Sobolev embeddings for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over non-doubling measure spaces

Abstract

In this paper we are concerned with Sobolev embeddings for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over non-doubling measure spaces. We deal with the boundedness of the generalized maximal operators, Sobolev's inequality, Trudinger's inequality and continuity for Riesz potentials of variable order. We prove the results in full generality: our function spaces will cover Morrey spaces, variable Lebesgue spaces as well as Lebesgue spaces with a Radon measure μ. Also, we can readily handle function spaces on metric measure spaces. Our results will cover the ones in our earlier paper [Sawano Y, Shimomura T. Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents. Collect. Math. 2013;64:313–350]. We shall show that the Trudinger exponential integrability is available in our generalized setting, whose counterpart on Lebesgue spaces was not considered in Mizuta et al. [Maximal functions, Riesz potentials and Sobolev embeddings on Musielak–Orlicz–Morrey spaces of variable exponent in Rⁿ. Rev. Mat. Complut. 2012;25:413–434.]

Keywords:

• Riesz potentials
• maximal functions
• Sobolev's inequality
• Trudinger's inequality
• continuity
• Musielak–Orlicz–Morrey spaces
• variable exponents
• non-doubling measure
• generalized smoothness

2000 Mathematics Subject Classification::

• Primary: 31B15
• Secondary: 46E35
• 26A33