Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent

References

• Samko , S. 2005 . On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators . Integral Transformations and Special Functions , 16 ( 5–6 ) : 461 – 482 .
   Web of Science ®Google Scholar
• Zhikov , V. V. 1987 . Averaging of function of calculus of variations and elasticity theory . Russian Mathematics (Izvestiya vuz) , 29 : 33 – 66 .
   Google Scholar
• Zhikov , V. V. 1995 . On Lavrentiev’s phenomenon . Russian Journal of Mathematical Physics , 3 ( 2 ) : 249 – 269 .
   Google Scholar
• Zhikov , V. V. 1997 . On some variational problems . Russian Journal of Mathematical Physics , 5 ( 1 ) : 105 – 116 .
   Web of Science ®Google Scholar
• Ružiĉka , M. 2000 . “ Electrorheological fluids: modeling and mathematical theory ” . Vol. 1748 , Berlin : Springer . Lecture Notes in Mathematics
   Google Scholar
• Diening , L. and Ružiĉka , M. 2003 . Calderon-Zygmund operators on generalized Lebesgue spaces L p(·) and problems related to fluid dynamics . Journal für die Reine und Angewandte Mathematik , 563 : 197 – 220 .
   Web of Science ®Google Scholar
• Nakano , H. 1951 . Topology of Linear Topological Spaces , Tokyo : Maruzen Co. Ltd .
   Google Scholar
• Musielak , J. and Orlicz , W. 1959 . On modular spaces . Studia Mathematica , 18 : 49 – 65 .
   Google Scholar
• Musielak , J. 1983 . “ Orlicz spaces and modular spaces. Lecture Notes in Mathematics ” . Vol. 1034 , Berlin : Springer-Verlag .
   Google Scholar
• Hudzik , H. and On , generalized Orlicz-Sobolev space . 1976 . Functiones et Approximatio Commentarli Mathematici , 4 : 37 – 51 .
   Google Scholar
• Sharapudinov , I. I. 1979 . The topology of the space . Matematicheskie Zametki , 26 ( 4 ) : 613 – 632 . (Russian)
   Google Scholar
• Sharapudinov , I. I. 1986 . The basis property of the Haar system in the space L p(t)([0, 1]) and the principle of localization in the mean . Matematicheskii Sbornik , : 275 – 283 . (Russian), (N.S.) 130(172), No. 2
   Google Scholar
• Samko , S. G. 1994 . “ Differentiation and integration of variable order and the spaces L p(x) ” . In Operator theory for complex and hypercomplex analysis 203 – 219 . Mexico City Contemporary Mathematics, 212, 1998, American Mathematical Society Providence, RI
   Google Scholar
• Samko , S. G. 1998 . Convolution type operators in L p(x) . Integral Transformations and Special Functions , 7 ( 1–2 ) : 123 – 144 .
   Web of Science ®Google Scholar
• Samko , S. G. 1998 . Convolution and potential type operators in L p(x)(R n ) . Integral Transformations and Special Functions , 7 ( 3–4 ) : 261 – 284 .
   Web of Science ®Google Scholar
• Kováčik , O. and Rákosník , J. 1991 . On spaces L p(x) and W k, p(x) . Czechoslovak Mathematical Journal , 41(116) ( 4 ) : 592 – 618 .
   Web of Science ®Google Scholar
• Edmunds , D. E. and Rákosník , J. Density of smooth functions in . Proceedings of the Royal Society of London . Vol. 437 , pp. 229 – 236 .
   Google Scholar
• Pick , L. and Ružiĉka , M. 2001 . An example of a space L p(x) on which the Hardy–Littlewood maximal operator is not bounded . Expositiones Mathematicae , 19 ( 4 ) : 369 – 371 .
   Google Scholar
• Edmunds , D. E. and Nekvinda , A. 2002 . Averaging operators on and L p(x) . Mathematical Inequalities & Applications , 5 ( 2 ) : 235 – 246 .
   Web of Science ®Google Scholar
• Fan , X. and Zhao , D. 1996 . Regularity of minimizers of variational integrals with continuous p(x)-growth conditions . Chinese Journal on Contemporary Mathematics , 17 ( 4 ) : 327 – 336 .
   Google Scholar
• Fan , X. and Zhao , D. 2000 . The quasi-minimizer of integral functionals with m(x) growth conditions . Nonlinear Analysis , 39 ( 7 ) : 807 – 816 .
