mm column (Waters Technology, USA). The mobile phase was a gradient of 0.1% formic acid in MiliQ water (A) and acetonitrile (B) at a flow rate of 0.3 mL/min. The injection volume was set at 10.0 μL. The elution gradient was set as follows: 0-5 min, 15% B; 5-10 min, 15-25% B; 10-20 min, 25-40% B; 20-25 min, 40-45% B; 25-30 min, 45-15% B. The MS detection was carried out by using electrospray ionization (ESI) on positive ionization mode with capillary temperature of 380 • C, sheath gas (N 2 ) flow rate of 48 arbitrary units (arb), auxiliary gas flow rate of 15 arb and ion spray voltage of 4.5 kV. The acquisition time was 30 min and full MS scans were acquired in the range of m/z 80-1500 with a mass resolution of 30,000. MS 1 and MS 2 data including retention time, quasi-molecular ions and fragment ions information were collected using Xcalibur software (Thermo Fisher Scientific, CA, USA) for data collection and analysis. These data were combined with the characteristic fragment ions, chemical structure, and reaction characteristics of Fis, Que, and Myr to analyze their metabolism in macrophagy cells.

Statistical analysis
All data were analyzed by GraphPad Prism 7 software and shown as mean ± SD. Significant differences between the experimental groups were determined through one-way ANOVA. p < 0.05 was regarded statistically significant. Every experiment was conducted for at least three times.

Toxicity of flavonols on RAW 264.7 cells
In order to evaluate whether flavonols have potential cytotoxicity at the texted concentrations, MTT assay was used to determine the effect of flavonols on the viability of RAW 246.7 cells. Cells were incubated with flavonols for 24 h and the results showed that the cell viability was significantly decreased when treated with over 20 uM Fis and over 40 uM Que comparing with the control group. Therefore, the safe concentrations of flavonols were Fis at 5, 10, and 20 μM, Que at 10, 20, and 40 μM, and Myr at 20, 40, 80 μM (Fig. 1B), which were used for the following experiments.

Flavonols inhibited LPS-induced NO release and pro-inflammatory proteins production in RAW264.7 cells
Excessive NO production is a typical sign of inflammation in activated RAW264.7 cells (Peng et al., 2020). The nitric oxide release in the LPS group was significantly higher than the control group (p < 0.0001) (data not shown). Results showed that three flavonols could dramatically inhibited NO release dose-dependently compared with the LPS group ( Fig. 2A). Specifically, the inhibitory rates were 6.84%, 17.92%, and 51.92% in Fis group of 5, 10, and 20 μM, 20.25%, 36.67%, and 59.97% in Que group of 10, 20, and 40 μM, and 14.47%, 17.71%, and 25.91% in Myr group of 20, 40, 80 μM, respectively. Regarding the same concentration of 20 μM for comparison, Fis has the strongest inhibition effect (51.92%) on NO release, and Que ranked the second. Myr showed the least inhibition effect (14.47%) at 20 μM, and it only had 25.91% inhibition even at its highest concentration at 80 μM. The iNOS protein played a vital role during inflammation (Fu et al., 2014). Western blotting was performed to evaluate whether the NO release is related to the modulation of iNOS expression. The result showed that LPS significantly increased the expression of iNOS protein, which could be reversed by the three flavonols with variable effects (Fig. 2D and E). Similarly, the expression levels of iNOS were inhibited to the greatest extent by Fis at 20 μM, followed by Que, while Myr had the lowest effect.
This result was consistent with that of NO inhibition.
In addition, great amounts of pro-inflammatory proteins were produced by activating macrophages when the inflammation triggers (Su et al., 2017). LPS strongly increased the expression levels of IL-6 and TNF-α, which were reduced by flavonols in dose-dependent manners ( Fig. 2B and C). Notably, these inflammatory mediators' expression was inhibited greatly by Fis group at 20 μM treatment. Taken together, the above data implied that the three flavonols effectively suppressed LPS-stimulated inflammatory responses in macrophages, among which Fis exhibited the strongest anti-inflammatory effect.

Flavonols suppressed LPS-induced ROS production in RAW264.7 cells
It was reported that inflammation is closely related to oxidative stress, with increased intracellular ROS levels. ROS was regarded as one of the inflammatory mediators (Ji et al., 2018). Flow cytometry analysis was performed to text the ROS levels under LPS (1 μg/mL, 12 h) induction. LPS stimulated excessive ROS production in cells compared with the control group, while the ROS production was decreased in the presence of all selected flavonols, especially in Fis group at 20 μM ( Fig. 3A and B). Moreover, the fluorescence of ROS in cells captured by Lecia DMI8 microscope showed the same tendency as the results of flow cytometry (Fig. 3C). These data confirmed that the three flavonols had the ability to reverse LPS-induced ROS production to some extent. Notably, Fis could almost completely inhibit the ROS production.

Flavonols inhibited the activation of MAPK signaling pathways in LPS-induced RAW264.7 cells
To further explore whether the anti-inflammatory effects of flavonols are associated with MAPK signaling pathways, the expressions of phosphorylation of JNK, ERK, p38, and MEK proteins were investigated by western blotting assays. RAW264.7 cells were pre-treated with flavonols and stimulated with LPS (1.0 μg/ml) for 30 min. As shown in Fig. 4A, the ratios of p-ERK/total ERK and p-MEK/total MEK were upregulated substantially by LPS but inhibited by Fis at 20 μM. However, the ratios of p-JNK/total JNK and p-p38/total p38 slightly decreased with no significant differences between LPS group and Fis group. The induced with LPS (1 μg/ml) for 12 h. The supernatant were used for (A) detection of NO release with Griess reagent and determination of (B) IL-6 and (C) TNF-α with ELISA kits. The cells were collected for analysis of (D, E) iNOS expression by western blotting. Data were mean ± SD. *P < 0.05, **P < 0.01, ***P < 0.001, and ****P < 0.0001 versus LPS, #### P < 0.0001 versus Control. ratios of p-ERK/total ERK and p-MEK/total MEK were remarkably lower in 20 μM and 40 μM Que groups than LPS group (Fig. 4B). However, the ratios f p-JNK/total JNK and p-p38/total p38 only slightly decreased in Que group with no significant differences compared to LPS group. Notably, Myr could down-regulated p-JNK, p-ERK, p-MEK, and p38 protein expressions apparently (Fig. 4C). These results suggest that antiinflammatory activities of Fis, Que, and Myr might be through blocking the activation of MAPK signaling pathways.

Flavonols inhibited the activation of NF-κB signaling pathways in LPS-induced RAW264.7 cells
NF-κB is a typical transcription factor for regulating inflammatory genes expression in LPS-induced inflammation (Ding et al., 2022). To further reveal the molecular mechanisms of inflammatory suppression of flavonols, the activity of NF-κB signaling pathway was investigated by Western blot. The expressions of phosphorylated-p65 and IκBα proteins  determined by western blotting. Values were mean ± SD. *P < 0.05, **P < 0.01, and ****P < 0.0001 versus LPS, #### P < 0.0001 versus Control.
were increased in LPS-induced RAW264.7 cells, but they were reversed by Fis treatment at the highest concentration at 20 μM (Fig. 5A). These proteins were also markedly declined at the middle and high dose of Que treatment (Fig. 5B). In addition, Myr could decrease these proteins expressions in a dose-dependent manner (Fig. 5C). The nuclear translocation of p65 protein is an important step for the function of NF-κB pathway, so the nuclear cell fractions were separated for the determination of p65 . Results showed that the nuclear expression levels of NF-κB p65 were dramatically increased in the LPS stimulation compared with control group, while flavonols pretreatment inhibited the LPS-induced nuclear accumulation of NF-κB p65. The inhibition effects were significant in Fis and Que groups at 20 μM ( Fig. 5D and E) and Myr group at 40 and 80 μM in (Fig. 5F). An immunofluorescence assay was performed to confirm the nuclear translocation of NF-κB p65. The red color intensity which presented the p65 fluorescence was accumulated in the nucleus after LPS induction, but it was reversed by flavonols treatment, which was nearly the same intensity as the control group (Fig. 5G). Overall, these data implied that three flavonols significantly inhibited the activation of LPS-stimulated NF-κB signaling pathway.

Metabolism of flavonols in RAW264.7 cells
To further reveal the anti-inflammatory effect of flavonols, it is necessary to explore the metabolism of flavonols in macrophages. Medium with flavonols were added into cells and collected them at different time points (5 min, 30 min, 6 h, and 16h) followed by UPLC-MS/MS analysis. In the first 30 min, the metabolism of the three flavonols were examined in macrophages. LPS was then added into cells after 4 h. The result showed that Fis, Que, and Myr quickly entered the cells at only 5 min and degraded at different times. As shown in Fig. 6A, the methylation reaction was speculated in Fis in the molecular weight (Mw) of 301.07 at 5 min. The intensity of methylated Fis was increased continuously and accumulated to the maximum value at 16 h. Glucuronidation of Fis (Mw: 463.09) was detected later. Finally, Fis was both methylated and glucuronidated according to the Mw of 477.10. Que was metabolized rapidly from 5 min to 30 min and mainly transformed into methylated Que (Mw: 317.06) (Fig. 6B). Myr was metabolized quickly as well and then transformed to methylated Myr (Mw: 333.06). However, methylated Myr product was almost undetectable at 30 min (Fig. 6C).

