2009
DOI: 10.1017/s0004972709000343
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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.

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Cited by 20 publications
(10 citation statements)
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“…Later many authors considered and sharpened this inequality with m = 1. We refer to [16][17][18][19] for related results.…”
Section: Remark 17mentioning
confidence: 99%
“…Later many authors considered and sharpened this inequality with m = 1. We refer to [16][17][18][19] for related results.…”
Section: Remark 17mentioning
confidence: 99%
“…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 .…”
Section: Fumi-yuki Maeda Yoshihiro Sawano and Tetsu Shimomuramentioning
confidence: 84%
“…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.…”
Section: Introductionmentioning
confidence: 90%