   Web of Science ®Google Scholar
• Diening , L. 2004 . Maximal function on generalized Lebesgue spaces L p(·) . Mathematical Inequalities & Applications , 7 ( 2 ) : 245 – 253 .
   Web of Science ®Google Scholar
• Nekvinda , A. 2004 . Hardy–Littlewood maximal operator on . Mathematical Inequalities and Applications , 7 ( 2 ) : 255 – 265 .
   Web of Science ®Google Scholar
• Cruz-Uribe , D. , Fiorenza , A. and Neugebauer , C. J. 2003 . The maximal function on variable L p spaces . Annales Academiae Scientiarium Fennieae Mathematics , 28 ( 1 ) : 223 – 238 .
   Google Scholar
• Diening , L. 2004 . Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p(·) and . Mathematische Nachrichten , 268 : 31 – 43 .
   Web of Science ®Google Scholar
• Capone , C. , Cruz-Uribe , D. and Fiorenza , A. “ The fractional maximal operator on variable Lp spaces ” . Preprint is available from http://www.iam.na.cnr.it/
   Google Scholar
• Kokilashvili , V. and Samko , S. 2003 . On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent . Zeitschrift für Analysis und ihre Anwendungen , 22 ( 4 ) : 899 – 910 .
   Web of Science ®Google Scholar
• Kokilashvili , V. and Samko , S. 2004 . Maximal and fractional operators in weighted L p(x) spaces . Revista Mathematica Iberoamericana , 20 ( 2 ) : 493 – 515 .
   Web of Science ®Google Scholar
• Kokilashvili , V. On a progress in the theory of integral operators in weighted Banach function spaces . Function Spaces, Differentiable Operators and Nonlinear Analysis proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands . Praha : Math Institutions of Academic Science in Czech Republick . May 28–June 2
   Google Scholar
• Cruz-Uribe , D. , Fiorenza , A. , Martell , J. M. and Perez , C. 2006 . The boundedness of classical operators on variable L p spaces . Annales Academiae Scientiarium Fennieae Mathematics , 31 : 239 – 264 .
   Google Scholar
• Samko , S. 2003 . Hardy-Littlewood-Stein-Weiss inequality in the Lebesgue spaces with variable exponent . Fractional Calculus & Applied Analysis , 6 ( 4 ) : 421 – 440 .
   Google Scholar
• Samko , S. and Vakulov , B. 2005 . Weighted Sobolev theorem with variable exponent . Journal of Mathematical Analysis and Applications , 310 : 229 – 246 .
   Web of Science ®Google Scholar
• Samko , S. , Shargorodsky , E. and Vakulov , B. 2007 . Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II . Journal of Mathematical Analysis and Applications , 325 ( 1 ) : 745 – 751 .
   Web of Science ®Google Scholar
• Edmunds , D. E. , Kokilashvili , V. and Meskhi , A. 2004 . A trace inequality for generalized potentials in Lebesgue spaces with variable exponent . Journal of Function Spaces and Applications , 2 ( 1 ) : 55 – 69 .
   Google Scholar
• Genebashvili , I. , Gogatishvili , A. and Kokilashvili , V . 1996 . Solution of two-weight problems for integral transforms with positive kernels . Georgian Mathematical Journal , 3 ( 1 ) : 319 – 342 .
   Google Scholar
• Genebashvili , I. , Gogatishvili , A. , Kokilashvili , V. and Krbec , M. 1998 . “ Weight theory for integral transforms on spaces of homogeneous type ” . In Pitman Monographs and Surveys in Pure and Applied Mathematics , Vol. 92 , Harlow : Longman .
   Google Scholar
• Edmunds , D. E. and Meskhi , A. 2002 . Potential-type operators in L p(x) spaces . Zeitschrift fur annales und ihre Anwendungen , 21 ( 3 ) : 681 – 690 .
   Web of Science ®Google Scholar
• Sawyer , E. T. 1982 . A characterization of a two–weight norm inequality for maximal operators . Studia Mathematica , 75 : 1 – 11 .
   Web of Science ®Google Scholar
• Wheeden , R. L. 1993 . A characterization of some weighted norm inequalities for the fractional maximal functions . Studia Mathematica , 107 : 251 – 272 .
   Web of Science ®Google Scholar
• Gogatishvili , A. and Kokilashvili , V. 1996 . Criteria of two-weight strong type inequalities for fractional maximal functions . Georgian Mathematical Journal , 3 ( 5 ) : 423 – 446 .