Discussion
RAW 264.7 macrophages are transformed from the Abelson leukemia virus, which is an ideal model for studying inflammatory effect. And LPS can fully activate macrophages to trigger inflammation, which is mediated by NO release (Hwang et al., 2019;Malayil et al., 2022). NO is a kind of free radicals to regulate human body functions, but excessive NO synthesized by iNOS will lead to autoimmune disorders, cytotoxicity, and inflammation (Zhang et al., 2019). In the present study, the Griess assay and Western blot analysis implied that the NO overproduction was related to iNOS overexpression when LPS triggered. Flavonols suppressed the production of NO, which were highly consistent with the down-regulation of iNOS protein. In addition, Fis treatment presented the greatest inhibition effect ( Fig. 2A, 2D-E). Excessive NO release could stimulate proinflammatory cytokines and these cytokines mediators could promote the NO production, which formed a positive feedback loop leading to increased inflammation . All selected flavonols markedly inhibited the proinflammatory cytokines of IL-6 and TNF-α ( Fig. 2B and C).
Oxidative stress played a critical role in inflammatory diseases, of which ROS level was an important mediator. Excessive ROS production disrupts the balance of oxidant/antioxidant in the human bodies, leading to damages of lipids, protein, and DNA (Wang et al., 2021b). DCFH-DA cell-permeable fluorescent probe is commonly used to determine intracellular ROS levels in cells or tissues by flow cytometry or confocal microscopy (Shen et al., 2018). The flow cytometry and fluorescence images showed that flavonols especially Fis dramatically decreased the ROS generations in LPS-induced macrophages, indicating flavonols could showed anti-inflammation on RAW264.7 cells by reducing oxidative stress (Fig. 3). The structure-activity relationship (SAR) could reveal the antioxidant characteristics of these structurally related flavonols (Zhong et al., 2022). The anti-oxidant potencies of dietary flavonoids are reported to be closely associated with the site and number of hydroxylations on the A and B rings. However, pyrogallol structure could form superoxide anions in the presence of stable free radicals, which could be good pro-oxidants (Acker et al., 1996). Thus, flavonoids containing a pyrogallol group (such as Myr) might have weaker antioxidant activity because of the counteraction effect. In addition, hydroxy group position would have influence on the stability of flavonols-protein complex, contributing to varying degrees of action on the downstream signaling pathways (Wang et al., 2021a). Taking together, these flavonoids, particularly Fis, can be good antioxidants to scavenge free radicals and ameliorate inflammation (Chen et al., 2019).
MAPKs and NF-κB signaling pathways are activated through LPS stimulation to regulate the expression of pro-inflammatory cytokines. MAPKs belong to serine-threonine protein kinases (Chen et al., 2017). The phosphorylation of MAPKs activates the transcription factor activator protein 1 (AP1), which then translocates to nucleus, causing the expression of pro-inflammatory genes such as TNF-α (Wu and Schauss, 2012). The main components of MAPK signaling pathways, i.e. JNK, ERK, p38, and MEK and their phosphorylation forms were examined. Flavonols significantly inhibited JNK, ERK, p38, and MEK phosphorylation in LPS-induced RAW264.7 cells (Fig. 4), which implied that the inhibition effect of MAPK may be involved in suppressing anti-inflammatory effect by flavonols. The transcription factor NF-κB has been proved to be an up-stream signal in inflammatory cascade reaction which can increase the chemokines and pro-inflammatory cytokines expression including IL-6, TNF-α, iNOS, etc. (Liu and Malik, 2006;Park et al., 2007). NF-κB exists in an inactive form in resting cells which is bound with IκB-α protein in the cytoplasm. However, the IκB-α would be activated and degrades upon LPS stimulation, then NF-κB would translocate into the nucleus and promote the expression of variable pro-inflammatory genes (Gilroy et al., 2004;Pham et al., 2017). Therefore, the effects of flavonols on NF-κB were examined. The results presented that flavonols remarkably attenuated the phosphorylation of IκB-α and p65 and inhibited the accumulation of NF-κB p65 in nucleus (Fig. 5). It has been demonstrated that MAPKs modulated NF-κB pathway. Inhibiting MAPK pathway could suppress the NF-κB activation (Surh et al., 2001;Zhong et al., 2012). We assume that the suppression of inflammatory cytokines and mediators of the three flavonols were probably via inhibiting MAPK and NF-κB signaling pathways. And the suppression of NF-κB might be, at least partially, attributed to the inhibition of MAPK. In addition to inhibiting the activation of MAPK and NF-κB signaling pathways, some signal cascades also involved in flavonoids anti-inflammation effects such as JAK/STAT signaling pathway and the inflammasome formation. Flavonoids could regulate the activities of other signaling molecules, such as tyrosine kinase, phosphoinositol kinase, and protein kinase C (Maleki et al., 2019). In addition, other binding targets of flavonoids were found in anti-inflammatory responses, including G-protein coupled receptors, estrogen receptors, and aryl hydrocarbon receptors (Safe et al., 2021). Based on our results, we hypothesized that the regulation of signaling pathways by the flavonols, could be attributed to the antioxidant property and direct interaction with related proteins in the pathways.
To further explore the anti-inflammation mechanisms of flavonols, the metabolism of the three flavonols in RAW264.7 cells was explored. Several metabolic products of Fis were formed by methylation and glucuronidation. Both Que and Myr were transformed into methylated products. The glucuronidation of Que and Myr was undetectable (Fig. 6). Methylation and glucuronidation were discovered for other flavonoids, such as catechin, luteolin, and catechin (Hai et al., 2020; with LPS (1 μg/ml) for 30 min. The protein expression levels of p-IκBα, IκBα, p-p65, and p65 were measured by western blotting from total cell lysates in Fis (A), Que (B), and Myr (C). The level of nuclear p65 in cells treated with Fis (D), Que (E), and Myr (F) was tested by western blotting. Lamin B was used as internal loading controls. Immunofluorescence staining of nuclear p65 was observed by confocal microscopy (G). Red fluorescence represented p65 protein and blue fluorescence (DAPI) indicated nuclei. Values were mean ± SD. *P < 0.05, **P < 0.01, and ***P < 0.001 versus LPS, # P < 0.05, ## P < 0.01, ### P < 0.001, and #### P < 0.0001 versus Control. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) Manach et al., 2004). Some studies have claimed that the metabolites and conjugates are bioactive forms of flavonols arising from their aglycone forms on absorption. One research has presented that the hydroxyl groups at 3 ′ site of Que was methylated to isorhamnetin. Isorhamnetin could absorbed more adequately and eliminated more slowly than parent Que form, which showed stronger antifibrotic effect on hepatic stellate cells than Que (Ganbolda et al., 2019). Also, methylated Myr could increase the activity of enhancing the life span from Caenorhabditis elegans (C. elegans). Methylated Myr enhanced stress resistance of C. elegans, which showed dependent on DAF-16 transcription factor (Büchter et al., 2015). The metabolism of Fis in vivo played a vital role in anticancer activities. Methylated Fis presented more cytotoxic than the parent Fis in tumor cells and glucuronidated Fis had longer terminal half-lives than Fis (Touil et al., 2011). These findings indicated that the anti-inflammatory effect of flavonols might be improved through adding glucuronide group and methyl group on functionally position sites. Thus, the methylated and glucuronidated derivatives of three flavonols are needed to be synthesized to test whether the derivatives can improve the biological effect and how they work against inflammation in further study.

Conclusion
This research proved that the flavonols Fis, Que and Myr exerted anti-inflammatory effects in LPS-induced RAW 264.7 cells. Fis, Que, and Myr can inhibit NO and ROS production in LPS-stimulated RAW264.7 cells. These flavonols reduced the expression levels of ROS, TNF-α and IL-6 and suppressed the activation of NF-κB and MAPK pathways by reducing the levels of phosphorylation of IκBα, p65, JNK, ERK, MEK, and the nuclear translocation of NF-κB p65 (Fig. 7). The three flavonols, particularly Fis, has the potential for the prevention or treatment of inflammation as an adjuvant medicine or food additive.

Data availability
The data presented in this study are available on request from the corresponding author.

Funding
This research was supported by the National Natural Science


s2fieldsofstudy:
Chemistry
Art
Physics
Computer Science

text:
Local and multilinear noncommutative de Leeuw theorems

Let $\Gamma<G$ be a discrete subgroup of a locally compact unimodular group $G$. Let $m\in C_b(G)$ be a $p$-multiplier on $G$ with $1 \leq p<\infty$ and let $T_{m}: L_p(\widehat{G}) \rightarrow L_p(\widehat{G})$ be the corresponding Fourier multiplier. Similarly, let $T_{m \vert_\Gamma}: L_p(\widehat{\Gamma}) \rightarrow L_p(\widehat{\Gamma})$ be the Fourier multiplier associated to the restriction $m|_{\Gamma}$ of $m$ to $\Gamma$. We show that $$ c( {\rm supp}( m|_{\Gamma} ) ) \Vert T_{m \vert_\Gamma}: L_p(\widehat{\Gamma}) \rightarrow L_p(\widehat{\Gamma}) \Vert \leq \Vert T_{m }: L_p(\widehat{G}) \rightarrow L_p(\widehat{G}) \Vert, $$ for a specific constant $0 \leq c(U) \leq 1$ that is defined for every $U \subseteq \Gamma$. The function $c$ quantifies the failure of $G$ to admit small almost $\Gamma$-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, $c(\Gamma) =1$ when $G$ has small almost $\Gamma$-invariant neighbourhoods. Our result thus extends the De Leeuw restriction theorem from [CPPR15] as well as De Leeuw's classical theorem [Lee65]. For real reductive Lie groups $G$ we provide an explicit lower bound for $c$ in terms of the maximal dimension $d$ of a nilpotent orbit in the adjoint representation. We show that $c(B_\rho^G) \geq \rho^{-d/2}$ where $B_\rho^G$ is the ball of $g\in G$ with $\Vert {\rm Ad}_g \Vert<\rho$. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear De Leeuw restriction theorem for pairs $\Gamma<G$ with $c(\Gamma) = 1$. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.

