Sobolev's inequality for Riesz potentials of functions in grand Musielak–Orlicz–Morrey spaces over nondoubling metric measure spaces

Abstract

In this paper we are concerned with Sobolev's inequality for Riesz potentials of functions in grand Musielak-Orlicz-Morrey spaces over nondoubling metric measure spaces. K E Y W O R D Sgrand Musielak-Orlicz-Morrey spaces, metric measure space, nondoubling measure, Riesz potentials, Sobolev's inequality, variable exponent M S C ( 2 0 1 0 ) Primary: 46E35, Secondary: 46E30 INTRODUCTIONGrand Lebesgue spaces were introduced in [18] for the study of the Jacobian. They play important roles also in the theory of partial differential equations (see [11,19,47], etc.). The generalized grand Lebesgue spaces appeared in [13], where the existence and uniqueness of the nonhomogeneous -harmonic equations were studied. The boundedness of the maximal operator on the grand Lebesgue spaces was studied in [10]. For variable exponent Lebesgue spaces, see [7,8]. In [20,27], grand Morrey spaces and generalized grand Morrey spaces were introduced. For Morrey spaces, we refer to [32,41]. Further, the boundedness of the maximal operator was studied on grand Morrey spaces of variable exponent in [12] and also on grand Musielak-Orlicz-Morrey spaces in [23]. Grand Musielak-Orlicz-Morrey spaces include various spaces (see Section 3 for the definition and Remark 3.2).For 0 < < , we define the Riesz potential of order for a locally integrable function on byThe classical Sobolev's inequality for Riesz potentials of -functions (see, e.g. [2, Theorem 3.1.4 (b)]) has been extended to various function spaces. For Morrey spaces, Sobolev's inequality was studied in [1,6,33,41], etc., for Morrey spaces of variable exponent in [3,14,15,28,30], etc., for grand Morrey spaces in [20,27], for grand Morrey spaces of variable exponent in [12] and also for Musielak-Orlicz-Morrey spaces in [22]. Recently, Sobolev's inequality has been extended to grand Musielak-Orlicz-Morrey spaces (see [23]).We denote by ( , , ) a metric measure space, where is a bounded set, is a metric on and is a nonnegative complete Borel regular outer measure on which is finite in every bounded set. For simplicity, we often write instead of ( , , ). For ∈ and > 0, we denote by ( , ) the open ball in centered at with radius and = sup{ ( , ) ∶ , ∈ }. We assume that ({ }) = 0 for ∈ and 0 < ( ( , )) < ∞ for ∈ and > 0 for simplicity.