ON THE WEIGHTED VARIABLE EXPONENT AMALGAM SPACE W(Lp(x), Lqm)

doi:10.1016/S0252-9602(14)60072-2

Abstract

Abstract:

In [4] , a new family W(L^p^(x), L^q_m) of Wiener amalgam spaces was defined and investigated some properties of these spaces, where local component is a variable exponent Lebesgue space L^p^(x) (R) and the global component is a weighted Lebesgue space L^q_m (R). This present paper is a sequel to our work [4]. In Section 2, we discuss necessary and sufficient conditions for the equality W(L^p^(x), L^q_m)= L^q (R) . Later we give some characterization of Wiener amalgam space W(L^p^(x), L^q_m). In Section 3 we define the Wiener amalgam space W(FL^p^(x), L^q_m) and investigate some properties of this space, where FL^p^(x) is the image of L^p^(x) under the Fourier transform. In Section 4, we discuss boundedness of the Hardy-Littlewood maximal operator between some Wiener amalgam spaces.

Key words: weighted Lebesgue space, variable exponent Lebesgue

CLC Number:

42B25

A. Turan GüRKANLI, Ismail AYDIN. ON THE WEIGHTED VARIABLE EXPONENT AMALGAM SPACE W(L^p^(x), L^q_m)[J].Acta mathematica scientia,Series B, 2014, 34(4): 1098-1110.

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References

[1] Acerbi E, Mingione G. Regularity results for stationary electrorheological fluids. Arch Ration Mech Anal, 2002, 164(3): 213–259

[2] Acerbi E, Mingione G. Gradient estimates for the p(x)-Laplacian system. J Reine Angew Math, 2005, 584: 117–148

[3] Ayd?n I, G¨urkanl?A T. On some properties of the spaces A^p^(x)_ω (Rⁿ) . Proc J Math Soc, 2009, 12(2): 141–155

[4] Ayd?n I, G¨urkanl?A T. Weighted variable exponent amalgam spaces W(L^p^(x), L^q_w). Glasnik Matematicki, 2012, 47(67): 165–174

[5] Bennett C, Sharpley R. Interpolation of Operators. Academic Press, Inc, 1988

[6] Chen Y, Levine S, Rao R. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math, 2006, 66(4): 1383–1406

[7] Cruz Uribe D, Fiorenza A, Neugebauer C J. Corrections to ”The maximal function on variable Lp spaces”. Ann Acad Sci Fenn Math, 2004, 29: 247–249

[8] Cruz Uribe D, Fiorenza A, Neugebauer C J. The maximal function on variable Lp spaces. Ann Acad Sci Fenn Math, 2003, 28: 223–238

[9] Diening L. Maximal function on generalized Lebesgue spaces Lp(.). Math Ineq Appl, 2004, 7: 245–253

[10] Diening L, H¨ast¨o P, Nekvinda A. Open problems in variable exponent Lebesgue and Sobolev spaces. In
FSDONA04 Proceedings (Milovy, Czech Republic, 2004, 38–58

[11] Diening L, H¨ast¨o P, Roudenko S. Function spaces of variable smoothness and integrability. J Funct Anal,
2009, 256(6): 1731–1768

[12] Edmunds D, Lang J, Nekvinda A. On Lp(x) norms. Proc R Soc Lond, Ser A, Math Phys Eng Sci, 1999, 455: 219–225

[13] Fan X -L. Global C1, , regularity for variable exponent elliptic equations in divergence form. J Differ Equ, 2007, 235(2): 397–417

[14] Feichtinger H G. Banach convolution algebras of Wiener type//Functions, Series, Operators. Proc Conf Budapest, Colloq Math Soc Janos Bolyai, 1980, 38: 509–524

[15] Feichtinger H G. Banach spaces of Distributions of Wiener’s type and Interpolation//Proc Conf Functional Analysis and Approximation (Oberwolfach August 1980). Internat Ser Numer Math, Boston: Birkhauser, 1981, 69: 153–165

[16] Feichtinger H G, Gr¨ochenig K H. Banach spaces related to integrable group representations and their atomic
decompositions I. J Funct Anal, 1989, 86: 307–340

[17] Feichtinger H G, G¨urkanli A T. On a family of weighted convolution algebras. Internat J Math and Math Sci, 1990, 13: 517–526

[18] Fischer R H, G¨urkanl? A T, Liu T S. On a Family of Wiener type spaces. Internat J Math Math Sci, 1996, 19: 57–66

[19] Fischer R H, G¨urkanl? A T, Liu T S. On a family of weighted spaces. Math Slovaca, 1996, 46: 71–82

[20] Fournier J J, Stewart J. Amalgams of Lpand ?q. Bull Amer Math Soc, 1985, 13: 1–21

[21] Gurka P, Harjulehto P, Nekvinda A. Bessel potential spaces with variable exponent spaces. Math Inequal Appl, 2007, 10(3): 661–676

[22] Harjulehto P, Hast¨o P, Latvala V. Minimizers of the variable exponent, non-uniformly convex Dirichlet energy. J Math Pures Appl, 2008, (9) 89(2): 174–197

[23] Heil C. An introduction to weighted Wiener amalgams//Wavelets and their applications (Chennai, January
2002). New Delhi: Allied Publishers, 2003: 183–216

[24] Holland F. Square-summable positive-definite functions on the real line//Linear Operators Approx II. Proc Conf Oberwolfach, ISNM 25, 1974: 247–257

[25] Holland F. Harmonic analysis on amalgams of Lp and ?q. J London Math Soc, 1975, 10(2): 295–305

[26] Kokilashvili V. On a progress in the theory of integral operators in weighted Banach function spaces//Function Spaces, Differential Operators and Nonlinear Analysis. Proceeding of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28-June 2, 2004. Math Inst Acad Sci Czech Republick, Praha, 2005: 152–175

[27] Kovacik O, Rakosnik J. On spaces Lp(x) and Wk,p(x). Czech Math J, 1991, 41(116): 592–618

[28] Orlicz W. ¨Uber konjugierte exponentenfolgen. Studia Math, 1931, 3: 200–212

[29] Pick L, Ruzicka M. An example of a space Lp(x) on which the Hardy-Littlewood maximal operator is not bounded. Expo Math, 2001, 19: 369–371

[30] Reiter H. Classical Harmonic Analysis and Locally Compact Groups. Oxford: Oxford Univ Press, 1968

[31] Ruzicka M. Electrorheological Fluids, Modeling and Mathematical Theory. Lecture Notes in Math, Vol 1748. Berlin: Springer-Verlag, 2000

[32] Samko S G. Convolution type operators in Lp(x). Integr Transform Special Funct, 1998, 7: 123–144

[33] Samko S. On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators. Integral Transforms Spec Funct, 2005, 16(5/6): 461–482

[34] Tchamitchian P. Generalisation des algebras de Beurling. Ann Inst Fourier (Grenoble), 1984, 34: 151–168

[35] Wiener N. Tauberian theorems. Ann Math, 1932, 33: 1–100

ON THE WEIGHTED VARIABLE EXPONENT AMALGAM SPACE W(L^p^(x), L^q_m)

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