On Variable Exponent Amalgam Spaces

Aydin, İsmail

doi:10.2478/v10309-012-0051-2

Abstract

We derive some of the basic properties of weighted variable exponent Lebesgue spaces L^p(.)_w (ℝⁿ) and investigate embeddings of these spaces under some conditions. Also a new family of Wiener amalgam spaces W(L^p(.)_w ;L^q_v) is defined, where the local component is a weighted variable exponent Lebesgue space L^p(.)_w (ℝ_n) and the global component is a weighted Lebesgue space L^q_v (ℝ_n) : We investigate the properties of the spaces W(L^p(.)_w ;L^q_v): We also present new Hölder-type inequalities and embeddings for these spaces.

References

[1] I. Aydın and A.T.G¨urkanlı, On some properties of the spaces Ap(x) ! (Rn) . Proceedings of the Jangjeon Mathematical Society, 12 (2009), No.2, pp.141-155.
Search in Google Scholar
[2] I. Aydın, Weighted variable Sobolev spaces and capacity, Journal of Function Spaces and Applications, Volume 2012, Article ID 132690, 17 pages, doi:10.1155/2012/132690.10.1155/2012/132690
Search in Google Scholar
[3] I.Aydın and A.T.G¨urkanlı, Weighted variable exponent amalgam spaces W ( Lp(x),Lqw ) , Glasnik Matematicki, Vol.47(67), (2012), 167-176.10.3336/gm.47.1.14
Search in Google Scholar
[4] D. Cruz Uribe and A. Fiorenza, LlogL results for the maximal operator in variable Lp spaces, Trans. Amer. Math. Soc., 361 (5), (2009), 2631-2647.10.1090/S0002-9947-08-04608-4
Search in Google Scholar
[5] D. Cruz Uribe, A. Fiorenza, J. M. Martell and C. Perez Moreno, The boundedness of classical operators on variable Lp spaces, Ann. Acad. Sci. Fenn., Math., 31(1), (2006), 239-264.
Search in Google Scholar
[6] L. Diening, Maximal function on generalized Lebesgue spaces Lp(:), Mathematical Inequalities and Applications, 7(2004), 245-253.10.7153/mia-07-27
Search in Google Scholar
[7] L. Diening, P. H¨ast¨o and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces. In FSDONA04 Proceedings (Milovy, Czech Republic, 2004), 38-58.
Search in Google Scholar
[8] L. Diening, P. H¨ast¨o, and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal., 256(6), (2009), 1731-1768.10.1016/j.jfa.2009.01.017
Search in Google Scholar
[9] D. Edmunds, J. Lang, and A. Nekvinda, On Lp(x) norms, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 455, (1999), 219-225.10.1098/rspa.1999.0309
Search in Google Scholar
[10] H. G. Feichtinger, Banach convolution algebras of Wiener type, In: Functions, Series, Operators, Proc. Conf. Budapest 38, Colloq. Math. Soc. Janos Bolyai, (1980), 509-524.
Search in Google Scholar
[11] H. G. Feichtinger, Banach spaces of Distributions of Wiener’s type and Interpolation, In Proc. Conf. Functional Analysis and Approximation, Oberwolfach August 1980, Internat. Ser. Numer. Math., 69:153-165. Birkhauser, Boston, 1981.10.1007/978-3-0348-9369-5_16
Search in Google Scholar
[12] H. G. Feichtinger and K. H. Gr¨ochenig, Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal., 86(1989), 307-340.10.1016/0022-1236(89)90055-4
Search in Google Scholar
[13] H. G. Feichtinger and A. T. G¨urkanli, On a family of weighted convolution algebras, Internat. J. Math. and Math. Sci., 13 (1990), 517-526.10.1155/S0161171290000758
Search in Google Scholar
[14] R. H. Fischer, A. T. G¨urkanlı and T. S. Liu, On a Family of Wiener type spaces, Internat. J. Math. and Math. Sci.,19 (1996), 57-66. 10.1155/S0161171296000105
Search in Google Scholar
[15] R. H. Fischer, A. T. G¨urkanlı and T. S. Liu, On a family of weighted spaces, Math. Slovaca, 46(1996), 71-82.
Search in Google Scholar
[16] J. J. Fournier and J. Stewart, Amalgams of Lpand ℓq, Bull. Amer. Math. Soc., 13 (1985), 1-21.10.1090/S0273-0979-1985-15350-9
Search in Google Scholar
[17] C. Heil, An introduction to weighted Wiener amalgams, In: Wavelets and their applications (Chennai, January 2002), Allied Publishers, New Delhi, (2003), p. 183-216.
Search in Google Scholar
[18] F. Holland, Square-summable positive-definite functions on the real line, Linear Operators Approx. II, Proc. Conf. Oberwolfach, ISNM 25, (1974), 247-257.10.1007/978-3-0348-5991-2_18
Search in Google Scholar
[19] F. Holland, Harmonic analysis on amalgams of Lpand ℓq, J. London Math. Soc. (2), 10, (1975), 295-305.10.1112/jlms/s2-10.3.295
Search in Google Scholar
[20] O. Kovacik and J. Rakosnik, On spaces Lp(x) and Wk;p(x), Czech. Math. J., 41(116), (1991), 592-618.10.21136/CMJ.1991.102493
Search in Google Scholar
[21] J. Musielak, Orlicz spaces and modular spaces, Springer-Verlag, Lecture Notes in Math., 1983.10.1007/BFb0072210
Search in Google Scholar
[22] W. Orlicz, ¨Uber konjugierte exponentenfolgen, Studia Math. 3, (1931), 200-212.10.4064/sm-3-1-200-211
Search in Google Scholar
[23] H. Reiter, Classical harmonic analysis and locally compact groups, Oxford University Press, Oxford, 1968.
Search in Google Scholar
[24] B. Sa˘gır: On functions with Fourier transforms inW(B, Y ), Demonstratio Mathematica, Vol. XXXIII, No.2, 355-363, (2000).
Search in Google Scholar
[25] S. G. Samko, Convolution type operators in Lp(x), Integr. Transform. and Special Funct.,7(1998), 123-144.10.1080/10652469808819191
Search in Google Scholar
[26] N. Wiener, Generalized Harmonic Analysis Tauberian Theorems, The M.I.T. Press, 1964.
Search in Google Scholar

On Variable Exponent Amalgam Spaces

Abstract

Paradigm

My account