Limiting case of the boundedness of fractional integral operators on nonhomogeneous space

Sawano, Yoshihiro; Sobukawa, Takuya; Tanaka, Hitoshi

doi:10.1155/jia/2006/92470

Cited by 16 publications

(13 citation statements)

References 10 publications

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“…6, we are concerned with the Morrey counterpart of Trudinger's type exponential integrability for U α(·),9 f . Our result contains the result of Trudinger [39] as well as those in [21,22,34]. For related results, see [2,[7][8][9][18][19][20]28,40].…”

Section: ( ) P(·)q(·) (X ·) Is Convex On [0 ∞) For Every X ∈ Xmentioning

confidence: 60%

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Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents

Sawano

Shimomura

2013

Collect. Math.

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The target is potential theory in connection with Morrey spaces on general metric measure spaces. The present paper is oritented to investigating Sobolev's inequality, Trudinger exponential integrability and continuity for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents. A counterexample shows that our results are reasonable. In addition to the example above, what is new about this paper is that everything can be developed once the underlying measure does not charge any point.

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Section: ( ) P(·)q(·) (X ·) Is Convex On [0 ∞) For Every X ∈ Xmentioning

confidence: 60%

“…We also establish Trudinger exponential integrability for Riesz potentials of functions in Morrey spaces of variable exponents in the non-doubling setting, as extensions of our earlier papers [21,22,34]. Further, we discuss continuity of Riesz potentials of variable order, which extends [7,12,21,23].…”

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confidence: 62%

Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents

Sawano

Shimomura

2013

Collect. Math.

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“…In the nonhomogeneous setting, the boundedness of I α on the Lebesgue spaces L p (µ) has been studied recently in [4,15]. Our goal here is to prove the boundedness of I α , and its generalized version, on the generalized nonhomogeneous Morrey space M p,φ (k, µ).…”

Section: Introductionmentioning

confidence: 99%

“…Using the following fact, it suffices for us to study the boundedness of I * α/n,κ . FACT 2 (Sawano et al [15]). We have I α f ≤ C I * α/n,κ f for all positive µ-measurable functions f .…”

Section: Introductionmentioning

confidence: 99%

Fractional Integral Operators in Nonhomogeneous Spaces

Gunawan¹,

Sawano²,

Sihwaningrum³

2009

Bull. Aust. Math. Soc.

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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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“…Orlicz-Morrey spaces unify Orlicz and Morrey spaces. Recently, Orlicz-Morrey spaces were used by Sawano, Sobukawa and Tanaka [34] to prove a Trudinger type inequality for Morrey spaces.…”

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Orlicz–Morrey spaces and the Hardy–Littlewood maximal function

Nakai

2008

Studia Math.

115

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Abstract. We prove basic properties of Orlicz-Morrey spaces and give a necessary and sufficient condition for boundedness of the Hardy-Littlewood maximal operator M from one Orlicz-Morrey space to another. For example, if f ∈ L(log L)(R n ), then M f is in a (generalized) Morrey space (Example 5.1). As an application of boundedness of M , we prove the boundedness of generalized fractional integral operators, improving earlier results of the author.1. Introduction. Orlicz spaces, introduced in [29,30], are generalizations of Lebesgue spaces L p . They are useful tools in harmonic analysis and its applications. For example, the Hardy-Littlewood maximal operator is bounded on L p for 1 < p ≤ ∞, but not on L 1 . Using Orlicz spaces, we can investigate the boundedness of the operator near p = 1 precisely (see Kita [14,15] and Cianchi [4]). It is known that the fractional integral operator I α is bounded from L p (R n ) to L q (R n ) for 1 < p < q < ∞ and −n/p+α = −n/q (the Hardy-Littlewood-Sobolev theorem). Trudinger [40] investigated the boundedness of I α near q = ∞. The Hardy-Littlewood-Sobolev theorem and Trudinger's result have been generalized by several authors: [28,37,38,5,4,23,24,25], etc. For the theory of Orlicz spaces, see [18,16,33].On the other hand, Morrey spaces were introduced in [19] to estimate solutions of partial differential equations, and studied in many papers. For the boundedness of the Hardy-Littlewood maximal operator and fractional integral operators, see [31,1,3,20].

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Limiting case of the boundedness of fractional integral operators on nonhomogeneous space

Abstract: We show the boundedness of fractional integral operators by means of extrapolation. We also show that our result is sharp.

Cited by 16 publications

References 10 publications

Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents

Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents

Fractional Integral Operators in Nonhomogeneous Spaces

Orlicz–Morrey spaces and the Hardy–Littlewood maximal function

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