Sobolev Embeddings for Generalized Riesz Potentials of Functions in Morrey Spaces over Nondoubling Measure Spaces

Yoshihiro Sawano,Tetsu Shimomura

doi:10.1155/2013/984259

Abstract

Our aim in this paper is to deal with the Sobolev embeddings for generalized Riesz potentials of functions in Morrey spaces over nondoubling measure spaces.

Highlights

We show that many endpoint results about the Adams theorem still hold in the nondoubling setting and that the integral kernel can be generalized to a large extent
In [1], in the setting of the Lebesgue measure, for 0 < α < n, recall that Adams considered and proved the boundedness of the fractional integral operator Iα given by f (y)
One of the reasons why the case when p = 1 is difficult is the failure of the boundedness of the HardyLittlewood maximal operator M

Summary

Introduction

We show that many endpoint results about the Adams theorem still hold in the nondoubling setting and that the integral kernel can be generalized to a large extent. In [1], in the setting of the Lebesgue measure, for 0 < α < n, recall that Adams considered and proved the boundedness of the fractional integral operator Iα given by f (y). The boundedness of Riesz potentials from L(1,φ)(G) to Orlicz-Morrey spaces was shown in [19] One of the reasons why the case when p = 1 is difficult is the failure of the boundedness of the HardyLittlewood maximal operator M In connection with this failure, we do not have Littlewood-Paley characterization. Despite the difficulty arising from harmonic analysis, the case when p = 1 occurs naturally As another evidence that the case when p = 1 is of importance, we recall that the space L1,λ(Rn) appears naturally in the following sharp maximal inequalities [23, Theorem 4.7], [24, Theorem 1.3], and [25, Theorem 1.2]: let 1 < p < ∞ and λ ∈

Notation and Terminologies

Main Results

Preliminary Lemmas

Proofs of the Theorems