Fractional Integral Operators in Nonhomogeneous Spaces

Abstract

We discuss here the boundedness of the fractional integral operator I α and its generalized version on generalized nonhomogeneous Morrey spaces. To prove the boundedness of I α , we employ the boundedness of the so-called maximal fractional integral operator I * a,κ . In addition, we prove an Olsentype inequality, which is analogous to that in the case of homogeneous type.2000 Mathematics subject classification: primary 42B20; secondary 42B35, 47G10, 31B10, 26A33.

Multilinear fractional integrals on Morrey spaces

“…Later many authors considered and sharpened this inequality with m = 1. We refer to [16][17][18][19] for related results.…”

Some norm inequalities in Musielak-Orlicz spaces

“…By an Olsen inequality, or a trace inequality, we mean an inequality of type (1.2) g · I α f X ≤ C g Y · f Z for some Banach function spaces X, Y and Z, where I α f is the Riesz potential (of order α) of f . There is a vast amount of literatures on Olsen inequalities [11,12,28,29,30,32,33,34,36]. We shall show that (1.2) holds with X = Z = L Φ (R N ) and a certain Morrey space Y .…”

Generalized Stummel Class and Morrey Spaces of Nonhomogeneous Type

“…Since 1970's, the doubling condition has been an important assumption in developing some theories in harmonic analysis. However, some recent results, see [4,7,8,12,13], show that doubling condition assumption is not as significant as thought before, that is some results can be obtained without doubling condition. The space (R d , µ) where the Radon measure µ does not satisfy the doubling condition is known as the nonhomogeneous space.…”