Estimates for singular integrals and extrapolation
arXiv:0704.1537v1  [math.CA]  12 Apr 2007
ESTIMATES FOR SINGULAR
INTEGRALS AND EXTRAPOLATION
Shuichi Sato
Abstract. In this note, we study singular integrals with rough kernels, which belong
to a class of singular Radon transforms. We prove certain estimates for the singular
integrals that are useful in an extrapolation argument. As an application, we prove
Lp boundedness of the singular integrals under a certain sharp size condition on their
kernels.
1. Introduction
Let Ω be a function in L1(Sn−1) satisfying
(1.1)
Ω(θ) dσ(θ) = 0,
where dσ denotes the Lebesgue surface measure on the unit sphere Sn−1 in Rn. In
this note we assume n ≥ 2. For s ≥ 1, let ∆s denote the collection of measurable
functions h on R+ = {t ∈ R : t > 0} satisfying
‖h‖∆s = sup
∫ 2j+1
|h(t)|s dt/t
where Z denotes the set of integers. We note that ∆s ⊂ ∆t if s > t. In this note
we always assume h ∈ ∆1. Let P (y) = (P1(y), P2(y), . . . , Pd(y)) be a polynomial
mapping, where each Pj is a real-valued polynomial on R
n. We consider a singular
integral operator of the form:
(1.2) T (f)(x) = p. v.
f(x− P (y))K(y) dy = lim
|y|>ǫ
f(x− P (y))K(y) dy,
for an appropriate function f on Rd, where K(y) = h(|y|)Ω(y′)|y|−n, y′ = |y|−1y.
Then, T (f) belongs to a class of singular Radon transforms. See Stein [17], Fan-Pan
[8] and Al-Salman-Pan [1] for this singular integral.
When h = 1 (a constant function), n = d and P (y) = y, we also write T (f) =
S(f). Let f̂(ξ) =
f(x)e−2πi〈x,ξ〉 dx be the Fourier transform of f , where 〈·, ·〉
Key words and phrases. Singular integrals, singular Radon transforms, maximal functions,
extrapolation.
2000 Mathematics Subject Classification. Primary 42B20, 42B25
Typeset by AMS-TEX
http://arxiv.org/abs/0704.1537v1
2 SHUICHI SATO
denotes the inner product in Rd. Then it is known that (Sf )̂ (ξ) = m(ξ′)f̂(ξ),
where
m(ξ′) = −
sgn(〈ξ′, θ〉) + log |〈ξ′, θ〉|
dσ(θ).
Using this, we can show that S extends to a bounded operator on L2 if Ω ∈
L logL(Sn−1), where L logL(Sn−1) denotes the Zygmund class of all those func-
tions Ω on Sn−1 which satisfy
|Ω(θ)| log(2 + |Ω(θ)|) dσ(θ) <∞.
Furthermore, if Ω ∈ L logL(Sn−1), by the method of rotations of Calderón-Zygmund
(see [2]) it can be shown that S extends to a bounded operator on Lp for all
p ∈ (1,∞).
When n = d and P (y) = y, R. Fefferman [10] proved that if h is bounded and Ω
satisfies a Lipschitz condition of positive order on Sn−1, then the singular integral
operator T in (1.2) is bounded on Lp for 1 < p < ∞. Namazi [13] improved this
result by replacing the Lipschitz condition by the condition that Ω ∈ Lq(Sn−1)
for some q > 1. In [7], Duoandikoetxea and Rubio de Francia developed methods
which can be used to study mapping properties of several kinds of operators in
harmonic analysis including the singular integrals considered in [13]. Also, see [6,
22] for weighted Lp boundedness of singular integrals, and [18, 19] for background
materials.
For the rest of this note we assume that the polynomial mapping P in (1.2)
satisfies P (−y) = −P (y) and P 6= 0. We shall prove the following:
Theorem 1. Let Ω ∈ Lq(Sn−1), q ∈ (1, 2] and h ∈ ∆s, s ∈ (1, 2]. Suppose Ω
satisfies (1.1). Let T be as in (1.2). Then we have
‖T (f)‖Lp(Rd) ≤ Cp(q − 1)
−1(s− 1)−1‖Ω‖Lq(Sn−1)‖h‖∆s‖f‖Lp(Rd)
for all p ∈ (1,∞), where the constant Cp is independent of q, s,Ω and h. Also, the
constant Cp is independent of polynomials Pj if we fix deg(Pj) (j = 1, 2, . . . , d).
In Al-Salman-Pan [1], the Lp boundedness of T was proved under the condition
that Ω is a function in L logL(Sn−1) satisfying (1.1) and h ∈ ∆s for some s > 1 ([1,
Theorem 1.3]). Also it is noted there that estimates like those in Theorem 1 (with
s being fixed) can be used to prove the same result by applying an extrapolation
method, but such estimates are yet to be proved (see [1, p. 156]). In [1], the authors
also considered singular integrals defined by certain polynomial mappings P which
do not satisfy the condition P (−y) = −P (y).
As a consequence of Theorem 1 we can give a different proof of [1, Theorem 1.3]
via an extrapolation method; in fact, we can get an improved result. For a positive
number a and a function h on R+, let
La(h) = sup
∫ 2j+1
|h(r)| (log(2 + |h(r)|))
dr/r.
We define a class La to be the space of all those measurable functions h on R+
which satisfy La(h) <∞. Also, let
Na(h) =
ma2mdm(h),
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 3
where dm(h) = supk∈Z 2
−k|E(k,m)| with E(k,m) = {r ∈ (2k, 2k+1] : 2m−1 <
|h(r)| ≤ 2m} for m ≥ 2, E(k, 1) = {r ∈ (2k, 2k+1] : |h(r)| ≤ 2}. We denote by Na
the class of all those measurable functions h on R+ such that Na(h) < ∞. Then
we readily see that Na(h) < ∞ implies La(h) < ∞. Conversely, if La+b(h) < ∞
for some b > 1, then Na(h) <∞. To see this, note that
2mma+b2−k|E(k,m)| ≤ C
E(k,m)
|h(r)| (log(2 + |h(r)|))
dr/r ≤ CLa+b(h)
for m ≥ 2; thus Na(h) ≤ 2d1(h) + CLa+b(h)
−b < ∞. By Theorem 1 and
an extrapolation method we have the following:
Theorem 2. Suppose Ω is a function in L logL(Sn−1) satisfying (1.1) and h ∈ N1.
Let T be as in (1.2). Then
‖T (f)‖Lp(Rd) ≤ Cp‖f‖Lp(Rd)
for all p ∈ (1,∞), where Cp is independent of polynomials Pj if the polynomials are
of fixed degree.
By Theorem 2 and the remark preceding it we see that T is bounded on Lp for
all p ∈ (1,∞) if Ω is as in Theorem 2 and h ∈ La for some a > 2.
When n = d, P (y) = y, Ω is as in Theorem 2 and h is a constant function, it is
known that T is of weak type (1, 1); see [5, 15]. Also, see [4, 9, 11, 12, 16, 20, 21]
for related results.
In Section 2, we shall prove Theorem 1. Applying the methods of [7] involving the
Littlewood-Paley theory and using results of [8, 14], we shall prove Lp estimates
for certain maximal and singular integral operators related to the operator T in
Theorem 1 (Lemmas 1 and 2). Lemma 1 is used to prove Lemma 2. By Lemma 2
we can easily prove Theorem 1. A key idea of the proof of Theorem 1 is to apply a
Littlewood-Paley decomposition adapted to a suitable lacunary sequence depending
on q and s for which Ω ∈ Lq(Sn−1) and h ∈ ∆s. The method of appropriately
choosing the lacunary sequence was inspired by [1], where, in a somewhat different
way from ours, a similar method was used to study several classes of singular
integrals.
We shall prove Theorem 2 in Section 3. Finally, in Section 4, we consider the
maximal operator
(1.3) T ∗(f)(x) = sup
N,ǫ>0
ǫ<|y|<N
f(x− P (y))K(y) dy
where P and K are as in (1.2). We shall prove analogs of Theorems 1 and 2 for the
operator T ∗.
Throughout this note, the letter C will be used to denote non-negative constants
which may be different in different occurrences.
2. Proof of Theorem 1
Let Ω, h be as in Theorem 1. We consider the singular integral T (f) defined
in (1.2). Let ρ ≥ 2 and Ek = {x ∈ R
n : ρk < |x| ≤ ρk+1}. Then T (f)(x) =
−∞ σk ∗ f(x), where {σk} is a sequence of Borel measures on R
d such that
(2.1) σk ∗ f(x) =
f(x− P (y))K(y) dy.
4 SHUICHI SATO
We note that
(σk ∗ f )̂ (ξ) = f̂(ξ)
e−2πi〈P (y),ξ〉K(y) dy.
We write
P (y) =
Qj(y), Qj(y) =
|γ|=N(j)
γ (aγ ∈ R
where Qj 6= 0, 1 ≤ N(1) < N(2) < · · · < N(ℓ), γ = (γ1, . . . , γn) is a multi-
index, yγ = y
1 . . . y
n and |γ| = γ1 + · · · + γn. Let βm = ρ
N(m) and αm =
(q − 1)(s− 1)/(2qsN(m)) for 1 ≤ m ≤ ℓ. Put P (m)(y) =
j=1Qj(y) and define a
sequence µ(m) = {µ
k } of positive measures on R
k ∗ f(x) =
x− P (m)(y)
|K(y)| dy
for m = 1, 2, . . . , ℓ. Also, define µ(0) = {µ
k } by µ
k = (
|K(y)| dy)δ, where δ
is Dirac’s delta function on Rd. For a sequence ν = {νk} of finite Borel measures
on Rd, we define the maximal operator ν∗ by ν∗(f)(x) = supk ||νk| ∗ f(x)|, where
|νk| denotes the total variation. We consider the maximal operators
m ≤ ℓ). We also write
= µ∗ρ.
Lj(ξ) = (〈aγ(j,1), ξ〉, 〈aγ(j,2), ξ〉, . . . , 〈aγ(j,rj), ξ〉),
where {γ(j, k)}
k=1 is an enumeration of {γ}|γ|=N(j) for 1 ≤ j ≤ ℓ. Then Lj is a
linear mapping from Rd to Rrj . Let sj = rankLj. There exist non-singular linear
transformations Rj : R
d → Rd and Hj : R
sj → Rsj such that
Rj(ξ)| ≤ |Lj(ξ)| ≤ C|Hjπ
Rj(ξ)|,
where πdsj (ξ) = (ξ1, . . . , ξsj ) is the projection and C depends only on rj (a proof
can be found in [8]). Let {σ
k } (0 ≤ m ≤ ℓ) be a sequence of Borel measures on
d such that
k ∗ f(x) =
x− P (m)(y)
K(y) dy
for m = 1, 2, . . . , ℓ, while σ
k = 0. Let ϕ ∈ C
0 (R) be supported in {|r| ≤ 1} and
ϕ(r) = 1 for |r| < 1/2. Define a sequence τ (m) = {τ
k } of Borel measures by
(2.2) τ̂
k (ξ) = σ̂
k (ξ)Φk,m(ξ)− σ̂
(m−1)
k (ξ)Φk,m−1(ξ)
for m = 1, 2, . . . , ℓ, where
Φk,m(ξ) =
j=m+1
βkj |Hjπ
Rj(ξ)|
if 0 ≤ m ≤ ℓ− 1 and Φk,ℓ = 1. Then σk = σ
m=1 τ
k . We note that
Φk,m(ξ)ϕ
βkm|Hmπ
Rm(ξ)|
= Φk,m−1(ξ) (1 ≤ m ≤ ℓ).
For 1 ≤ m ≤ ℓ, let T
ρ (f) =
k ∗ f . Then T =
m=1 T
For p ∈ (1,∞) we put p′ = p/(p − 1) and δ(p) = |1/p − 1/p′|. Let θ ∈ (0, 1).
Then we have the following Lp estimates for (µ(m))∗ and T
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 5
Lemma 1. For p > 1 + θ and 0 ≤ j ≤ ℓ, we have
(2.3)
(µ(j))∗(f)
Lp(Rd)
≤ C(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s
1− ρ−θ/(2q
)−2/p
‖f‖Lp(Rd).
Lemma 2. For p ∈ (1 + θ, (1 + θ)/θ) and 1 ≤ m ≤ ℓ, we have
‖T (m)ρ (f)‖Lp(Rd) ≤ C(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s
1− ρ−θ/(2q
)−1−δ(p)
‖f‖Lp(Rd).
The constants C in Lemmas 1 and 2 are independent of q, s ∈ (1, 2], Ω ∈
Lq(Sn−1), h ∈ ∆s, ρ and the coefficients of the polynomials Pk (1 ≤ k ≤ d).
