A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces

Sawano, Yoshihiro; Sugano, Satoko; Tanaka, Hitoshi

doi:10.1155/2009/835865

Cited by 40 publications

(27 citation statements)

References 7 publications

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“…The idea of decomposing the fractional integral operator, as we did in this paper, dates back to [, Section ]. The estimate which we used in :

\sum_{Q \in {scriptD}_{k j}} ρ^{*} (ℓ (Q)) \int_{3 Q} f (y) d y \leq C \tilde{ρ} (ℓ (Q_{k j})) \int_{3 Q_{k j}} f (y) d y

goes back to [, (2.19)] and [, p. 6488].…”

Section: Proof Of Theorem and Historical Remarksmentioning

confidence: 99%

Generalized weighted Morrey spaces and classical operators

Nakamura

2016

Mathematische Nachrichten

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Hardy-Littlewood maximal operator, generalized fractional maximal operator, generalized fractional integral operator, singular integral operator MSC (2010) Primary: 42B25, Secondary: 42B35We introduce the notion of generalized weighted Morrey spaces and investigate the boundedness of some operators in these spaces, such as the Hardy-Littlewood maximal operator, generalized fractional maximal operators, generalized fractional integral operators, and singular integral operators. We also study their boundedness in the vector-valued setting.

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“…The idea of decomposing the fractional integral operator, as we did in this paper, dates back to [, Section ]. The estimate which we used in :

\sum_{Q \in {scriptD}_{k j}} ρ^{*} (ℓ (Q)) \int_{3 Q} f (y) d y \leq C \tilde{ρ} (ℓ (Q_{k j})) \int_{3 Q_{k j}} f (y) d y

goes back to [, (2.19)] and [, p. 6488].…”

Section: Proof Of Theorem and Historical Remarksmentioning

confidence: 99%

Generalized weighted Morrey spaces and classical operators

Nakamura

2016

Mathematische Nachrichten

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“…The generalized Morrey spaces were extensively studied in Nakai [24,25], Kurata-Nishigaki-Sugano [16] and Sawano-Sugano-Tanaka [37,38]. Let 0 < r < ∞ …”

Section: Embeddings On Generalized Morrey Spacesmentioning

confidence: 99%

On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

Sawano

Wadade

2012

J Fourier Anal Appl

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Our main purpose in this article is to establish a Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space H M the growth order r 1− 1 p as r → ∞. The inequality (GN) implies that the growth order as r → ∞ is linear, which might look worse compared to the case of the critical Sobolev space. Communicated by Hans Triebel. Y. Sawano ( ) J Fourier Anal ApplHowever, we investigate the optimality of the growth order and prove that this linear order is best-possible. Furthermore, as several applications of the inequality (GN), we shall obtain a Trudinger-Moser type inequality and a Brézis-Gallouët-Wainger type inequality in the critical Sobolev-Morrey space.

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“…Before stating our results, let us recall some definitions and properties of the spaces that we are going to use (see e.g., and references therein).…”

Section: Orlicz–morrey Spacesmentioning

confidence: 99%

A new Beale–Kato–Majda criteria for the 3D magneto‐micropolar fluid equations in the Orlicz–Morrey space

Gala

Sawano

Tanaka

2012

Math Methods in App Sciences

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In this work, we are interested in the study of regularity for the three‐dimensional magneto‐micropolar fluid equations in Orlicz–Morrey spaces. If the velocity field satisfies uMathClass-rel∈L21MathClass-bin−r()0MathClass-punc,TMathClass-punc;scriptMnormalL2logPnormalL3r()double-struckR3 with rMathClass-rel∈(0MathClass-punc,1) and PMathClass-rel>1 or the gradient field of velocity satisfies MathClass-rel∇uMathClass-rel∈L22MathClass-bin−r()0MathClass-punc,TMathClass-punc;scriptMnormalL2logPnormalL3r()double-struckR3 with rMathClass-rel∈(0MathClass-punc,2) and PMathClass-rel>1MathClass-punc, then we show that the solution remains smooth on [0,T]. In view of the embedding L3rMathClass-rel⊂scriptMp3rMathClass-rel⊂scriptMnormalL2logPnormalL3r with 2 < p < 3 ∕ r and P > 1, we see that our result extends the result of Yuan and that of Gala. Copyright © 2012 John Wiley & Sons, Ltd.

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A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces

Abstract: We show some inequalities for generalized fractional integral operators on generalized Morrey spaces. We also show the boundedness property of the generalized fractional integral operators on the predual of the generalized Morrey spaces.

Cited by 40 publications

References 7 publications

Generalized weighted Morrey spaces and classical operators

Generalized weighted Morrey spaces and classical operators

On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

A new Beale–Kato–Majda criteria for the 3D magneto‐micropolar fluid equations in the Orlicz–Morrey space

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