   Google Scholar
• Sawyer , E. T. 1982 . “ Two-weight norm inequalities for certain maximal and integral operators ” . In Harmonic analysis, Minneapolis, Minn. 1981, Lecture Notes in Math , Vol. 908 , 102 – 127 . Berlin-New York : Springer .
   Google Scholar
• Sawyer , E. T. 1984 . A two-weight weak type inequality for fractional integrals . Transactions of American Mathematical Society , 281 : 339 – 345 .
   Google Scholar
• Gabidzashvili , M. and Kokilashvili , V. 1989 . “ Two weight weak type inequalities for fractional type integrals ” . Prague Preprint, No. 45, Mathematical Institute Czech Academic Science
   Google Scholar
• Kokilashvili , V. and Krbec , M. 1991 . “ Weighted inequalities in Lorentz and Orlicz spaces ” . New Jersey, London, Hong Kong, , Singapore : World Scientific .
   Google Scholar
• Sawyer , E. T. and Wheeden , R. L. 1992 . Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces . American Journal of Mathematics , 114 : 813 – 874 .
   Google Scholar
• Sawyer , E. T. , Wheeden , R. L. and Zhao , S. 1996 . Weighted norm inequalities for operators of potential type and fractional maximal functions . Potential Analysis , 5 : 523 – 580 .
   Web of Science ®Google Scholar
• Edmunds , D. E. , Kokilashvili , V. and Meskhi , A. 2002 . Bounded and Compact Integral Operators , Dordrecht, Boston, London : Kluwer Academic Publishers .
   Google Scholar
• Muckenhoupt , B. and Wheeden , R. L. 1974 . Weighted norm inequalities for fractional integrals . Transactions of the American Mathematial Society , 192 : 261 – 276 .
   Web of Science ®Google Scholar
• Carleson , A. 1958 . An interpolation problem for bounded analytic functions . American Journal of Mathematics , 80 : 921 – 930 .
   Web of Science ®Google Scholar
• Hörmander , L. L ^p estimates for (pluri-) subharmonic functions. L ^p estimates for (pluri-) subharmonic functions , 20 65 – 78 .
   Google Scholar
• Verbitsky , I. E. 1992 . Weighted norm inequalities for maximal operators and Pisier’s theorem on factorization through L p∞ . Integral Equations Operator Theory , 15 ( 1 ) : 124 – 153 .
   Web of Science ®Google Scholar
• Coifman , R. R. and Fefferman , C. 1974 . Weighted norm inequalities for maximal functions and singular integrals . Studia Mathematica , 51 : 241 – 249 .
   Web of Science ®Google Scholar
• Franchi , B. and Wheeden , R. L. 1999 . Some remarks about Poincare type inequalities and representation formulas in metric spaces of homogeneous type . Journal of Inequalities and Applications , 3 ( 1 ) : 65 – 89 .
   Web of Science ®Google Scholar
• Strömberg , J. O. and Torchinsky , A. 1989 . “ Weighted Hardy spaces, Lecture Notes in Mathematics ” . Vol. 1381 , Berlin : Springer Verlag .
   Google Scholar
• Fefferman , C. and Stein , E. 1971 . Some maximal inequalities . American Journal of Mathematics , 93 : 107 – 115 .
   Web of Science ®Google Scholar
• García–Cuerva , J. and Rubio de Francia , J. L. 1985 . “ Weighted norm inequalities and related topics. North-Holland Mathematics Studies 116, Mathematical Notes 104 ” . Amsterdam : North-Holland Publishing Co .
   Google Scholar
• Martin-Reyes , F. J. and de la Torre , A. 1993 . Two weight norm inequalities for fractional one-sided maximal operators . Proceedings of American Mathematical Society , 117 ( 2 ) : 483 – 489 .
   Web of Science ®Google Scholar
• Adams , D. R. 1973 . A trace inequality for generalized potentials . Studia Mathematica , 48 : 99 – 105 .
   Web of Science ®Google Scholar
• Hedberg , L. 1972 . On certain convolution inequalities . Proceedings of American Mathematical Society , 36 : 505 – 510 .
   Web of Science ®Google Scholar
• Sobolev , S. L. 1938 . On a theorem in functional analysis . Matematicheskii Sbornik , 4 ( 46 ) : 471 – 497 . (N.S.), (Russian); English transl., 1963, American Mathematical Society, Translations (2), 34, 39–68
   Google Scholar