We prove Lemma 2 first, taking Lemma 1 for granted for the moment. Let A =
(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s and B =
1− β−θαmm
1− ρ−θ/(2q
. Then, we
have the following estimates:
(2.4) ‖τ
k ‖ ≤ c1A (‖τ
k ‖ = |τ
k |(R
(2.5) |τ̂
k (ξ)| ≤ c2A
βkm|Lm(ξ)|
(2.6) |τ̂
k (ξ)| ≤ c3A
βk+1m |Lm(ξ)|
(2.7)
(τ (m))∗(f)
≤ CpAB
2/p‖f‖p for p > 1 + θ,
for some constants ci (1 ≤ i ≤ 3) and Cp, where we simply write ‖f‖Lp(Rd) = ‖f‖p.
Now we prove the estimates (2.4)–(2.7). First we see that
k ‖ ≤ C
k ‖+ ‖σ
(m−1)
(2.8)
≤ C‖Ω‖1
∫ ρk+1
|h(r)| dr/r ≤ C(log ρ)‖Ω‖1‖h‖∆1.
From this (2.4) follows. To prove (2.5), define
F (r, ξ) =
Ω(θ) exp(−2πi〈ξ, P (m)(rθ)〉) dσ(θ).
Then, via Hölder’s inequality, for s ∈ (1, 2] we see that
k (ξ)| =
∫ ρk+1
h(r)F (r, ξ) dr/r
(2.9)
∫ ρk+1
|h(r)|s dr/r
)1/s(
∫ ρk+1
|F (r, ξ)|
)1/s′
≤ C(log ρ)1/s‖h‖∆s‖Ω‖
(s′−2)/s′
∫ ρk+1
|F (r, ξ)|
)1/s′
We need the following estimates for the last integral:
6 SHUICHI SATO
Lemma 3. Let 1 < q ≤ 2 and Ω ∈ Lq(Sn−1). Then there exists a constant C > 0
independent of q, ρ,Ω and the coefficients of the polynomial components of P (m)
such that
∫ ρk+1
|F (r, ξ)|
dr/r ≤ C(log ρ)
βkm|Lm(ξ)|
)−1/(2q′N(m))
‖Ω‖2q.
Proof. Take an integer ν such that 2ν < ρ ≤ 2ν+1. By the proof of Proposition 5.1
of [8] we have
∫ ρk+1
|F (r, ξ)|
dr/r =
∣F (ρkr, ξ)
dr/r ≤
∫ 2j+1
∣F (ρkr, ξ)
∣F (2jρkr, ξ)
)2/q′
2jN(m)ρkN(m)|Lm(ξ)|
)−1/(2N(m)q′)
‖Ω‖2q
≤ C(log ρ)
ρkN(m)|Lm(ξ)|
)−1/(2N(m)q′)
‖Ω‖2q.
This completes the proof of Lemma 3.
By (2.9) and Lemma 3 we have |σ̂
k (ξ)| ≤ CA
βkm|Lm(ξ)|
. Also, we have
(m−1)
k ‖ ≤ CA by (2.8). We can prove the estimate (2.5) by using these estimates
in the definition of τ
k in (2.2) and by noting that ϕ is compactly supported.
Next, to prove (2.6), using (1.1) when m = 1, we see that
k (ξ)| ≤
k (ξ)− σ̂
(m−1)
k (ξ)
Φk,m(ξ)
(Φk,m(ξ) − Φk,m−1(ξ)) σ̂
(m−1)
k (ξ)
≤ C‖Ω‖1β
m |Lm(ξ)|
∫ ρk+1
|h(r)| dr/r + C‖σ
(m−1)
m|Lm(ξ)|
≤ C(log ρ)‖Ω‖1‖h‖∆1β
m |Lm(ξ)|,
where to get the last inequality we have used (2.8). By this and (2.8), we have
k (ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1
βk+1m |Lm(ξ)|
for all c ∈ (0, 1], which implies (2.6). Finally, the estimate (2.7) follows from Lemma
1 since
(τ (m))∗(f)
(µ(m))∗(|f |)
(µ(m−1))∗(|f |)
≤ CAB2/p‖f‖p
for p > 1 + θ, where the first inequality can be seen by change of variables and a
well-known result on maximal functions (see [8]).