Introduction
In his seminal paper Karel de Leeuw [Lee65] proved several fundamental theorems about Fourier multipliers on the Euclidean space R n . Recall that m ∈ L ∞ (R n ) is called a p-multiplier if the map T m on L 2 (R n ) determined by F 2 (T m f ) = mF 2 (f ) with F 2 the unitary Fourier transform extends to a bounded map on L p (R n ). Using Pontrjagin duality the definition of p-multipliers may be extended to any locally compact abelian group G and we call m ∈ L ∞ (G) a p-multiplier if the map T m on L 2 ( G) determined by F 2 (T m f ) = mF 2 (f ) extends to a bounded map on L p ( G). Recall here that in case of R n we have that L p ( R n ) is isomorphic to L p (R n ) canonically by conjugating with F 2 .
One of De Leeuw's most important contributions is the restriction theorem. Let H < R n be any subgroup equipped with the discrete topology. If m ∈ C b (R n ) is a p-multiplier then m| H ∈ C b (H) is a p-multiplier. Moreover, A particular case occurs when H = R n disc is R n equipped with the discrete topology. Then R n disc is the Bohr compactification of R n . This special case is usually referred to as the compactification theorem and is essentially the strongest theorem proved in [Lee65]. Its proof consists of first showing (1.1) when H is discrete in the Euclidean topology inherited from R n combined with a lattice approximation theorem. These theorems became standard tools in harmonic analysis and are the first instances of so-called transference results.
The emergence of noncommutative integration has created a realm in which Fourier multipliers can be interpreted naturally for any locally compact group. Here function algebras on the Pontrjagin dual group are replaced by group algebras and their C * -and von Neumann closures. The theory of noncommutative L p -spaces [Nel74], [PiXu03] and realization of Fourier transforms [Kun58], [Coo10], [Cas13] leads to a natural notion of p-multipliers. In this interpretation G plays the role of the frequency side and for m ∈ L ∞ (G) the Fourier multiplier T m acts on a noncommutative space, namely the noncommutative L p -space of the group von Neumann algebra of G.
For p = ∞ the relevance of Fourier multipliers on group von Neumann algebras was already recognized in the fundamental results of Haagerup on approximation properties [Haa78], as well as the works by De Cannière-Haagerup [CaHa85] and Cowling-Haagerup [CoHa89] on approximation properties of real rank one simple Lie groups.
There is a significant interest in understanding the class of p-multipliers beyond p = ∞. Already in commutative analysis this has triggered an enormous machinery of harmonic analysis including the study of singular integrals, Calderón-Zygmund theory, Hörmander-Mikhlin multipliers, et cetera, see e.g. [Gra08], [Gra09], [Ste70]. The study of multipliers is further motivated by their applications to convergence properties of Fourier series, the structure of Banach spaces and the analysis of pseudo-differential operators.
In the noncommutative situation several efforts have been made in the understanding of pmultipliers. In [JMP14] multipliers associated with cocycles on locally compact groups were constructed; such multipliers are compositions m = m • b with b : G → H a cocycle into a finite dimensional orthogonal representation (H, π) with additional Hörmander-Mikhlin conditions on m. Further results in this direction have been obtained in [JMP18], [JGP17]. Note that if G has property (T) then a well-known theorem by Delorme-Guichardet [Del77], [Gui72] (see [BHV08, Sections 2.2, 2.12]) shows that every cocycle b is bounded and therefore the multiplier m = m • b has a certain oscillatory behaviour. Note that if G = SL(n, R) the lack of nontrivial orthogonal representations limits this method. However, concrete multipliers on SL(n, R) have been constructed in [PRS] where a version of the Hörmander-Mikhlin multiplier theorem was obtained for the Lie group SL(n, R), the real n × n-matrices with determinant 1. Here the Hörmander-Mikhlin conditions are formulated in the natural differentiable structure of the Lie group. In the discrete setting Cotlar's identity has successfully been applied [MeRi17], [MRX] to obtain Hilbert transforms and Hörmander-Mikhlin multipliers on free groups.
It is natural to ask which relations there are between p-multipliers on G and p-multipliers on the discrete subgroups of G. A cornerstone theorem was obtained in [CPPR15] where the De Leeuw restriction theorem was obtained for any pair Γ < G with Γ a discrete subgroup of a (for simplicity unimodular) locally compact group G such that G has small almost Γ-invariant neighbourhoods (see Definition 3.1). If Γ is amenable then G always has small almost Γ-invariant neighbourhoods [CPPR15,Theorem 8.7]. In particular the theorem is applicable to nilpotent locally compact groups.
There are several deep open problems that concern the relation of multipliers to their restrictions on subgroups beyond the cases covered by [CPPR15]. For instance, the existence of explicit p-multipliers on SL(n, Z), n ≥ 3 remains an open problem. In particular, there does not seem to be a genuine example of a p-multiplier with compact support or with sufficient decay. This paper does not directly solve this problem but it provides a potential entry as it would be sufficient to construct multipliers with sufficient bounds on SL(n, R); this also closely relates to the Calderón-Torchinsky problem mentioned in [PRS,Section 5.C]. Other questions are raised by the approximation and rigidity properties of noncommutative L p -spaces obtained in [JuRu03], [Laf11], [Laa13], [LaSa18], [PRS, Theorem B].
Another direction that is initiated in this paper is the theory of multilinear Fourier multipliers on noncommutative L p -spaces. In commutative harmonic analysis this became a large topic including milestone results by Lacey-Thiele on the bilinear Hilbert transform [LaTh99] and the development of multilinear Calderón-Zygmund theory [GrTo02]. Also very recently, several results in the semicommutative/vector-valued setting have been obtained [AmUr20], [DLMV20a], [DLMV20b]. In the noncommutative setting multilinear Schur multipliers and operator integrals have already provided very deep applications such as the resolution of Koplienko's conjecture [PSS13]. The theory of Fourier multipliers -which by transference is intimately related to Schur multipliers [Neu11], [CaSa15] -seems to be undeveloped beyond abelian groups in the multilinear setting. We shall prove the multilinear versions of De Leeuw's restriction, lattice approximation and compactification theorem. These were proved in the abelian (bilinear) setting in [Bla05], [Rod13].
We now summarize the main results of this paper. We shall introduce a quantified version of having small almost invariant neighbourhoods. More precisely, we consider the following definition.
Definition 1.1. Let G be a locally compact group with (left) Haar measure µ. Let F ⊆ G be arbitrary and let V ⊆ G be relatively compact of non-zero measure. We introduce the quantity For a neighbourhood basis V of the identity of G we set δ F (V) = lim inf V ∈V δ F (V ). Then set δ F to be the supremum of δ F (V) over all symmetric neighbourhood bases V of the identity.
We prove the following local version of the noncommutative De Leeuw restriction theorem [CPPR15, Theorem A]. We call this theorem 'local' since we obtain an estimate that is controlled by a subset U of Γ and which holds for all symbols m ∈ C b (G) such that the restriction m| Γ is supported on U .
From this point let Γ be a discrete subgroup of any locally compact unimodular group G.
Theorem A. Let m ∈ C b (G). Then for every 1 ≤ p < ∞ we have that where c(U ) = inf{δ 1 2 F | F ⊆ U, F finite} is defined for every U ⊆ Γ. Theorem A holds as well if the bounds of both multipliers are replaced by their complete bounds (see proof of Proposition 3.16). The strength of Theorem A is that the constant c can be determined explicitly in many interesting situatons. We have c(Γ) = 1 if and only if G has small almost Γ-invariant neighbourhoods. Hence, our theorem recovers [CPPR15,Theorem A]. In Sections 8, 9 and 10 we find very natural lower estimates on δ F in the full generality of real reductive Lie groups. For ρ ≥ 1 let B G ρ = {g ∈ G | Ad g ≤ ρ} where Ad is the adjoint representation.
Theorem B. Let G be a real reductive Lie group. For ρ > 1 we have δ B G ρ ≥ ρ −d/2 where d is the maximal dimension of a nilpotent adjoint orbit.
In order to prove Theorem B we first reduce it to the connected adjoint group of G, which is semisimple. We then construct a symmetric neighbourhood basis V g ε,R , ε > 0, R > 0 of 0 in the Lie algebra g such that This implies Theorem B. The neighbourhood basis is constructed in such a way that ε>0 V g ε,R is the intersection of the nilpotent cone N ⊆ g with the ball of radius R around the origin. Note that the union of the nilpotent orbits O X of maximal dimension d is dense in N . Using results of Harisch-Chandra and Barbasch-Vogan on limiting orbit integrals, we show that the scaling behaviour of µ(exp(V g ε,R )) is governed by the scaling behaviour of the Liouville form of O X . Since the KKS symplectic form ω KKS scales as ρ under dilation, the Liouville form 1 (d/2)! ∧ d/2 ω KKS scales as ρ d/2 .
In Example 3.14 we consider the concrete case Γ = SL(n, Z), G = SL(n, R). We discuss how Theorems A and B give an ansatz to construct multipliers on Γ = SL(n, Z).
Next we start the analysis of multilinear multipliers. Most efforts are required for the restriction theorem.
Theorem C. Let G be a locally compact unimodular group. Let Γ < G be a discrete subgroup such that c(Γ) = 1. Let m ∈ C b (G ×n ). Then for every 1 ≤ p, p 1 , . . . , p n < ∞ with p −1 = n j=1 p −1 j we have that Theorem C hinges on the intertwining property of Lemma 4.6 whose proof is very delicate and which requires several new ideas compared to its linear counterpart. The global idea is to approximate m with multipliers m k that can be expressed in terms of nested compositions of linear ∞-multipliers. We then construct asymptotically isometric maps that intertwine the Fourier multipliers associated with m k | Γ ×n and m k . The condition c(Γ) = 1 is used at several places in the proof, whereas in the linear case it is only needed to construct asymptotically isometric maps. This also explains why we do not obtain a 'local' version of Theorem C as well.
We further obtain multilinear versions of the lattice approximation theorem (Theorem 5.2) as well as the multilinear compactification theorem and the periodization theorem (Section 6). We use these theorems to construct examples of multilinear Fourier multipliers on the Heisenberg group. These are the first genuine examples of multilinear Fourier multipliers on a nonabelian group.
Structure of the paper. Section 2 contains preliminaries on noncommutative L p -spaces of group von Neumann algebras. Section 3 proves Theorem A. We also establish approximate embeddings of the non-commutative L p -spaces assocatied with an inclusion of groups Γ < G. Section 4 defines multilinear Fourier multipliers and proves Theorem C. The proofs of the remaing De Leeuw theorems are contained in Sections 5 and 6; this concerns the lattice approximation, compactification and periodization. Section 7 contains examples on the Heisenberg group. In Sections 8, 9, 10 we prove Theorem B; we have postponed the proof of Theorem B to the end of the paper so that all Lie theoretic arguments are presented separately.
Acknowledgement. BJ and LM are grateful to Tobias Diez for interesting discussions and for providing crucial references. MC wishes to thank Adrián González-Pérez, Javier Parcet andÉric Ricard for useful discussions and communication.
2. Non-commutative L p -spaces of group von Neumann algebras N denotes the natural numbers starting from 1. We denote N ≥0 = N ∪ {0}.
2.1. Assumptions on groups. All groups G in this paper are assumed to be locally compact, second countable and unimodular. Though our results can be stated without the second countability assumption, it significantly simplifies the exposition of our proofs as we can work with neighbourhood bases of the identity instead of shrinking nets of neighbourhoods. When we say that Γ is a discrete subgroup of G we mean that it is discrete in the topology of G. Any discrete subgroup Γ < G is then countable.
For s, t ∈ G we write Ad s (t) = sts −1 . We denote µ for the Haar measure on G. We use the shorthand notation ds = dµ(s) for integrals with respect to µ. A set V ⊆ G is called symmetric if V = V −1 . A neighbourhood basis of the identity is called symmetric if it consists of symmetric sets.
2.2. Von Neumann algebras. A von Neumann algebra M is a unital * -subalgebra of bounded operators on a Hilbert space that is closed in the strong topology. M is called semi-finite if it admits a faithful normal trace τ : M + → [0, ∞]. M is finite in case there exists such τ with τ (1) = 1, i.e. τ extends to a state on M .For general von Neumann algebra theory and noncommutative integration we refer to [Tak02], [StZs79].
2.3. Group von Neumann algebras. Let C b (G) be the continuous bounded functions G → C and let C c (G) be the subspace of compactly supported functions. Let L p (G), 1 ≤ p < ∞ be the Banach space of functions G → C that are p-integrable, meaning f p : We further set be the left regular representation. Set λ(f ) = G f (s)λ(s)d(s) where the integral is convergent in the strong topology. We have We set the group von Neumann algebra where span denotes the strong closure of the linear span. There exists a unique normal semi-finite faithful weight ϕ G on L(G) that is defined as follows. For x ∈ L(G) we have otherwise.
ϕ G is tracial, meaning that for all x ∈ L(G), ϕ G (x * x) = ϕ G (xx * ), iff G is unimodular, which we will always assume. ϕ G is a state if and only if G is discrete.
2.4. Crossed products. Let M be a semi-finite von Neumann algebra with a trace τ M and let Γ be a discrete group acting on M ⊆ B(H) via a trace preserving action θ : Γ → Aut(M ). For This has a natural tracial weight τ which extends τ M on {ι(x) : x ∈ M } and τ Γ on 1 ⊗ λ(g) : g ∈ Γ (c.f. [Tak02]).
2.5. Non-commutative L p -spaces. The results in this section can be found in [Kun58], [Nel74], [Ter81], [PiXu03], [Cas13]. For an exponent p ∈ [1, ∞] we will write p ′ ∈ [1, ∞] for the conjugate exponent set by 1 p + 1 p ′ = 1. Let M be a von Neumann algebra equipped with a normal, semi-finite, faithful trace τ . For 1 ≤ p < ∞, set We define L p (M, τ ) as the completion of {x ∈ M | x p < ∞} with respect to · p . Alternatively, L p (M, τ ) may be described as the space of all closed, densely defined operators x that are affiliated with M and for which x p p := τ (|x| p ) < ∞. The latter description is more concrete but requires the introduction of affiliated operators and extension of the trace thereon; this shall not be used further in this paper. We define L ∞ (M, τ ) to be M with the operator norm. Then L p (M, τ ), 1 ≤ p ≤ ∞ is a Banach space and we have a Hölder inequality The Hölder inequality is moreover sharp in the sense that indeed if x = u|x| is the polar decomposition of x then the supremum is attained at y = |x| p q . The trace τ may be extended from M ∩ L 1 (M, τ ) to L 1 (M, τ ) linearly and continuously in · 1 . For x ∈ L p (M, τ ) and y ∈ L p ′ (M, τ ) we have in particular that yx ∈ L 1 (M, τ ) and we have a pairing (2.4) y, x p ′ ,p = τ (yx).
We have x p = x * p for any x ∈ L p (M, τ ). In case (M, τ ) is (L(G), ϕ G ) with G a unimodular locally compact group -so that ϕ G is tracial -we simply write L p ( G) for L p (L(G), ϕ G ). If G is abelian the latter space is isomorphic to the usual L p -space of the Pontrjagin dual group G. For 2 ≤ p ≤ ∞ and f ∈ C c (G) we have λ(f ) ∈ L p ( G) with λ(f ) p ′ ≤ f p (see [Kun58], [Coo10], [Cas13]) and such elements are dense. We denote C c (G) * 2 = C c (G) * C c (G) for the second convolution power of C c (G). The elements λ(f ), f ∈ C c (G) * 2 are dense in L p ( G), 1 ≤ p < ∞. The same densities hold for p = ∞ but then in the σ-weak topology of L ∞ ( G) = L(G). In each case the frequency support of λ(f ) is by definition the support of f . For φ, f ∈ λ(C c (G) * 2 ) we have It follows directly from (2.2) that the Plancherel identity holds: In case G = Γ is discrete the elements with finite frequency support are given by the group algebra which is dense in L p ( Γ) for any 1 ≤ p < ∞. Here c 00 (Γ) denotes the finitely supported functions Γ → C.
Remark 2.1. There are notions of non-commutative L p associated with an arbitrary von Neumann algebra due to Haagerup [Ter81] and Connes-Hilsum [Hil81]. This allows for the study of de Leeuw theorems for non-unimodular groups, see for instance [CPPR15,Section 8]. However, to keep the current paper broadly accessible we will work within the realm of unimodular groups and tracial von Neumann algebras.
2.6. Fourier multipliers. We say that a function m In particular this encompasses that λ(mf ) ∈ L q ( G). We briefly say p-multiplier in case p = q. By the Plancherel identity the space of 2-multipliers with continuous symbol is C b (G).
Remark 2.2. Let 2 ≤ p < ∞. Suppose that m ∈ C b (G) is a p-multiplier. Then under the pairing (2.5) the dual of T m : L p ( G) → L p ( G) is the p ′ -multiplier given by T m ∨ : L p ′ ( G) → L p ′ ( G).

Theorem A: The local linear De Leeuw restriction theorem
The aim of this section is to prove Theorem A. We also introduce the quantified version of local almost invariant neighbourhoods in Definition 1.1 which we shall determine for real reductive Lie groups in Section 8. We fix again a locally compact unimodular group G which we assume to be second countable. Let Γ < G be a discrete subgroup. Recall that µ is the Haar measure of G.
At this stage recall that δ F as was defined in Definition 1.1.  Remark 3.3. G has small invariant neighbourhoods (notation G ∈ [SIN] Γ ) with respect to Γ if there exists a neighbourhood basis V of the identity of G such that Ad s (V ) = V, s ∈ Γ.
Clearly then G ∈ [SAIN] Γ . Indeed, we may replace V ∈ V by V ∩ V −1 to obtain a symmetric neighbourhood basis with the desired properties.
Remark 3.4. Suppose that G ∈ [SAIN] Γ . Since G is first countable pick any neighbourhood basis (U i ) i∈N of the identity of G. Since G is second countable, Γ is countable. Let F i ⊆ Γ, i ∈ N be an increasing sequence of finite subsets whose union is Γ. For every i ∈ N we may inductively pick a relatively compact open neighbourhood V i of the identity in G such that 1 − 1 This construction has the advantage that we may fix a single neighbourhood basis V independent of the finite set F .
3.1. Asymptotic embeddings for 2 ≤ p < ∞. The aim of this section is to construct contractive maps L p ( Γ) → L p ( G) that are asymptotic embeddings. In case G ∈ [SAIN] Γ these maps are asymptotically isometric. Our results therefore generalize [CPPR15, Claim A].
is positive definite. Moreover A ≥ δ F (V )I where I = (1) s,t∈F is the matrix with all entries equal to 1.
Proof. Let V 0 := s∈F Ad s (V ) and define ξ : As usual we view C[Γ] ⊆ L(Γ) ⊆ L(G) as subalgebras of each other naturally.
Proposition 3.6. Let x ∈ C[Γ] with finite frequency support F ⊆ Γ. Let V be a relatively compact symmetric neighbourhood of the identity of G and assume that sV, s ∈ F are disjoint.
where we use the notation Φ ∞,V (x) = x. Then for every 2 ≤ p ≤ ∞ we have Proof. Write x = s∈F x s λ(s). By Remark 3.7 below and the Plancherel identity Since sV, s ∈ F are disjoint and using once more the Plancherel identity, This gives the upper estimate (3.1) for p = 2. For p = ∞ the estimate (3.1) is trivial. From the three lines lemma (similar to Stein's interpolation theorem for analytic families of maps [Ste56]) we then have (3.1) for all 2 ≤ p ≤ ∞.
Remark 3.7. In Proposition 3.6 since V = V −1 it follows that k V is self-adjoint. Hence h V commutes with u V and u V is a self-adjoint partial isometry with u 2 V being the support projection of h V . We shall repeatedly make use of these observations without further reference.
Proposition 3.8. Let x ∈ C[Γ] with finite frequency support F ⊆ Γ. For V a symmetric neighbourhood basis of the identity of G we have for every 2 < p < ∞ Proof. For p = 2 the proof is (3.2) and for p = ∞ the statement is obvious as L(Γ) is a von Neumann subalgebra of L(G). So we assume that 2 < p < ∞. Let 2 < q < ∞ be the L 2 -conjugate of p determined by p −1 + q −1 = 2 −1 . Let ε > 0 and let y ∈ C[Γ] be such that y Lq ( Γ) ≤ 1 and We may consider a tail of the symmetric basis V so that each V ∈ V is relatively compact and so that each of the following families are disjoint: (1) sV, s ∈ F ; (2) sV, s ∈ F y . Moreover, we may assume that (3) s 1 V t 1 and s 2 V t 2 are disjoint whenever s 1 , s 2 ∈ F , t 1 , t 2 ∈ F y with s 1 t 1 = s 2 t 2 .
The frequency support of y * is F y . Since we assumed that sV, s ∈ F y are disjoint it follows from (3.1) -but then applied to y -that, . Using (2.3) and (3.4) we find for V ∈ V, We have By the Plancherel identity and disjointness assumption (3) in the first paragraph of this proof, .

Asymptotic intertwiners.
We will need to make use of the following proposition to prove our De Leeuw restriction theorems: both in the local linear and in the multilinear setting.
Proof. For 2 ≤ q < p < ∞ the proposition is exactly Claim B of [CPPR15]. The case 1 ≤ p < q ≤ 2 the same proof holds with trivial modifications with only one remark taken into account. Namely, at the place where [CPPR15, Corollary 1.4] was used (which is stated only for 2 ≤ q < ∞) we need to use [CPR18, Lemma 3.1] instead (which extends this corollary for 1 < q < 2). Note that [CPR18, Lemma 3.1] was only stated for finite von Neumann algebras, but its proof remains valid in the semi-finite setting exactly as it is stated.
Proof. For p = 2 the statement is obvious. By complex interpolation the logarithm of the norms on both sides of (3.8) is a convex function in p and hence certainly continuous. So the case p = 1 follows from the case 1 < p < 2 by approximation. The case 1 < p < 2 follows by duality from the case 2 < p < ∞, see Remark 2.2.
The frequency support F of T m| Γ (x) is finite and contained in supp(m| Γ ). Let ε > 0. Let V be a symmetric neighbourhood basis shrinking to the identity of G such that By Proposition 3.9 the last summand goes to 0. Hence using again Proposition 3.8, Letting ε ց 0 concludes the proof as C[Γ] is dense in L p ( Γ).
Remark 3.12. In Theorem 3.11 if moreover G ∈ [SAIN] Γ then δ F = 1 for any F ⊆ Γ finite and we recover [CPPR15, Theorem A]. In particular our theorem also generalizes the restriction theorem from [Lee65].
Remark 3.13. Use the notation of Theorem 3.11. Now let N be any (semi-finite) von Neumann algebra and consider the tensor product L p (N ) ⊗ L p ( G), which is equipped with the L p norm induced by the tensor product of the weights on N and L(G). We write L p ( G; L p (N )) for L p (N )⊗ L p ( G). Consider the vector-valued extension T N m of T m given by L p ( G; L p (N )) → L p ( G; L p (N )), Let R be the hyperfinite II 1 -factor. We will say that T m is completely bounded if T R m is bounded and write T m cb = T R m . We still have (3.8) in the cb-norm since we may apply the theorem Example 3.14. Consider G = SL(n, R) which has real rank n − 1. Set Γ = SL(n, Z). As a particular case of Theorem 8.1 proved below (see Remark 8.2) we obtain that we verify next as a sanity check. This shows that our methods provide an entry to construct multipliers on SL(n, Z).
In [Mau07, Corollary 1.1 and Example 4.2] it is shown that Indeed, let a be a maximally noncompact Cartan subalgebra of the Lie algebra g of G and let λ 1 , . . . , λ n ∈ a * be the natural weights described in [Mau07, Example 4.1, 4.2]. The weights occurring in the adjoint representation are λ i − λ j , 1 ≤ i, j ≤ n and C is defined as the convex hull of these. For β the sum of all positive roots (see [Mau07]; so not the usual half-sum) we find, If n is even β/D is on the face of C given by the convex hull of the weights If n is odd then β/D is on the face of C given by the convex hull of the weights So we have determined D and E as in [Mau07] and [Mau07, Corollary 1.1] yields (3.10). Note that D and E in this example are written as d and e in [Mau07] which differs from d as in Theorem B.
3.4. Asymptotic isometries for 1 ≤ p < 2 under the [SAIN] condition. In order to prove the multilinear theorems we prove two lemmas that in particular extend the results of Section 3.1 under the [SAIN] condition. Recall that the group C * -algebra C * r (Γ) is defined as the norm closure of C[Γ] in B(ℓ 2 (Γ)).
Proof. By approximation and taking linear combinations it suffices to prove the lemma in case x = λ(r) with r ∈ G. Then [CPPR15, Corollary 1.4] shows that for 2 ≤ p < ∞, . (3.12) Taking the limit over V ∈ V this converges to 0 by the [SAIN] condition, c.f. Remark 3.4. Now for any 1 ≤ p < ∞, . By Proposition 3.6 and (3.12) this expression converges to 0.
For completeness we note that the crucial inequality behind (3.12) is Ando's inequality [CPPR15, Lemma 2.2] from which it could also be proved directly using the methods in [CPPR15, Section 2].
and assume x = y * y. Then we have, by Lemma 3.15, Proposition 3.8 and similarly, by Lemma 3.15, .
So we may conclude the proof from the self-adjoint case. Now for general x ∈ C * r (Γ) we may replace x and h V by respectively, ) respectively and conclude the proof; it follows that the first part of the proof also holds in this 2 × 2-matrix amplification. One way to see this is using the fact that the group von Neumann algebra of the infinite permutation group S ∞ is the hyperfinite II 1 factor, R. We now replace the inclusion Γ < G by S ∞ × Γ < S ∞ × G. The corresponding von Neumann algebras are then R ⊗ L(Γ) and R ⊗ L(G). Since the matrix algebras embed into R, we may view the matrix amplification of x as an element of L p (R ⊗ L ∞ ( Γ)).

