Journal of the Mathematical Society of Japan

Sobolev's inequality for Riesz potentials in Lorentz spaces of variable exponent

Yoshihiro MIZUTA and Takao OHNO

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In the present paper we discuss the boundedness of the maximal operator in the Lorentz space of variable exponent defined by the symmetric decreasing rearrangement in the sense of Almut [ 1]. As an application of the boundedness of the maximal operator, we establish the Sobolev inequality by using Hedberg's trick in his paper [ 10].

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J. Math. Soc. Japan, Volume 67, Number 2 (2015), 433-452.

First available in Project Euclid: 21 April 2015

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Primary: 31B15: Potentials and capacities, extremal length 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

maximal functions Lorentz space of variable exponent Riesz potential Sobolev embeddings Sobolev's inequality


MIZUTA, Yoshihiro; OHNO, Takao. Sobolev's inequality for Riesz potentials in Lorentz spaces of variable exponent. J. Math. Soc. Japan 67 (2015), no. 2, 433--452. doi:10.2969/jmsj/06720433.

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  • B. Almut, Rearrangement inequalities, Lecture notes, June 2009.
  • B. Almut, Cases of equality in the Riesz rearrangement inequality, Ann. of Math. (2), 143 (1996), 499–527.
  • B. Almut and H. Hichem, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal., 233 (2006), 561–582.
  • C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics 129, Academic Press, Inc., Boston, MA, 1988.
  • D. Cruz–Uribe, L. Diening and P. Hästö, The maximal operator on weighted variable Lebesgue spaces, Fract. Calc. Appl. Anal., 14 (2011), 361–374.
  • D. Cruz–Uribe, A. Fiorenza and C. J. Neugebauer, The maximal function on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math., 28 (2003), 223–238; Ann. Acad. Sci. Fenn. Math., 29 (2004), 247–249.
  • L. Diening, Maximal functions in generalized $L^{p(\cdot )}$ spaces, Math. Inequal. Appl., 7 (2004), 245–254.
  • L. Ephremidze, V. Kokilashvili and S. Samko, Fractional, maximal and singular operators in variable exponent Lorentz spaces, Fract. Calc. Appl. Anal., 11 (2008), 407–420.
  • P. Hästö and L. Diening, Muckenhoupt weights in variable exponent spaces, preprint.
  • L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc., 36 (1972), 505–510.
  • F.-Y. Maeda, Y. Mizuta and T. Ohno, Approximate identities and Young type inequalities in variable Lebesgue–Orlicz spaces $L^{p(\cdot)}(\log L)^{q(\cdot)}$, Ann. Acad. Sci. Fenn. Math., 35 (2010), 405–420.
  • Y. Mizuta, Potential theory in Euclidean spaces, Gakkōtosho, Tokyo, 1996.
  • Y. Mizuta and T. Shimomura, Weighted Sobolev inequality in Musielak–Orlicz space, J. Math. Anal. Appl., 388 (2012), 86–97.