Let {ψk}
−∞ be a sequence of functions in C
∞((0,∞)) such that
supp(ψk) ⊂ [β
m , β
ψk(t)
2 = 1, |(d/dt)jψk(t)| ≤ cj/t
j (j = 1, 2, . . . ),
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 7
where the constants cj are independent of βm. Define an operator Sk by (Sk(f)) (̂ξ) =
Rm(ξ)|
f̂(ξ) and let
j (f) =
k ∗ Sj+k(f)
Then by Plancherel’s theorem and the estimates (2.4)–(2.6) we have
j (f)
D(j+k)
k (ξ)|
2|f̂(ξ)|2 dξ
(2.10)
≤ CA2 min
1, β−2αm(|j|−2)m
D(j+k)
|f̂(ξ)|2 dξ
≤ CA2 min
1, β−2αm(|j|−2)m
‖f‖22,
where D(k) = {β−k−1m ≤ |Hmπ
Rm(ξ)| ≤ β
Applying the proof of Lemma in [7, p. 544] and using the estimates (2.4) and
(2.7), we can prove the following.
Lemma 4. Let u ∈ (1 + θ, 2]. Define a number v by 1/v− 1/2 = 1/(2u). Then we
have the vector valued inequality
(2.11)
k ∗ gk|
≤ (c1Cu)
1/2AB1/u
where the constants c1 and Cu are as in (2.4) and (2.7), respectively.
By the Littlewood-Paley theory we have
j (f)‖p ≤ cp
k ∗ Sj+k(f)|
,(2.12)
|Sk(f)|
≤ cp‖f‖p,(2.13)
where 1 < p < ∞ and cp is independent of βm and the linear transformations
Rm, Hm. Suppose that 1 + θ < p ≤ 4/(3− θ). Then we can find u ∈ (1 + θ, 2] such
that 1/p = 1/2 + (1 − θ)/(2u). Let v be defined by u as in Lemma 4. Then by
(2.11)–(2.13) we have
(2.14) ‖V
j (f)‖v ≤ CAB
1/u‖f‖v.
Since 1/p = θ/2 + (1− θ)/v, interpolating between (2.10) and (2.14), we have
j (f)‖p ≤ CAB
(1−θ)/u min
1, β−θαm(|j|−2)m
‖f‖p.
8 SHUICHI SATO
It follows that
‖T (m)ρ (f)‖p ≤
j (f)‖p ≤ CAB
(1−θ)/u(1− β−θαmm )
−1‖f‖p
(2.15)
≤ CAB2/p‖f‖p,
where we have used the inequality
−θαm(|j|−2)
1− β−θαmm
We also have ‖T
ρ (f)‖2 ≤
j (f)‖2 ≤ CAB‖f‖2 by (2.10), since B ≥
(1− β−αmm )
. By duality and interpolation, we can now get the conclusion of
Lemma 2.
Next, we give a proof of Lemma 1. We prove Lemma 1 by induction on j. Now
we assume (2.3) for j = m − 1, 1 ≤ m ≤ ℓ, and prove (2.3) for j = m. Let
ϕ ∈ C∞0 (R) be as above. Define a sequence η
(m) = {η
k } of Borel measures on R
k (ξ) = ϕ
βkm|Hmπ
Rm(ξ)|
(m−1)
k (ξ).
Then by (2.3) with j = m− 1, we have
(2.16)
(η(m))∗(f)
(µ(m−1))∗(f)
≤ CAB2/p‖f‖p
for p > 1 + θ. Furthermore, we have the following:
k ‖+ ‖µ
k ‖ ≤ C‖µ
(m−1)
k ‖+ ‖µ
k ‖ ≤ C‖Ω‖1
∫ ρk+1
|h(r)| dr/r
(2.17)
≤ C(log ρ)‖Ω‖1‖h‖∆1 ≤ CA,
k (ξ)− η̂
k (ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1
βk+1m |Lm(ξ)|
(2.18)
βk+1m |Lm(ξ)|
(2.19) |µ̂
k (ξ)| ≤ CA
βkm|Lm(ξ)|
(2.20) |η̂
k (ξ)| ≤ C(log ρ)‖h‖∆1‖Ω‖1
βkm|Lm(ξ)|
βkm|Lm(ξ)|
To see (2.18) we note that
k (ξ)−η̂
k (ξ)| ≤ |µ̂
k (ξ)−µ̂
(m−1)
k (ξ)|+
βkm|Hmπ
Rm(ξ)|
(m−1)
k (ξ)
Thus arguing as in the proof of (2.6), we have the first inequality of (2.18). The
estimate (2.19) follows from the arguments used to prove (2.5). Also, we can see
the first inequality of (2.20) by the definition of η
k and (2.17).
