Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces
In this paper, we study the boundedness of some sublinear operators with rough kernels, satisfied by most of the operators in classical harmonic analysis, on the generalized weighted grand Morrey spaces. More specifically, we show that the sublinear operators with rough kernels are bounded on these spaces under the conditions that the operators and the kernel functions satisfy some size conditions, and the operators are bounded on Lebesgue spaces. This is done by exploiting the well-known boundedness of sublinear operators with rough kernels on Lebesgue spaces, a more explicit decomposition of the generalized weighted grand Morrey spaces and the good properties of the weight functions and the kernel functions. Through combining some properties of Ap weight with the relevant lemmas of operators with rough kernel, we obtain the boundedness for sublinear operators with rough kernels on weighted grand morrey spaces. Furthermore, using the equivalent norm and the properties of BMO functions, an application of the boundedness of the sublinear operators with rough kernels to the corresponding commutators formed by certain operators and BMO functions are also considered. And the boundedness of commutator is obtained by the lemma of function BMO.
Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces, Pure and Applied Mathematics Journal.
Vol. 8, No. 1,
2019, pp. 18-29.
C. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.
F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Math., Appl., 7 (7) (1987), 273-279.
J. García-Cuerva and J. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Stud., North-Holland, Amsterdam, 1985.
L. Grafakos, Classical and modern Fourier analysis, Pearson Education, 2004.
B. Muckenhoupt and R. Wheeden, Weighted bounded mean oscillation and the Hilbert transform, Studia Math., 54 (1976), 221-237.
Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr., 282 (2009), 219-231.
A. Meskhi, Maximal functions, potentials and singular integrals in grand Morrey spaces, Complex Var Elliptic Equ., 56 (2011), 1003-1019.
L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta Math., 92 (1997), 249–258.
C. Capone and A. Fiorenza, On small Lebesgue spaces, J. Funct. Spaces Appl., 3 (2005), 73-89.
A. Fiorenza, Duality and reflexivity in grand Lebesgue spaces, Collect. Math., 51 (2) (2000), 131-148.
A. Fiorenza, B. Gupta and P. Jain, The maximal theorem in weighted grand Lebesgue spaces, Studia Math., 188 (2) (2008), 123-133.
A. Fiorenza and G. Karadzhov, Grand and small Lebesgue spaces and their analogs, Zeitschrift fur Analysis und ihre Anwendungen, 23 (4) (2004), 657-681.
T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal., 119 (1992), 129-143.
V. Kokilashvili, A. Razmadze and M. Aleksidze, Boundedness criteria for singular integrals in weighted grand lebesgue spaces, J. Math. Sci., 170 (1) (2010), 20-23.
J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc., 336 (1993), 869-880.
Y. Ding, D. S. Fan and Y. B. Pan, Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana Univ. Math. J., 48 (1999), 1037-1055.
Y. Ding, S. Z. Lu and K. Yabuta, On commutators of Marcinkiewicz integrals with rough kernel, J. Math. Anal. Appl., 275 (2002), 60-68.
D. S. Fan and Y. B. Pan, Singular integral operators with rough kernel supported by subvarieties, Amer. J. Math., 119 (1997), 799-839.
S. Z. Lu, D. C. Yang and G. E. Hu, Herz type spaces and their applications, Science press, 2008.
A. Seeger, Singular integral operators with rough convolution kernels, J. Amer. Math. Soc., 9 (1996), 95-105.
P. Sjögren and F. Soria, Rough maximal functions and rough singualr integral operators applied to integrable radial functions, Rev. Mat. Iberoam., 13 (1997), 1-18.
G. E. Hu, S. Z. Lu and D. C. Yang, Boundedness of rough singular integral operators on homogeneous Herz spaces, J. Australia Math. Soc., 66 (1999), 201-223.
F. Soria and G. Weiss, A remark on singular integrals and power weights, Indiana Univ. Math., J., 43 (1994), 187-204.
D. Watson and R. Wheeden, Norm estimates and representations for Calderón-Zygmund operators using averages over starlike sets, Trans. Amer. Math. Soc., 351 (1999), 4127-4171.
H. Ojanen, Weighted norm inequalities for rough singular integrals, Doctor Dissertation, New Jerse., 1999.
X. W. Li and D. C. Yang, Boundedness of some sublinear operators on Herz spaces, Illinois J. of Math., 40 (1996), 484-501.
D. X. Chen, X. L. Chen and Z. W. Fu, CBMO estimates for the commutators with rough kernel on homogeneous Morrey-Herz spaces, Acta. Math. Sinica (Chinese Series), 52 (2009), 861-872.
F. Gürbüz, Parabolic sublinear operators with rough kernel generated by parabolic Calderón-Zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces, Open Math., 14 (1) (2016), 300-323.
Q. R. Zhang, L. Zhang and S. G. Shi, Boundedness for sublinear operators on generalized weighted grand Morrey spaces, Acta. Math., 35A (3) (2015), 503-514.
B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc., 16 (1971), 249-258.
S. Chanillo, D. Wastom and R. Wheeden, Some integral and maximal operators related to starlike sets, Studia Math., 107 (1993), 223-255.
Y. Ding and S. Z. Lu, Weighted norm inequalities for fractional integrals with rough kernel, Canad. J. Math., 50 (1998), 29-39.
Y. Ding and S. Z. Lu, Homogeneous fractional integrals on Hardy spaces, Tohoku Math. J., 52 (2000), 153-162.
S. Z. Lu and Y. Zhang, Criterion on LP-boundedness for a class of oscillatory singular integrals with rough kernels, Rev. Mat. Iberoam., 8 (1992), 201-219.
F. John and L. Nirenberg, On solutions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426.
C. Perez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128 (1995), 163-185.
R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.