Akbulut, Ali ; Guliyev, Vagif ; Mustafayev, Rza
On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces. (English). Mathematica Bohemica, vol. 137 (2012), issue 1, pp. 27-43
MSC: 42B20, 42B25, 42B35 | MR 2978444 | Zbl 1250.42038 | DOI: 10.21136/MB.2012.142786
generalized Morrey space; maximal operator; Hardy operator; singular integral operator
In the paper we find conditions on the pair $(\omega _1,\omega _2)$ which ensure the boundedness of the maximal operator and the Calderón-Zygmund singular integral operators from one generalized Morrey space $\mathcal {M}_{p,\omega _1}$ to another $\mathcal {M}_{p,\omega _2}$, $1<p<\infty $, and from the space $\mathcal {M}_{1,\omega _1}$ to the weak space $W\mathcal {M}_{1,\omega _2}$. As applications, we get some estimates for uniformly elliptic operators on generalized Morrey spaces.
[1] Burenkov, V. I., Guliyev, H. V.: Necessary and sufficient conditions for boundedness of the maximal operator in the local Morrey-type spaces. Studia Mathematica 163 (2004), 157-176. DOI 10.4064/sm163-2-4 | MR 2047377
[2] Burenkov, V. I., Guliyev, H. V., Guliyev, V. S.: Necessary and sufficient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces. J. Comput. Appl. Math. 208 (2007), 280-301. DOI 10.1016/j.cam.2006.10.085 | MR 2347750
[3] Burenkov, V. I., Guliyev, V. S., Serbetci, A., Tararykova, T. V.: Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces. Doklady Ross. Akad. Nauk. 422 (2008), 11-14. MR 2475077
[4] Burenkov, V. I., Gogatishvili, A., Guliyev, V. S., Mustafayev, R. Ch.: Boundedness of the fractional maximal operator in Morrey-type spaces. Complex Var. Elliptic Equ. 55 (2010), 739-758. MR 2674862
[5] Burenkov, V., Gogatishvili, A., Guliyev, V., Mustafayev, R.: Boundedness of the fractional maximal operator in local Morrey-type spaces. Preprint, Institute of Mathematics, AS CR, Praha (2008), 20. MR 2674862
[6] Calderón, A. P., Zygmund, A.: Singular integral operators and differential equations. Amer. J. Math. 79 (1957), 901-921. DOI 10.2307/2372441 | MR 0100768
[7] Carro, M., Pick, L., Soria, J., Stepanov, V. D.: On embeddings between classical Lorentz spaces. Math. Ineq. & Appl. 4 (2001), 397-428. MR 1841071 | Zbl 0996.46013
[8] Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rend. Math. 7 (1987), 273-279. MR 0985999 | Zbl 0717.42023
[9] Fazio, G. D., Ragusa, M. A.: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112 (1993), 241-256. DOI 10.1006/jfan.1993.1032 | MR 1213138 | Zbl 0822.35036
[10] Guliyev, V. S.: Integral operators on function spaces on homogeneous groups and on domains in ${\mathbb R}^n$. Doctoral dissertation, Moskva, Mat. Inst. Steklov (1994), 329 Russian.
[11] Guliyev, V. S.: Function spaces, integral operators and two weighted inequalities on homogeneous groups. Some applications. Baku, Elm. (1999), 332 Russian.
[12] Guliyev, V. S.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. 2009, Art. ID 503948 20. MR 2579556 | Zbl 1193.42082
[13] Kurata, K., Sugano, S.: A remark on estimates for uniformly elliptic operators on weighted $L_p$ spaces and Morrey spaces. Math. Nachr. 209 (2000), 137-150. DOI 10.1002/(SICI)1522-2616(200001)209:1<137::AID-MANA137>3.0.CO;2-3 | MR 1734362 | Zbl 0939.35036
[14] Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces. Harmonic Analysis S. Igari ICM 90 Satellite Proceedings, Springer, Tokyo (1991), 183-189. MR 1261439 | Zbl 0771.42007
[15] Morrey, C. B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43 (1938), 126-166. DOI 10.1090/S0002-9947-1938-1501936-8 | MR 1501936 | Zbl 0018.40501
[16] Murata, M.: On construction of Martin boundaries for second order elliptic equations. Pub. Res. Instit. Math. Sci. 26 (1990), 585-627. DOI 10.2977/prims/1195170848 | MR 1081506 | Zbl 0726.31009
[17] Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces. Math. Nachr. 166 (1994), 95-103. DOI 10.1002/mana.19941660108 | MR 1273325
[18] Li, H. Q.: Estimations $L_p$ des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal. 161 (1999), 152-218. DOI 10.1006/jfan.1998.3347 | MR 1670222 | Zbl 0929.22005
[19] Peetre, J.: On convolution operators leaving ${\mathcal L}^{p,\lambda}$ spaces invariant. Ann. Mat. Appl. IV. Ser. 72 (1966), 295-304. MR 0209917
[20] Shen, Z. W.: $L_p$ estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45 (1995), 513-546. DOI 10.5802/aif.1463 | MR 1343560
[21] Smith, H. F.: Parametrix construction for a class of subelliptic differential operators. Duke Math. J. 63 (1991), 343-354. DOI 10.1215/S0012-7094-91-06314-3 | MR 1115111 | Zbl 0777.35002
[22] Stein, E. M.: Harmonic analysis: Real variable methods, orthogonality, and oscillatory integrals. Princeton Univ. Press, Princeton, NJ (1993). MR 1232192 | Zbl 0821.42001
[23] Sugano, S.: Estimates for the operators $V^{\alpha} (-\Delta+V)^{-\beta}$ and $V^{\alpha} \nabla (-\Delta+V)^{-\beta}$ with certain nonnegative potentials $V$. Tokyo J. Math. 21 (1998), 441-452. MR 1663618
[24] Thangavelu, S.: Riesz transforms and the wave equations for the Hermite operators. Commun. Partial Differ. Equations 15 (1990), 1199-1215. DOI 10.1080/03605309908820720 | MR 1070242
[25] Zhong, J. P.: Harmonic analysis for some Schrödinger type operators. PhD thesis, Princeton University (1993). MR 2689454