Theorem C: The multilinear De Leeuw restriction theorem
In this section we define multilinear Fourier multipliers and prove the multilinear De Leeuw restriction theorem. 4.1. Multilinear maps. Let X 1 , . . . , X n and Y be Banach spaces. Let T : X 1 × . . . × X n → Y be a multilinear map, meaning that it is linear in each of its variables. The norm of T is then defined as 4.2. Multilinear Fourier multipliers. Let G be a locally compact unimodular group. Let m : G ×n → C be a bounded measurable function. Consider the tuple (p 1 , . . . , p n ) with 1 ≤ p i < ∞ and let also 1 ≤ p < ∞. We say that m is a (p 1 , . . . , p n ) → p multiplier if there exists a bounded n-linear map for all f 1 , . . . , f n ∈ C c (G) * 2 . By mild abuse of terminology we also refer to the map T m as the (p 1 , . . . , p n ) → p multiplier and m is then explicitly referred to as the symbol of T m . If p is the conjugate determined by p −1 = n j=1 p −1 j then we simply speak about a (p 1 , . . . , p n )-multiplier in analogy with the linear case n = 1. n is also called the order of the multiplier. 4.3. Reduction formulae. We prove a number of lemmas that reduce the analysis of n-linear Fourier multipliers to Fourier multipliers of lower order.
For 1 ≤ k ≤ n let (q 1 , . . . , q k ), 1 ≤ q j < ∞ be a consummation of (p 1 , . . . , p n ), 1 ≤ p j < ∞ at indices 1 = i 1 < . . . < i k ≤ n. Suppose that m : G ×k → C is bounded and measurable and set Then m is a (q 1 , . . . , q k )-multiplier iff m is a (p 1 , . . . , p n )-multiplier and moreover, for Proof. Let f 1 , . . . f n ∈ C c (G) * 2 . Then, using the invariance of the Haar integral for the second equality and (2.1), So if T m is bounded it follows by this expression and the Hölder inequality that also T m is bounded.
and taking the supremum over all such f i we conclude the proof.
Then m is a (p 1 , . . . , p k ) → p-multiplier if and only if so is m i ( · ; r, t, r ′ ). In that case for Proof. The proof of (4.1) is similar to Lemma 4.2, namely one verifies the equality for x j = λ(f j ), f j ∈ C c (G) * 2 by an elementary computation and then concludes by continuity. Since multiplication with λ(s), s ∈ G determines an isometry on L p j ( G) that is strongly continuous in s by [JuSh05, Lemma 2.3] the final statements follow as well.
There are several formulae available for nested Fourier multipliers of which the following lemma is a particular case.
Then m is a (p 1 , . . . , p n )-multiplier and for Proof. Again the identity (4.2) can directly be verified for The boundedness of T m then follows as all T m j are bounded together with the Hölder inequality.

4.4.
Proof of Theorem C. The idea behind the following proof is that we approximate a multiplier by multipliers that decompose as a nested iteration of linear multipliers. Then using the analysis of linear multipliers we may prove the desired result. We first give the core of the proof of Theorem 4.5 and prove the intertwining property that is needed for it afterwards in Lemma 4.6.
Theorem 4.5 (Theorem C). Let G be a locally compact unimodular group. Let Γ < G be a discrete subgroup such that G ∈ [SAIN] Γ . Let m ∈ C b (G ×n ). Then for every 1 ≤ p, p 1 , . . . , p n < ∞ with p −1 = n j=1 p −1 j we have that Proof. In case n = 1 the theorem was proved in [CPPR15, Theorem A] or Remark 3.12. We therefore assume that n ≥ 2. Consequently 1 < p 1 , . . . , p n < ∞. We note however that our proof remains valid for n = 1 provided that 1 < p = p 1 < ∞. Let (ϕ k ) k be a net of positive definite functions in the Fourier algebra A(G) = L 2 (G) * L 2 (G) with ϕ k 1 = 1 and compact support shrinking to the identity of G. Recall that elements of A(G) are continuous and p-multipliers for all 1 ≤ p ≤ ∞ (see [CaHa85]). For any function (4.5) Since m k → m pointwise and all x j have finite frequency support we find that Let V be a symmetric neighbourhood basis of the identity of G as in Remark 3.4. By Proposition 3.16, Now by Lemma 4.6 -which we prove below -the limit over k and V in the second summand exists and is 0. Therefore, By Lemma 4.3 we find that T m k ≤ T m . Hence, together with (3.1) of Proposition 3.6, we conclude that x j Lp j ( Γ) .
Since the elements x j ∈ C[Γ] are dense in L p ( Γ) we conclude the proof.
Lemma 4.6. In the proof of Theorem 4.5, in particular with n ≥ 2, we have that If n = 1 the statement also holds in case 1 < p = p 1 < ∞.
Proof. Recall that all x j have finite frequency support and therefore by the triangle inequality it suffices to consider the case where x j = λ(r j ), r j ∈ Γ. Let ζ : G → R ≥0 be a compactly supported continuous, positive definite function in the Fourier algebra A(G) with ζ(e) = 1, so that T ζ is a contraction for all 1 ≤ p ≤ ∞. For 1 ≤ j ≤ n, s ∈ G let (4.6) ζ j (s) = ζ(r −1 j s).
As observed in the first paragraph the latter expression converges to 0 in k, so we find that lim k lim V ∈V A k,V = 0.
Next we treat the summand C k,V . Define, using the notation (4.4), Since by similar arguments to the ones used in Lemmas 4.2 and 4.4, we find We have see also Proposition 3.6. Recall (4.6) and the fact that ζ(e) = 1 and x j = λ(r j ). By Lemma 3.15 and using Proposition 3.9 we see that for every 1 ≤ j ≤ n − 2, Hence from (4.8), (4.9) Now going back to the original quantity we have to estimate, we find by Lemma 4.3, Taking limits in (4.9) yields, This expression goes to 0 when taking the limit in k. This concludes the estimate for C k,V . We now prove that for every k we have lim V B k,V = 0. For what follows, we will assume, by scaling ϕ k if necessary, that T ϕ k is a contraction on L p ( G); we may do this since at this point we need to prove convergence in V for fixed k and our arguments do not rely on the earlier assumption ϕ k 1 = 1 anymore. Set the function When we see ψ k for fixed t 1 , . . . , t n as a function of the variables s 1 , . . . , s n we shall write ψ k ( · ; t 1 , . . . , t n ). By the last expression of (4.5) we see that We justify that the integrand is dominated by an integrable function that does not depend on V . Indeed, note that T ψ k ( · ;t 1 ,...,tn) and T ψ k ( · ;t 1 ,...,tn)| Γ ×n are contractive (this follows from the expansions (4.11) and (4.12) below) and in combination with Proposition 3.6 it follows that the integrand on the second line of (4.10) is at most 2 for any t 1 , . . . , t n ∈ G. Since m(ζ 1 , . . . , ζ n ) is integrable, the integral (4.10) is majorized by G ×n 2|(m(ζ 1 , . . . , ζ n ))(t −1 1 t 2 , . . . , t −1 n−2 t n−1 , t −1 n )|dt 1 . . . t n = 2 m(ζ 1 , . . . , ζ n ) L 1 (G ×n ) < ∞.
So if we can show that the limit in V of the integrand of (4.10) converges to 0 pointwise then by the Lebesgue dominated convergence theorem we conclude that for all k we have lim V B k,V = 0. Introduce the short hand notation for the multipliers T j = T ϕ k ( · t j ) and S j = T ϕ k ( · t j )| Γ for 1 ≤ j ≤ n − 1. We also set T n = T ϕ k (tn · ) and S n = T ϕ k (tn · )| Γ For any y j ∈ L p j ( G) we have by Lemma 4.4, (4.11) T ψ k ( · ;t 1 ,...,tn) (y 1 , . . . , y n ) = T 1 (y 1 (T 2 (y 2 . . . T n−1 (y n−1 ) . . .)))T n (y n ).
Now fix y j := y j, for a while V shall be fixed and at the very end of the proof we will take a limit in V . Set 1 < q j < ∞ by q −1 j = n−1 i=j p −1 i . Note that q −1 1 + p −1 n = p −1 . Then set for 0 ≤ j ≤ n − 1, ) . . .))).

