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SETSCI - Volume 4 (6) (2019)
ISAS WINTER-2019 (ENS) - 4th International Symposium on Innovative Approaches in Engineering and Natural Sciences, Samsun, Turkey, Nov 22, 2019

On Some Properties of Lorentz-Sobolev Spaces with Variable Exponent
İsmail Aydın1*
1Sinop University, Sinop, Turkey
* Corresponding author:
Published Date: 2019-12-22   |   Page (s): 9-11   |    213     7

ABSTRACT In recent years there has been an increasing interest in the study of various mathematical problems with variable exponent Lebesgue spaces. There are also a lot of published papers in these spaces. Spaces of weakly differentiable functions, so called Sobolev spaces, play an important role in modern Analysis. The theory of variable exponent Sobolev spaces is useful theoretical tool to study the variable exponent problems, such as solutions of elliptic and parabolic partial differentiable equations, calculus of variations, nonlinear analysis, capacity theory and compact embeddings. Moreover, several authors studied some continuous embeddings from Sobolev spaces to Lorentz spaces. These kinds of embedding results are very interesting and valuable in analysis, and there are many applications of them in various fields. In this paper we define variable exponent Lorentz-Sobolev spaces and prove the boundedness of maximal function in these spaces. Also we will show that there is a continuous embedding between variable exponent Lorentz-Sobolev spaces and variable exponent Lorentz spaces under some conditions.
KEYWORDS Variable exponent Lorentz and Sobolev spaces, Maximal function, embedding
REFERENCES R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.
A. Alvino, `Sulla diseguaglianza di Sobolev in spazi di Lorentz Boll.un.mat.Ital. 14, 148-156, 1975.
A. Alvino, `Un caso limite della diseguaglianza di Sobolev in spazi di Lorentz', Rend. Acad. Sci. Napoli XLIV, 105-112.
H. Brezis and S. Waigner, `A note on limiting cases of Sobolev embeddings and convolution inequalities', Commun. Partia Diff. Eq., 5, 773-789, 1980.
D. Cruz Uribe and A. Fiorenza, J. M. Martell and C. Perez Moreno, The boundedness of classical operators on variable L^p spaces, Ann. Acad. Sci. Fenn., Math., 31(1), 239-264, 2006.
D. Cruz Uribe and A. Fiorenza, LlogL results for the maximal operator in variable L^p spaces, Trans. Amer. Math. Soc., 361 (5), 2631-2647, 2009.
D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and harmonic analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013.
L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal., 256(6), 1731-1768, 2009.
L. Diening, Maximal function on generalized Lebesgue spaces L^(p(.)), Math. Inequal. Appl. 7, No 2, 245-253, 2004.
L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L^(p(.)) and W^(k,p(.)), Math. Nachr. 268, 31-43, 2004.
L. Diening, P. Hästö and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces. In FSDONA04 Proc. (Milovy, Czech Republic), 38-58, 2004.
J. Duoandikoetxea, Fourier Analysis, Grad. Studies in Math. 29, Amer. Math. Soc., Providence, 2000.
D. E. Edmunds, J. Lang and A. Nekvinda, A. On L^(p(x)) norms, Proc. R. Soc. Lond. A 455, 219-225, 1999.
L. Ephremidze, V. Kokilashvili and S. Samko, Fractional, maximal and singular operators in variable exponent Lorentz spaces, Fract. Calc. Appl. Anal. 11, 4, 1-14, 2008.
P. Hajlasz and J. Onninen: On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 29, 167-176, 2004.
F. Helein, Regularite des appications faiblement harmoniques entre une surface et une variete riemannienne, C. R. Acad. Sci, series I.
M. Jiang, A theorem on embedding W^(s,p) into L^(np/(n-sp),p) (R^n ) , Acta Mathematica Scientia, Vol.14, No.3, pp. 313-317, 1994.
O. Kovacik and J. Rakosnik, On spaces L^(p(x)) and W^(k,p(x)), Czech Math J. 41(116), 592-618, 1991.
O. Kulak and A. T. Gurkanlı, Bilinear Multipliers of Weighted Lorentz Spaces and Variable Exponent Lorentz Spaces, Math. and Statis. 5(1), 5-18, 2017.
J. Maly and W.P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Amer. Math. Soc. (Providence, RI), 1997.
R. O'Neil, `Convolution inequalities and L(p; q) spaces', Duke. math. J. 80, 129-142, 1980.

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