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 9
Since ‖(µ(m))∗(f)‖∞ ≤ CA‖f‖∞, by taking into account an interpolation, it
suffices to prove (2.3) with j = m for p ∈ (1+θ, 2]. Define a sequence ν(m) = {ν
of Borel measures by ν
k = µ
k − η
k . Let
gm(f)(x) =
k ∗ f(x)
(2.21) (µ(m))∗(f) ≤ gm(f) + (η
(m))∗(|f |).
Thus, by (2.16), to get (2.3) with j = m it suffices to prove ‖gm(f)‖p ≤ CAB
2/p‖f‖p
for p ∈ (1 + θ, 2] with an appropriate constant C. By a well-known property of
Rademacher’s functions, this follows from
(2.22)
U (m)ǫ (f)
≤ CAB2/p‖f‖p
for p ∈ (1 + θ, 2], where U
ǫ (f) =
k ǫkν
k ∗ f with ǫ = {ǫk}, ǫk = 1 or −1, and
the constant C is independent of ǫ.
The estimate (2.22) is a consequence of the following:
Lemma 5. We define a sequence {pj}
1 by p1 = 2 and 1/pj+1 = 1/2+(1−θ)/(2pj)
for j ≥ 1. (We note that 1/pj = (1 − a
j)/(1 + θ), where a = (1 − θ)/2, so {pj} is
decreasing and converges to 1 + θ.) Then, for j ≥ 1 we have
U (m)ǫ (f)
≤ CjAB
2/pj ‖f‖pj .
Proof. Let
j (f) =
ǫkSj+k
k ∗ Sj+k(f)
Then by Plancherel’s theorem and the estimates (2.17)–(2.20), as in (2.10) we have
(2.23)
j (f)
≤ CAmin
1, β−αm(|j|−2)m
‖f‖2.
It follows that
ǫ (f)
j (f)‖2 ≤ CAB‖f‖2. If we denote by A(s)
the assertion of Lemma 5 for j = s, this proves A(1).
Now we derive A(s+1) from A(s) assuming that A(s) holds, which will complete
the proof of Lemma 5 by induction. Using (2.21), we see that
(ν(m))∗(f) ≤ (µ(m))∗(|f |) + (η(m))∗(|f |) ≤ gm(|f |) + 2(η
(m))∗(|f |).
Note that A(s) implies ‖gm(f)‖ps ≤ CAB
2/ps‖f‖ps . By this and (2.16) we have
(2.24)
(ν(m))∗(f)
≤ ‖gm(|f |)‖ps + 2
(η(m))∗(|f |)
≤ CAB2/ps‖f‖ps .
10 SHUICHI SATO
By (2.17), (2.23) and (2.24), we can now apply the arguments used in the proof of
(2.15) to get A(s+ 1). This completes the proof of Lemma 5.
Now we prove (2.22) for p ∈ (1 + θ, 2]. Let {pj}
1 be as in Lemma 5. Then we
have pN+1 < p ≤ pN for some N . Thus, interpolating between the estimates of
Lemma 5 for j = N and j = N + 1, we have (2.22). This proves (2.3) for j = m.
Finally, we can easily see that (µ(0))∗(f) ≤ C(log ρ)‖Ω‖1‖h‖∆1|f | (see (2.17)),
which implies the estimate (2.3) for j = 0. Therefore, by induction we have (2.3)
for all 0 ≤ j ≤ ℓ. This completes the proof of Lemma 1.
Now we can prove Theorem 1. Since θ ∈ (0, 1) is arbitrary, by taking ρ = 2q
in Lemma 2 we have
(f)‖p ≤ Cp(q − 1)
−1(s− 1)−1‖Ω‖q‖h‖∆s‖f‖p
for all p ∈ (1,∞). This completes the proof of Theorem 1, since T =
m=1 T
3. Proof of Theorem 2
Theorem 2 can be proved by Theorem 1 and an extrapolation argument. Let
T (f) be the singular integral in (1.2). We also write T (f) = Th,Ω(f). We fix
q ∈ (1, 2], Ω ∈ Lq(Sn−1), p ∈ (1,∞) and a function f with ‖f‖p ≤ 1 and put
S(h) = ‖Th,Ω(f)‖p. Then we have the following subadditivity:
(3.1) S(h+ k) ≤ S(h) + S(k).
Set E1 = {r ∈ R+ : |h(r)| ≤ 2} and Em = {r ∈ R+ : 2
m−1 < |h(r)| ≤ 2m} for
m ≥ 2. Then, applying Theorem 1, we see that
(3.2) S (hχEm) ≤ C(q − 1)
−1(s− 1)−1‖Ω‖q‖hχEm‖∆s
for s ∈ (1, 2], where χE denotes the characteristic function of a set E. Now we follow
the extrapolation argument of Zygmund [23, Chap. XII, pp. 119–120]. First, note
‖hχEm‖∆1+1/m ≤ 2
mdm/(m+1)m (h)
for m ≥ 1, where dm(h) is as in Section 1. Using this and (3.2) we see that
S (hχEm) ≤ C(q − 1)
−1‖Ω‖q
m‖hχEm‖∆1+1/m
≤ C(q − 1)−1‖Ω‖q
m2mdm/(m+1)m (h).
Recalling the definition of Na(h), we have
m2mdm/(m+1)m (h) =
dm(h)<3−m
m2mdm/(m+1)m (h) +
dm(h)≥3−m
m2mdm/(m+1)m (h)
m2m3−m
2/(m+1) +
m2mdm(h)3
m/(m+1) ≤ C(1 +N1(h)).
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 11
Therefore, by (3.1) we see that
(3.3) S(h) ≤
S (hχEm) ≤ C(q − 1)
−1‖Ω‖q (1 +N1(h)) .
Next, fix h ∈ N1, p ∈ (1,∞) and f with ‖f‖p ≤ 1 and let R(Ω) = ‖Th,Ω(f)‖p.
Put em = σ(Fm) for m ≥ 1, where Fm = {θ ∈ S
n−1 : 2m−1 < |Ω(θ)| ≤ 2m} for
m ≥ 2 and F1 = {θ ∈ S
n−1 : |Ω(θ)| ≤ 2}. We decompose Ω as Ω =
m=1 Ωm,
where Ωm = ΩχFm − σ(S
n−1)−1
Ω dσ. We note that
Ωm dσ = 0, ‖Ωm‖r ≤
m for 1 < r <∞. Now, by (3.3) and the subadditivity of R(Ω) we see that
R(Ω) ≤
R(Ωm) ≤ C (1 +N1(h))
m‖Ωm‖1+1/m
≤ C (1 +N1(h))
m2mem/(m+1)m = C (1 +N1(h))
em<3−m
em≥3−m
≤ C (1 +N1(h))
m2m3−m
2/(m+1) +
m2mem3
m/(m+1)
≤ C (1 +N1(h)))
|Ω(θ)| log(2 + |Ω(θ)|) dσ(θ)
This completes the proof of Theorem 2.
4. Estimates for maximal functions
For the maximal operator T ∗ in (1.3) we have a result similar to Theorem 1.
Theorem 3. Let q ∈ (1, 2], s ∈ (1, 2] and Ω ∈ Lq(Sn−1), h ∈ ∆s. Suppose Ω
satisfies (1.1). Then we have
‖T ∗(f)‖Lp(Rd) ≤ Cp(q − 1)
−1(s− 1)−1‖Ω‖Lq(Sn−1)‖h‖∆s‖f‖Lp(Rd)
for all p ∈ (1,∞), where Cp is independent of q, s, Ω and h.
As Theorem 1 implies Theorem 2, we have the following as a consequence of
Theorem 3.
Theorem 4. Let Ω be a function in L logL(Sn−1) satisfying (1.1) and h ∈ N1.
‖T ∗(f)‖Lp(Rd) ≤ Cp‖f‖Lp(Rd)
for all p ∈ (1,∞).
As in the cases of Theorems 1 and 2, the constants Cp of Theorems 3 and 4 are
also independent of polynomials Pj if we fix deg(Pj) (j = 1, 2, . . . , d). When Ω is
as in Theorem 4 and h ∈ ∆s for some s > 1, the L
p boundedness of T ∗ was proved
in [1]. When n = d, P (y) = y, Ω ∈ Lq for some q > 1 and h is bounded, the Lp
boundedness of T ∗ is due to [3].
We use the following to prove Theorem 3.
12 SHUICHI SATO
Lemma 6. Let τ (m) = {τ
k } (1 ≤ m ≤ ℓ), where the measures τ
k are as
in (2.2). Let θ ∈ (0, 1) and let positive numbers A = (log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s,
1− β−θαmm
be as above. We define
(4.1) T ∗ρ,m(f)(x) = sup
j ∗ f(x)
Then, for p ∈ (2(1 + θ)/(θ2 − θ + 2), (1 + θ)/θ) =: Iθ we have
‖T ∗ρ,m(f)‖p ≤ CA
B1+δ(p) +B2/p+1−θ/2
‖f‖p,
where C is independent of q, s ∈ (1, 2], Ω ∈ Lq(Sn−1), h ∈ ∆s, ρ and the coefficients
of the polynomials Pj (1 ≤ j ≤ d).
Proof. Let T
ρ (f) =
k ∗ f be as in Lemma 2. Let a function ϕ be as in the
definition of τ
k in (2.2). Define ϕk by ϕ̂k(ξ) = ϕ
βkm|Hmπ
Rm(ξ)|
. Let δ be
the delta function as above. Following [8], we decompose
j ∗ f = ϕk ∗ T
ρ (f)− ϕk ∗
j ∗ f
+ (δ − ϕk) ∗
j ∗ f
It follows that
(4.2) T ∗ρ,m(f) ≤ sup
ϕk ∗ T
ρ (f)
j (f),
whereN
j (f) = supk
k−j−1 ∗ f
+supk
(δ − ϕk) ∗
j+k ∗ f
. By Lemma
2 we have
(4.3)
ϕk ∗ T
ρ (f)
≤ CAB1+δ(p)‖f‖p for p ∈ (1 + θ, (1 + θ)/θ).
Also, by (2.7) we see that
(4.4) ‖N
j (f)‖r ≤ CAB
2/r‖f‖r for r > 1 + θ.
On the other hand, we have
j (f) ≤
(δ − ϕk) ∗
j+k ∗ f
k−j−1 ∗ f
Therefore, by the estimates (2.5), (2.6) and Plancherel’s theorem, as in [8, p. 820]
we see that
(4.5) ‖N
j (f)‖2 ≤ CAβ
1− β−2αmm
)−1/2
‖f‖2.
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 13
For p ∈ Iθ we can find r ∈ (1 + θ, 2(1 + θ)/θ) such that 1/p = (1 − θ)/r + θ/2, so
an interpolation between (4.4) and (4.5) implies that
(4.6) ‖N
j (f)‖p ≤ CAB
2(1−θ)/r
1− β−2αmm
)−θ/2
β−αmθjm ‖f‖p.
Therefore, by (4.2), (4.3) and (4.6), for p ∈ Iθ we have
‖T ∗ρ,m(f)‖p ≤ CA
B1+δ(p) +B2(1−θ)/r+1
1− β−2αmm
)−θ/2
‖f‖p.
This implies the conclusion of Lemma 6, since
1− β−2αmm
≤ B and 2(1−θ)/r+
θ/2 + 1 = 2/p+ 1− θ/2.
Proof of Theorem 3. Note that T ∗(f) ≤ 2T ∗0 (f) + 2µ
ρ(|f |), where T
0 (f) is defined
by the formula in (4.1) with {τ
j } replaced by the sequence {σj} of measures in
(2.1) and µ∗ρ = (µ
(ℓ))∗ is as in Lemma 1. We note that T ∗0 (f) ≤
m=1 T
ρ,m(f).
Now, Lemma 6 implies that
‖T ∗ρ,m(f)‖p ≤ C(log ρ)
1− ρ−θ/(2q
‖Ω‖q‖h‖∆s‖f‖p
for p ∈ Iθ. By using this with ρ = 2
q′s′ , since θ ∈ (0, 1) is arbitrary, we can conclude
′s′ ,m
(f)‖p ≤ Cp(q − 1)
−1(s− 1)−1‖Ω‖q‖h‖∆s‖f‖p
for p ∈ (1,∞). Also, by Lemma 1 µ∗ρ satisfies a similar estimate when ρ = 2
q′s′ .
Collecting results, we have Theorem 3.
Remark. Let
M(f)(x) = sup
|y|<t
|f(x− P (y))||Ω(y′)||h(|y|)| dy.
It is easy to see that M(f) ≤ Cµ∗ρ(f), where C is independent of ρ ≥ 2. Therefore,
by Lemma 1 we can prove results similar to Theorems 1 and 2 for the maximal
operator M . In [1], Lp boundedness of M was proved under the condition that
Ω ∈ L logL(Sn−1) and h ∈ ∆s for some s > 1. When n = d, P (y) = y, it is known
that M is of weak type (1, 1) if Ω ∈ L logL(Sn−1) and h is bounded (see [5, 4]).
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Department of Mathematics
Faculty of Education
Kanazawa University
Kanazawa, 920-1192
Japan
E-mail address: shuichi@kenroku.kanazawa-u.ac.jp