By definition
R n−1,V =T 1 (y 1 (T 2 (y 2 . . . T n−1 (y n−1 ) . . .))), and these expressions should be compared to (4.11) and (4.12). Also let R denote the term S 1 (x 1 (S 2 (x 2 . . . S n−1 (x n−1 ) . . .))). S j and T j are contractions and therefore we record at this Then -in view of (4.10) -we estimate using the triangle inequality V and that T n = T ϕ k ( · tn) is a multiplier both on L 1 and L ∞ . Therefore by the Hölder inequality and Proposition 3.9, It remains to show that the final summand in (4.13) goes to 0 in V . For 1 ≤ j ≤ n − 1, set We estimate, (4.14) Since T j is a contraction the first summand goes to 0 in V by Lemma 3.15. For the second summand we note that T j = T ϕ k ( · t j ) is a multiplier both on L 1 ( G) and L ∞ ( G). Since 1 < q j < ∞ we get that also the second summand in (4.14) goes to 0 by Proposition 3.9.
Remark 4.7. If in Lemma 4.6 we have x 1 = . . . = x n = 1 then the proof holds for any symmetric neighbourhood basis V not necessarily witnessing the [SAIN] Γ condition. Remark 4.8. As in Remark 3.13, we note that Theorem C also has a completely bounded version. Use the notation of Theorem 4.5 and let N be any (semi-finite) von Neumann algebra. The vectorvalued extension T N m of T m is given by L p 1 ( G; L p 1 (N ))×L p 2 ( G; ; L p 2 (N ))×. . .×L pn ( G; L pn (N )) → L p ( G; L p (N )), Let R be the hyperfinite II 1 -factor. As in the linear case, we will say that T m is completely bounded if T R m is bounded and write T m cb = T R m . Then we still have (4.3) in the cb-norm.

Multilinear lattice approximation
The aim of this section is to prove a multilinear version of [CPPR15, Theorem C].
Definition 5.1. A locally compact group G is approximable by discrete subgroups (notation G ∈ ADS) if there exists a sequence of discrete lattices (Γ j ) j∈N of G with associated fundamental domains X j which form a neigbourhood basis of the identity.
By passing to a subsequence we may assume that all X j in Definition 5.1 are relatively compact. In other words Γ j is cocompact. The following theorem shows that for ADS groups the norm of Fourier multipliers can be approximated by looking at the restriction of the symbol to appropriate lattices.
Theorem 5.2. Let G ∈ ADS with approximating cocompact lattices (Γ j ) j∈N . Let m ∈ C b (G ×n ) and let 1 ≤ p, p 1 , . . . , p n < ∞ be such that p −1 = n i=1 p −1 i . Then, Before proving the theorem we introduce the auxiliary maps from [CPPR15, Theorem C]. Let X j be shrinking relatively compact fundamental domains for the inclusions of Γ j in G. For s ∈ G we shall write γ j (s) for the unique element in Γ j such that s ∈ γ j (s)X j . Let h j = λ(1 X j ). For x ∈ C[Γ j ] we set the elements in L ∞ ( G) by The proof of [CPPR15, Theorem C, p. 19-20] shows that Φ Since x has compact frequency support these summations are finite. The proof of [CPPR15, Theorem C, p. 19-20] shows that Ψ (p) j extends to a contraction L p ( G) → L p ( Γ). Now set Recall that we defined φ ∨ (s) = φ(s −1 ) for φ a function on G. The proof of Theorem 5.2 hinges on the following lemma.
Hence we see that this term goes to 0 proving (5.3).
Proof of Theorem 5.2. Let x 1 , . . . , x n be as in Lemma 5.3 and assume x i Lp i ( G) ≤ 1. Lemma 5.3 shows (for p = 1 this requires Kaplansky's density theorem), Then, as Φ This concludes the proof.
We note that we may use the lattice approximation theorem to obtain a stronger version of the restriction theorem. The following follows directly by combining Theorem 4.5 and Theorem 5.2.
Theorem 5.4. Let H be a subgroup of a locally compact group G. Let H ∈ ADS and G ∈ [SAIN] H . Let m : G ×n → C be a bounded continuous symbol giving rise to a multilinear (p 1 , p 2 , . . . , p n )-multiplier on G with 1 ≤ p, p i < ∞ and p −1 = n i=1 p −1 i . Then

Other multilinear de Leeuw theorems: periodization and compactification
In this section we prove multilinear versions of the periodization and compactification theorem. The proofs in this section are very similar to the linear case in [CPPR15] or to other proofs in this paper. Therefore, we only give a sketch of the proofs.
Given a locally compact group G, let G disc denote the same group equipped with the discrete topology. The compactification theorem relates the (p 1 , p 2 , . . . , p n )-boundedness of a Fourier multiplier on G with the (p 1 , p 2 , . . . , p n )-boundedness of a Fourier multiplier on G disc . We recall that if G is abelian G disc is known as the Bohr compactification of G. We refer to the discussion in [CPPR15, Section 6] for why the hypotheses in the following theorem are the most natural ones.
Theorem 6.1. Let G be a locally compact group and let m ∈ C b (G ×n ). Let 1 < p, p 1 , . . . , p n < ∞, Proof. (i) Let (Γ j ) j∈N be a sequence of approximating cocompact lattices as in the hypothesis of Theorem 5.2. Then, Now Γ j < G disc is an inclusion of two discrete groups so that the natural inclusion C[Γ] ⊆ C[G disc ] extends to an isometric inclusion L p ( Γ j ) ⊆ L p ( G disc ). Therefore, T m| Γ ×n j is just a restriction of T m and hence (ii) This is essentially a restriction result for the subgroup G disc . As such, the proof is similar to Theorem 4.5 and we sketch its main difference here. From [CPPR15,Theorem 8.7] we know that the amenability of G disc implies that G ∈ [SAIN] G . Now for F ⊂ G finite let Γ F be the smallest (not necessarily closed) subgroup of G containing F . Then Γ F is countable and we may still perform the construction in Remark 3.4 with Γ replaced by Γ F . Hence we obtain a local neighbourhood basis V of the identity of G such that for every F ′ ⊆ Γ F finite we have lim V ∈V δ F ′ (V ) = 1. Then Proposition 3.6 still holds for any x ∈ C[ Γ F ] with V ∈ V sufficiently small.
By [CPPR15,Lemma 6.1] the amenability of G disc implies that for any x ∈ C[G disc ] we have x C * r (G disc ) = x C * r (G) . So naturally we have an isometric embedding C * r (G disc ) ⊆ L ∞ ( G). Consequently, the proof of Proposition 3.6 still holds, where for the p = ∞ case the latter fact is used. Since Γ F is a discrete subgroup of G disc we have C * r (Γ F ) ⊆ C * r (G disc ) which thus is included in L ∞ ( G). Then Lemma 3.15 and Proposition 3.16 hold for any x ∈ C * r (Γ F ). This suffices then to run the proofs of Theorem 4.5 and Lemma 4.6 by putting F as in the previous paragraph equal to the union of the frequency supports of x 1 , . . . , x n occurring in the proof; note that products of the x i 's then have frequency support contained in Γ F . The proof of the periodization theorem is a direct modification of the linear case (Theorem D (iii) and (iv) in [CPPR15]).
Theorem 6.2. Let G be a locally compact group and let H be a closed normal subgroup of G. Let m q ∈ C b ((G/H) ×n ) and let m π ∈ C b (G n ) denote its H-periodization m π (g 1 , . . . , g n ) = m (g 1 H, g 2 H, . . . , g n H). Then, the following inequalities hold for 1 < p, Proof. (i) was proved in the linear case by Saeki [Sae70] and in the multilinear case by [Rod13]. The proof of (ii) remains essentially unchanged from the linear case proved in [CPPR15,Section 7]. As in [CPPR15, Section 7] we note that the operator defined by is a central projection of L(G) such that λ(s)Π = Π = Πλ(s) for all s ∈ H. Moreover, this lets us define a normal * -homomorphism π preserves the Plancherel weight on L(G/H), so it induces an isometry π p : L p ( G/H) → L p ( G)Π for 1 ≤ p ≤ ∞. Since Π is a central projection in L(G), a similar computation as in [CPPR15] now shows us that π p • T mq (x 1 , . . . , x n ) = (G/H) n m q (g 1 H, g 2 H, . . . , g n H) x 1 (g 1 H) . . . x n (g n H)λ(g 1 . . . g n )Πdµ G/H (g 1 H) . . . dµ G/H (g n H) . This concludes (ii) since π p is an isometry for all 1 ≤ p ≤ ∞.

Example: Multilinear multipliers on the Heisenberg group
We provide an example of multilinear multipliers on a non-abelian group. Let be the (2 + 1)-dimensional Heisenberg group. We shall see H as R 3 with the group law H is unimodular and its Haar measure is just the usual Lebesgue measure of R 3 . There is an action R R 2 by x·(y, z) = (y, z +xy). Then H = R 2 ⋊R. Recall our definition of the completely bounded norm from Remark 4.8.
Set the unitary U = s∈Λ P s ⊗ λ(s) ∈ B(ℓ 2 (Λ)) ⊗ L ∞ ( Λ). Then set the trace preserving *homomorphism, Then π determines an isometry As described in Remark 3.13, L p ( Γ) ⊗ S p here refers to the tensor product equipped with the L p norm given by the tensor products of the weights on Γ and N. Since Γ is amenable we have a trace preserving embedding L(Γ) ⊆ R by Connes' theorem [Con76] where R is the hyperfinite II 1 factor. Also R ⊗ M k (C) embeds into R ⊗ R ≃ R in a trace preserving way. Thereforeapproximating S p with L p (M k (C)) -since T m is completely bounded we may extend T m to a vector valued map and we have T vec m • (π p 1 × π p 2 ) = π p • S M . Therefore S M is bounded by the norm of the right hand side of (7.1).
By the Følner condition the limit of the fraction is 1 and so this expression equals T M (x, y), z p,p ′ . Since j p,α are contractions (see [Gon18]), this shows (for p = 1 with Kaplansky's density theorem) that the norm of T M is bounded above by the norm of S M , which in turn is bounded by T vec m cb .
We have for 1 ≤ p, p 1 , j ∈ N, viewed as a cocompact discrete subgroup of H with ∪ j∈N H j dense in H. Set Γ j = j −1 Z×j −2 Z and Λ j = j −1 Z. The action R R 2 described above restricts to an action Λ j Γ j and H j = Γ j ⋊ Λ j . We now have by Theorem 5.2, Proposition 7.1 and Theorem 4.5 (see Remark 4.8), Remark 7.3. We note that the hypotheses of Theorem 7.2 are satisfied with finite norms in (7.3) if m obeys a suitable Hörmander-Mikhlin type condition, see [DLMV20a], [DLMV20b].
Remark 7.4. The methods in this section in fact work for more general semi-direct products of the form H = G ⋊ R n for an action by automorphisms of R n on a locally compact amenable group G provided that G is ADS and the approximating groups Γ j of G are invariant under the action of 1 j Z n ⊆ R n .

