Abstract
Our aim in this paper is to give Sobolev's inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces. The main result is oriented to the outrange of the well-known Adams theorem. MSC: Primary 31B15; secondary 46E35; 26A33
Highlights
The boundedness of fractional integral operators on Morrey spaces is known as the Adams theorem
Many endpoint results have been obtained for this theorem, and in this paper we extend them to generalized Morrey spaces with variable exponent attaining the value over non-doubling measure spaces
We aim to understand how the fractional integral operators behave in generalized Morrey spaces with variable exponent attaining the value over non-doubling measure spaces
Summary
The boundedness of fractional integral operators on Morrey spaces is known as the Adams theorem. In Morrey observed that a weaker regularity sufficed in order that the solutions in elliptic differential equations were smooth [ ]. This observation grew up to be a useful tool for partial differential equations in general. Nowadays, his technique turned out to be a wide theory of function spaces called Morrey spaces; see [ ]. We are oriented to building up a theory of a metric measure space (X, d, μ) for which the notion of dimension is not equipped with μ, where d is a distance function and μ is a.
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