Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 389-416

SOBOLEV'S THEOREM AND DUALITY FOR HERZ-MORREY SPACES OF VARIABLE EXPONENT

Yoshihiro Mizuta and Takao Ohno

Hiroshima Institute of Technology, Department of Mechanical Systems Engineering
2-1-1 Miyake Saeki-ku Hiroshima 731-5193, Japan; y.mizuta.5x 'at' it-hiroshima.ac.jp

Oita University, Faculty of Education and Welfare Science
Dannoharu Oita-city 870-1192, Japan; t-ohno 'at' oita-u.ac.jp

Abstract. In this paper, we consider the Herz-Morrey space H_{{x_0}}^{p(\cdot),q,\omega}(G) of variable exponent consisting of all measurable functions f on a bounded open set G \subset Rⁿ satisfying

||f||_{H_{{x_0}}^{p(\cdot),q,\omega}(G)} = (\int_0^{2d_G} (\omega(x₀,r) ||f||_{L^{p(\cdot)}(B(x₀,r)
\setminus B(x₀,r/2))})^qdr/r)^1/q < \infty,

and set H^{p(\cdot),q,\omega}(G) = \bigcap_{x_0 \in G} H_{{x_0}}^{p(\cdot),q,\omega}(G). Our first aim in this paper is to give the boundedness of the maximal and Riesz potential operators in H^{p(\cdot),q,\omega}(G) when q = \infty. In connection with H_{{x_0}}^{p(\cdot),q,\omega}(G) and H^{p(\cdot),q,\omega}(G), let us consider the families \underlineH_{{x_0}}^{p(\cdot),q,\omega}(G), \underlineH^{p(\cdot),q,\omega}(G), \overlineH_{{x_0}}^{p(\cdot),q,\omega}(G) and \tildeH_{{x_0}}^{p(\cdot),q,\omega}(G). Following Fiorenza-Rakotoson [18], Di Fratta-Fiorenza [17] and Gogatishvili-Mustafayev [19], we next discuss the duality properties among these Herz-Morrey spaces.

2010 Mathematics Subject Classification: Primary 31B15, 46E35.

Key words: Herz-Morrey spaces of variable exponent, maximal functions, Riesz potentials, Sobolev's inequality, Trudinger's inequality, duality.

Reference to this article: Y. Mizuta and T. Ohno: Sobolev's theorem and duality for Herz-Morrey spaces of variable exponent. Ann. Acad. Sci. Fenn. Math. 39 (2014), 389-416.

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doi:10.5186/aasfm.2014.3913