Theorem B: Lower bounds on δ for reductive Lie groups
In the following sections 8 through 10, we proceed with the proof of Theorem B. Let G be a real reductive Lie group with Lie algebra g, Cartan involution θ, maximal compact subgroup K, and invariant bilinear form B, cf. [Kna96, §VII.2]. The inner product B θ (x, y) := −B(x, θy) on g endows End(g) with the operator norm A → A . For ρ ≥ 1, we denote by (8.1) B G ρ := g ∈ G ; Ad g ≤ ρ the preimage under the adjoint representation Ad : G → End(g) of the closed ball of radius ρ around the origin. The aim of this section is to prove the following lower bound on δ B G ρ in terms of the radius.
Theorem 8.1 (Theorem B). If d is the maximal dimension of a nilpotent orbit of G in g, then As the adjoint orbit O X of a nilpotent element X ∈ g is a symplectic manifold, the coefficient d/2 in (8.2) is an integer. Since T X O X = g/g X is the quotient of g by the centralizer g X of X, the maximal dimension d can be expressed as where N ⊆ g is the nilpotent cone of g. In particular, d = 0 if g is compact or abelian, d = dim(g) − rk(g) if g is split (or quasisplit [Rot72, Thm 5.1]), and d = 2(dim C (g) − rk C (g)) if g is complex. In particular, d = n(n − 1) for SL(n, R) and GL(n, R), and d = 2n(n − 1) for SL(n, C) and GL(n, C).
Let g = k ⊕ p be the Cartan decomposition of g, let a be a maximal abelian subspace of p, and let A be the corresponding analytic subgroup. If g = k 1 ak 2 is the KAK-decomposition of g ∈ G ([Kna96, §VII.3]), then Ad g = Ad a since both B and θ are invariant under Ad(K). Let g = g 0 ⊕ λ∈Σ g λ be the restricted root space decomposition, where the sum runs over the set Σ ⊆ a * of restricted roots. Since A is simply connected, the set B G ρ can be equivalently described as (8.4) B G ρ = Kρ P K := K exp log(ρ)P K, where P ⊆ a is the polygon P = {h ∈ a ; α(h) ≤ 1 ∀ α ∈ Σ}. From this description, the following result easily follows. 8.1. A neighbourhood basis. The reductive Lie algebra g decomposes as the direct sum g = g 0 ⊕ z of the maximal semisimple ideal g 0 = [g, g] and the centre z. The former admits the Cartan decomposition g 0 = k 0 ⊕ p 0 , with k 0 = k ∩ g 0 and p 0 = p ∩ g 0 .
Let B z r ⊆ z and B g 0 r ⊆ g 0 be the open balls of radius r with respect to the inner product B θ in z and g 0 , respectively. Both are K-invariant, and the global Cartan decomposition G = K exp(p) implies that B z r ⊆ z is invariant under all of G. We are interested in the neighbourhood basis (8.5) V g ε,R,r := Ad G (B g 0 ε ) ∩ B g 0 R × B z r , obtained by intersecting the G-invariant neighbourhoods with the bounded sets B g 0 R × B z r . It will be convenient to write V g ε,R,r = V g 0 ε,R × B z r , where V g 0 ε,R is the bounded open subset of g 0 defined by Remark 8.4 (K-invariance). Since B g 0 ε is K-invariant, the global Cartan decomposition G = K exp(p) yields Ad G (B g 0 ε ) = Ad exp(p) (B g 0 ε ). Further, since z acts trivially on g 0 , we have Ad G (B g 0 ε ) = Ad exp(p 0 ) (B g 0 ε ) for p 0 = p ∩ g 0 . It follows that V g ε,R,r = V g 0 ε,R × B z r is the product of the K-invariant set V g 0 ε,R ⊆ g 0 that depends only on the restriction of B θ to g 0 , and the G-invariant set B z r ⊆ z that depends only on the restriction of B θ to z. In particular the sets V g ε,R,r ⊆ g are K-invariant, and they depend only on the triple (g, θ, B), not on the Lie group G.
The lower bound (8.2) will be established by calculating where µ is a Haar measure on G. Since V g ε,R,r = −V g ε,R,r , the sets exp(V g ε,R,r ) constitute a symmetric neighbourhood basis of the identity. It follows that , so in order to establish (8.2), it suffices to prove that δ 0 8.2. Relation between Haar measure and Lebesgue measure. The first step is to reformulate this in terms of the Lebesgue measure on the Lie algebra g. Let Vol G be a left invariant volume form on G, so that integrating against Vol G corresponds to a left Haar measure µ on G. Let Vol g be a constant volume form on g, corresponding to a Lebesgue measure Λ on g. We normalize these volume forms in such a way that Vol g agrees with exp * Vol G at the origin in g. Then exp * Vol G = νVol g , where the density ν of exp * Vol G with respect to the Lebesgue measure satisfies ν(0) = 1. We show that ν can be chosen arbitrarily close to 1 on U ε ⊆ g for small ε.
Proposition 8.5. The density is given by ν(x) = det(Φ x ), where Φ : g → End(g) is the left logarithmic derivative of the exponential map. The function ν : g → R is smooth, G-invariant, and equal to 1 on z ⊆ g. Moreover, there exists a constant c g 0 > 0 (depending only on g 0 ) such that for ε sufficiently small, ν − 1 ∞ ≤ c g 0 ε uniformly on U ε .
Proof. Let Φ : g → End(g) be the left logarithmic derivative of the exponential map, defined by Φ x (y) : Then for all x, y 1 , . . . , y n ∈ g, where the last two steps use that Vol G is left invariant, and that exp * Vol G agrees with Vol g at the origin. It follows that ν(x) = det(Φ x ). Since exp : g → G is equivariant with respect to the adjoint action on g and the conjugate action on G, its logarithmic derivative satisfies Φ Ad g (x) = Ad g • Φ x • Ad g −1 . In particular, ν(x) = det(Φ x ) is invariant under the adjoint action.
In fact, the left logarithmic derivative Φ x : g → g of the exponential map is explicitly given (cf. [Fa08, Thm. 2.1.4]) by the convergent power series Note that the determinant of the real linear map Φ x : g → g is equal to the determinant over C of the complexification Φ C x : g C → g C . In terms of the Jordan-Chevalley decomposition ad x = ad xs + ad xn into a semisimple and a nilpotent element of g C , we thus find where µ i are the eigenvalues of ad x as a complex linear transformation of g C . In particular, ν(x) depends only on the second factor in g = z ⊕ g 0 , and we can write ν(x) = ν 0 (ad x ) for a smooth function ν 0 : g 0 → R with ν 0 (0) = 1. It follows that for any c g 0 > ∇ν 0 (0) , there exists an ε 0 > 0 such that ν 0 − 1 ≤ c g 0 ε uniformly on B g 0 ε for all ε < ε 0 . The uniform estimate for ν on U ε now follows from G-invariance.
Since exp is equivariant under the adjoint action on g and G, this allows us to express δ 0 B G ρ in terms of the Lebesgue measure on g, .
Proposition 8.6. The number δ 0 B G ρ depends on G only through the maximal semisimple ideal , and the restriction of B θ to g 0 .
Proof. By Remark 8.4 the sets V g ε,R,r are K-invariant. If g = k exp(p) is the global Cartan decomposition of g ∈ G = K exp(p), we thus have Further, since z acts trivially on g, the decomposition p = p 0 + p z of p ∈ p with respect to p = p 0 ⊕ (z ∩ p) yields Since B G ρ is K-invariant (Proposition 8.3), we have k exp(p) ∈ B G ρ if and only if exp(p) ∈ B G ρ , which is the case if and only if exp(p 0 ) ∈ B G ρ . But since Ad exp(p 0 ) acts by the identity on second factor of g = g 0 ⊕ z, we have Ad g exp(p 0 ) ≤ ρ for the adjoint action on g if and only Ad g 0 exp(p 0 ) ≤ ρ for the adjoint action on g 0 . If G 0 denotes the connected adjoint group of the semisimple Lie algebra g 0 , we thus have where Λ g and Λ g 0 denote the Lebesgue measure on g and g 0 , respectively.
8.3. Proof of Theorem 8.1. By Proposition 8.6, it suffices to prove Theorem 8.1 for the case where G is the adjoint group of the semisimple Lie algebra g 0 . The proof hinges on the following lemma.
Lemma 8.7 (Key Lemma). Let G be a connected, real reductive Lie group with semisimple Lie algebra g. Then for all R > 0 and all ρ > 1, we have The proof of this lemma requires a rather detailed discussion of limits of orbital integrals, and will be deferred to Section 10. Assuming Lemma 8.7, the proof of Theorem 8.1 is quite straightforward.
Proof of Theorem 8.1, assuming Lemma 8.7. In view of (8.3), the maximal dimension d of the nilpotent orbits is the same in g and g 0 . By Proposition 8.6, we may therefore assume without loss of generality that G is a connected, real reductive Lie group with semisimple Lie algebra g.
Since Ad g ≤ ρ, we have Ad The strategy to prove Lemma 8.7 is as follows. Since the closure of an adjoint orbit O x through x ∈ g contains 0 if and only if x is nilpotent, the set ε>0 V ε,R is the intersection of the nilpotent cone N with the unit ball B g R (0). The union of the nilpotent orbits O X of maximal dimension is a dense open subset of the nilpotent cone. Using results of Harish-Chandra and Barbasch-Vogan on limiting orbit integrals, we will show that as ε approaches 0, the volume of V ε,R with respect to the Lebesgue measure on g scales with R in the same way as the Liouville volume of the symplectic manifold O X ∩ B g R (0). Since the Kostant-Kirillov-Souriau symplectic form ω KKS O X on the cone O X scales as R under dilation, the corresponding Liouville volume form Vol KKS O X scales as R d/2 , yielding the factor ρ d/2 in Lemma 8.7.

Limits of orbital measures
In the remainder of this section, we focus on the proof of Lemma 8.7. From now on, we assume that the Lie algebra g is semisimple, and that the invariant bilinear form B is the Killing form κ. We will generally denote generic elements of g by x, y, z, elements of a Cartan subalgebra h ⊂ g by h, and elements of the nilpotent cone N ⊂ g by X, Y, Z. For a subset A ⊆ g, we denote the centralizer and the normalizer in g by Z g (A) and N g (A) respectively. On the group level, we similarly define Z G (A) := {g ∈ G : Ad g y = y ∀y ∈ A}, N G (A) := {g ∈ G : Ad g A ⊂ A}.
9.1. Regular elements of Lie algebras. We recall the notion of regularity in g. For x ∈ g consider the characteristic polynomial Let k(x) be the minimal index so that a k(x) (x) = 0. If k(x) = min y∈g k(y), we say x is regular. If A ⊂ g is any subset of g, we define A reg to be the set of regular elements in A. If h is a Cartan subalgebra (CSA) and C is a connected component of h reg , we call C an open Weyl chamber of h.
Remark 9.1. Our notion of regularity is different from a common definition where x is regular if its centralizer has minimal dimension among the centralizers of all x ′ ∈ g. Our current definition is used, for example, in [Bou05].
We will use the following standard properties of regular elements: Lemma 9.2. Let g be a real Lie algebra.
i) Regular elements in g lie in a unique CSA given by their centralizer. ii) If a single element in an adjoint orbit O x ⊂ g is regular, then all elements are.
iii) The set of regular elements is dense and open in g, and its complement in g is a Lebesgue null set. Vol Ox defines a Radon measure µ Ox on g. In particular, it is finite on compact subsets of g. On nilpotent orbits, these measures are homogeneous: Lemma 9.3. Let A ⊂ g be a Borel set, let X ∈ g a nilpotent element, and let k = dim O X /2. Then for all ρ > 0, we have Proof. By the Jacobson-Morozov theorem, the nilpotent orbit O X is a cone, i.e. ρ · O X = O X for all ρ > 0. Thus we have Denote by m ρ : O X → O X the multiplication by ρ. By definition of the KKS form we have for all X ′ ∈ O X and y, z ∈ g: as required.
The measure µ Ox on the adjoint orbit O x through x ∈ g yields the distribution D Ox on g defined by is an oriented inner product space of even dimension 2k, then the Pfaffian of a skew-symmetric linear map A : where Vol ∈ ∧ 2k V * is the volume form associated to the inner product B on the oriented vector space V , and ω A (v, w) = B(v, Aw) is the 2-form associated to A with respect to the inner product B. With V = h ⊥ and A = θ • ad h , this yields Recall that the Pfaffian is related to the determinant by Since the determinant is the product of the eigenvalues over C, we can determine |π(h)| from the eigenvalues of ad h on the complexification (h ⊥ ) C ∼ = g C /h C . In view of the root space decomposi- where the first map is the natural embedding and the second map the projection of the direct sum (9.4) onto the first direct summand. Since the Slodowy slice intersects O x transversally, the composition of these two maps is surjective. Using T x O x ∼ = ad x g and T X O X ∼ = Ad X g, this surjective map induces the following exact sequence: We obtain a canonical volume form on O x ∩ S X as the quotient of the KKS volume forms on O x and O X , which in turn gives rise to the measure m x,X . In [Har12, Chapter 2], the following limit of orbits is defined for all x ∈ g: One may think of this set as the limit of the orbits O εx as ε approches zero, hence as the orbits approach the nilpotent cone. Let us first show that every nilpotent orbit arises, in this sense, as a limit of regular orbits: Lemma 9.5. Every nilpotent orbit O X lies in the set N x for some regular x ∈ g.
Proof. The idea of this proof is essentially due to [BaVo80,p.48]. If X is nilpotent, choose an sl 2 -triple {X, Y, H} with X as the nilpositive and H as the semisimple element. Consider the associated Slodowy slice S X = X + Z g (Y ). Since G · S X is an open neighbourhood of O X , and since the set of regular elements is dense in g, there exists a regular element x ∈ G · S X . In other words, there exist g ∈ G and V ∈ Z g (Y ) with The centralizer Z g (Y ) is stable under ad H , hence we can decompose it into the eigenspaces of ad H on this space. Due to the structure theory of finite-dimensional sl 2 -modules (cf. [Bou05, Sec VIII.2]), the eigenvalues λ on these eigenspaces are all nonpositive: Consider then the element g t := exp − 1 2 log(t)H ∈ G. Then we have But this means that X ∈ N ∩ t>0 Ad G (tx) = N x .
Remark 9.6. A closer inspection of such orbital limits is given in [FrMe]. Their definition of N x coincides with the one given here by [FrMe,Remark 3.6].
We will need the following asymptotic expression for the orbital integrals. It is proven in [Har12, Cor 2.3], relying on [BaVo80, Theorem 3.2].
Definition 9.7. For x ∈ g, we define where the minimum is taken over all adjoint orbits O X contained in N x .
Theorem 9.8. Let g be a real, reductive Lie algebra, and let x ∈ g. Then for all f ∈ C ∞ c (g) we have where the sum is taken over the set of nilpotent orbits O X ⊂ N x which are of maximal dimension among all orbits contained in N x , and the volume of S X ∩ O x is calculated with respect to the measure m x,X .
Remark 9.9. If C is an open Weyl chamber of some CSA h, one actually has N h = N h ′ for all h, h ′ ∈ C by [Har12, Cor 2.4] (originally attributed to [BaVo80]). In particular, the number m in Theorem 9.8 is equal for all h in one such open Weyl chamber. In this case we also write m(C) for the m associated to any h ∈ C.
9.4. Uniform Orbital Limits. Using results of Harish-Chandra and Varadarajan, we will show that the convergence in Theorem 9.8 is uniform on certain subsets of h. For an open subset C ⊆ V of a vector space V , we denote by C k (C, R) the space of functions u : C → R that are k times continuously differentiable on C, and whose derivatives of order at most k extend continuously to the closure. The following result follows from [HC57, Thm 3] (see also [Var77,Thm I.3.23]).
Theorem 9.10. Let h ⊂ g be a Cartan subalgebra, C ⊂ h an open Weyl chamber, and let f ∈ C ∞ c (g, R). Then the function u f : C → R defined by Remark 9.11. In fact, u f extends to a Schwartz function on the connected component of the non-zero sets of the singular imaginary roots.
Remark 9.12. The definition of the invariant integral in [HC57] is, up to a positive scalar, equivalent to ours by Proposition 9.4.  Fix again an open Weyl chamber C of some Cartan subalgebra h ⊂ g. From Theorem 9.8, Theorem 9.10, and the fact that integrals over compact domains commute with uniform limits, it immediately follows that for all compact K ⊂ h, all f ∈ C ∞ c (g) and all continuous functions w : h → R. In particular, the integral on the right-hand side is well-defined. A similar statement holds with D O X (f ) replaced by µ O X (B R (0)).
Corollary 9.14. Let h ⊂ g be a Cartan subalgebra, K ⊂ h a compact subset, and C ⊂ h an open Weyl chamber. Let w : h → R be a continuous function, and let R > 0. Then with the sum over the O X as in Theorem 9.8, and the right hand side is integrable.
Let k = dim O X /2 for any of the orbits O X in the sum. Since w can be written as the difference of two nonnegative functions, we may assume without loss of generality that w is nonnegative. Then, for all n ∈ N, we have (9.8) Note that µ O X (B R (0)) = µ O X (B R (0)) for all nilpotent elements X. Indeed, since O X is a cone, the boundary ∂B R (0) intersects O X transversally, and the intersection O X ∩ ∂B R (0) ⊂ O X is either empty (when X = 0) or a submanifold of codimension at least one (when X = 0). Its Liouville measure is thus zero by Sard's Theorem. Finally, the statement follows by taking the monotone limits n → ∞ in (9.8).
10. Proof of the key Lemma 8.7 concluding Theorem B In all that follows, we will fix a maximal set of mutually nonconjugate, θ-stable CSAs h 1 , . . . , h n of g, with associated Cartan subgroups H 1 , . . . , H n ⊂ G defined by H i := Z G (h i ). We denote by W j := N G (h j )/H j the Weyl group associated to h j . We will need the following Lie algebraic version of the well-known Harish-Chandra integral formula [HC65, Lemma 41], see e.g. [Var77, Part I, Section 3, Lemma 2].
Lemma 10.1. Let g be a reductive Lie algebra with connected adjoint group G, and let f ∈ L 1 (g, Λ g ) be an integrable function on g. Then, for j = 1, . . . , n, there are G-invariant volume forms Vol G/H j on G/H j , so that Here, |π j | is a product of positive roots of (g C , (h j ) C ) as in Proposition 9.4. Proof. To see that the right hand side is indeed a disjoint union, suppose that Ad g h = Ad g ′ h ′ for regular elements h ∈ h i , h ′ ∈ h j and g, g ′ ∈ G. Since h and h ′ are regular, h i = Z g (h) and h j = Z g (h ′ ). Since h is conjugate to h ′ , h i is conjugate to h j . As the various CSA's are mutually nonconjugate, we conclude that h i = h j . ⊂: Let Ad g x ∈ (V ε,R ) reg with g ∈ G and x ∈ B ε (0). Since x is regular, it lies in a unique CSA, and is conjugate to an element h ∈ h j for some j, i.e. ∃g ′ ∈ G : Ad g ′ x = h ∈ h j . By Lemma 9.2, the orbit of a regular element consists only of regular elements. Hence h ∈ (V h j ε ) reg and Ad g(g ′ ) −1 h = Ad g x ∈ B R (0). Hence Ad g x lies in the right hand side. ⊃: Let Ad g h lie in the right hand side. Because h ∈ Ad G B ε (0), we can write h = Ad g ′ x for some g ′ ∈ G and x ∈ B ε (0). But then Ad g h = Ad gg ′ x ∈ Ad G B ε (0), and since h was regular, so is every element in its orbit, and we have Ad g h ∈ (V ε,R ) reg .
It follows that ad * h = − ad θh . Since [h, θh] = 0, the operator ad h : g → g is normal, and its operator norm ad h with respect to the inner product κ θ satisfies ad h = max λ∈Spec(ad h ) |λ| .
Let g ∈ G such that Ad g (h) ∈ B ε (0). Since ad Adg h is conjugate to ad h , it is a normal operator with the same eigenvalues, so in particular ad h = ad Adg h . So since the operator norm is bounded on B ε (0), it is bounded on Ad G (B ε (0)) ∩ h as well, and the latter is a bounded subset of h.
Recall that V ε,R differs from (V ε,R ) reg only in a Lebesgue null set. Also, we can write every (h j ) reg as the union of its open Weyl chambers C j,r . Hence, we can use Lemma 10.1 and the decomposition To simplify, we will fix a single CSA h j and a single open Weyl chamber C j,r , and suppress the indices: H := H j , π := π j , h := h j , Vol G/H := Vol G/H j , C := C j,r . For now, let us look at the single summand of (10.2) corresponding to h and C.
Lemma 10.4. Fix the notation as in (10.3), let m := m(h) as in Definition 9.7 for an arbitrary h ∈ C, and R > 0 arbitrary. Then there is some c = 0 which does not depend on R so that where the sum is taken over all nilpotent orbits O X contained in N h for an arbitrary h ∈ C, cf. Remark 9.9. Proof. Recall the notation µ O h for the measure defined in Section 9, and that π(h) was defined in Equation In the last step, we used that V h ε = εV h 1 , that dΛ h (εh) = ε dim h dΛ h (h), and that π(εh) = ε |∆ + | π(h). By Corollary 9.14 and compactness of V h 1 ∩ C (Lemma 10.3), we have: This shows the statement.
Finally, we use this to prove the key lemma: Proof of key Lemma 8.7. Let O X ⊂ g be a nilpotent orbit of dimension d. By Lemma 9.5 and Remark 9.9, there is some CSA h ⊂ g and some open Weyl chamber C ⊂ h reg so that O X ⊂ N h for all h ∈ C. Since O X is of maximal dimension, the number m := m(x) from Definition 9.7 is minimal among m(x ′ ) for all x ′ ∈ g reg . Then, by (10.2) and Lemma 10.4, there are numbers c j,r = 0, independent of R, such that where the sum over the O X is carried out over all nilpotent orbits O X ⊂ N h of dimension d. Note that this sum is independent of the choice of h ∈ C j,r for fixed j and r, cf. Remark 9.9. All summands in the above sum are positive as they arise from volumes of subsets of g, and thus the total sum is nonzero. Lastly, the sum is homogeneous of degree d/2 in R by Lemma 9.3. Hence we may consider their quotient and conclude the proof: