Herz type Besov and Triebel-Lizorkin spaces with variable exponent

Abstract

The Herz type Besov and Triebel-Lizorkin spaces with variable exponent are introduced. Then characterizations of these new spaces by maximal functions are given.

Keywords

Variable exponent / Herz space / Besov space / Triebel-Lizorkin space / equivalent norm / maximal function

References

[1] Almeida A, Hasanov J, Samko S. Maximal and potential operators in variable exponent Morrey spaces. Georgian Math J, 2008, 15: 198-208

[2] Almeida A, Höstö P. Besov spaces with variable smoothness and integrability. J Funct Anal, 2010, 258: 1628-1655

[3] Andersen K, John R. Weighted inequalities for vector-valued maximal functions and singular integrals. Studia Math, 1980, 69: 19-31

[4] Cruz-Uribe D, Fiorenza A, Martell C, Pérez C. The Boundedness of classical operators on valible L^p spaces. Ann Acad Sci Fenn Math, 2006, 31: 239-264

[5] Cruz-Uribe D, Fiorenza A, Neugebauer C. The maximal function on variable L^p spaces. Ann Acad Sci Fenn Math, 2003, 28: 223-238

[6] Diening L. Maximal function on generalized Lebesgue spaces L^p(x). Math Inequal Appl, 2004, 7: 245-253

[7] Diening L. Maximal functions on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull Sci Math, 2005, 129: 657-700

[8] Diening L, Harjulehto P, Hästö P, Růžička M. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, Vol 2017. Berlin: Springer-Verlag, 2011

[9] Diening L, Hästö P, Roudenko S. Function spaces of variable smoothness and integrability. J Funct Anal, 2009, 256: 1731-1768

[10] Fefferman C, Stein E. Some maximal inequalities. Amer J Math, 1971, 93: 107-115

[11] Frazier M, Jawerth B. Decomposition of Besov spaces. Indiana Univ Math J, 1985, 34: 777-799

[12] Gurka P, Harjulehto P, Nekvinda A. Bessel potential spaces with variable exponent. Math Inequal Appl, 2007, 10: 661-676

[13] Harjulehto P, Hästö P, Le U V, Nuortio M. Overview of differential equations with non-standard growth. Nonlinear Anal, 2010, 72: 4551-4574

[14] Hästö P. Local-to-global results in variable exponent spaces. Math Res Lett, 2009, 16(2): 263-278

[15] Hernández E, Yang D. Interpolation of Herz spaces and applications. Math Nachr, 1999, 205: 69-87

[16] Izuki M. Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal Math, 2010, 36: 33-50

[17] Izuki M. Boundedness of commutators on Herz spaces with variable exponent. Rend Circ Mat Palermo, 2010, 59: 199-213

[18] Izuki M. Vector-valued inequalities on Herz spaces and characterizations of Herz-Sobolev spaces with variable exponent. Glas Mat, 2010, 45: 475-503

[19] Kempka H. 2-Microlocal Besov and Triebel-Lizorkin spaces of variable integrability. Rev Mat Complut, 2009, 22: 227-25

[20] Kempka H. Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces. J Funct Spaces Appl, 2010, 8: 129-165

[21] Kováčik O, Rákosník J. On spaces L^p(x) and W^k,p(x). Czechoslovak Math J, 1991, 41: 592-681

[22] Li X, Yang D. Boundedness of some sublinear operators on Herz spaces. Illinois J Math, 1996, 40: 484-501

[23] Lu S. Multipliers and Herz type spaces. Sci China Series A: Math, 2008, 51(10): 1919-1936

[24] Lu S, Yabuta K, Yang D. The Boundedness of some sublinear operators in weighted Herz-type spaces. Kodai Math J, 2000, 23: 391-410

[25] Lu S, Yang D. The decomposition of the weighted Herz spaces and its application. Sci China Series A: Math, 1995, 38: 147-158

[26] Lu S, Yang D, Hu G. Herz Type Spaces and Their Applications. Beijing: Science Press, 2008

[27] Nekvinda A. Hardy-Littlewood maximal operator on L^p(x). Math Inequal Appl, 2004, 7: 255-265

[28] Peetre J. On spaces of Triebel-Lizorkin type. Ark Mat, 1975, 13: 123-130

[29] Rychkov V S. On a theorem of Bui, Paluszyński, and Taibleson. Proc Steklov Inst Math, 1999, 227 : 280-292

[30] Stein E, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton Univ Press, 1971

[31] Tang L, Yang D. Boundedness of vector-valued operators on weighted Herz spaces. Approx Theory & Its Appl, 2000, 16: 58-70

[32] Ullrich T. Continuous characterizations of Besov-Lizorkin-Triebel spaces and new interpretations as coorbits. J Funct Spaces Appl, 2012, doi: 10.1155/2012/163213

[33] Xu J. A discrete characterization of the Herz-type Triebel-Lizorkin spaces and its applications. Acta Math Sci Ser B Engl Ed, 2004, 24(3): 412-420

[34] Xu J. Equivalent norms of Herz type Besov and Triebel-Lizorkin spaces. J Funct Spaces Appl, 2005, 3(1): 17-31

[35] Xu J. The relation between variable Bessel potential spaces and Triebel-Lizorkin spaces. Integral Transforms Spec Funct, 2008, 19(8): 599-605

[36] Xu J. Variable Besov spaces and Triebel-Lizorkin spaces. Ann Acad Sci Fen Math, 2008, 33: 511-522

[37] Xu J. An atomic decomposition of variable Besov and Triebel-Lizorkin spaces. Armen J Math, 2009, 2(1): 1-12

[38] Xu J, Yang D. Applications of Herz-type Triebel-Lizorkin spaces. Acta Math Sci Ser B Engl Ed, 2003, 23(3): 328-338

[39] Xu J, Yang D. Herz-type Triebel-Lizorkin spaces (I). Acta Math Sinica (Engl Ser), 2005, 21: 643-654

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg