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January 2020 Lipschitz estimates for rough fractional multilinear integral operators on local generalized Morrey spaces
I. Ekincioglu, C. Keskin, R. V. Guliyev
Tbilisi Math. J. 13(1): 47-60 (January 2020). DOI: 10.32513/tbilisi/1585015219

Abstract

We obtain the Lipschitz boundedness for a class of fractional multilinear operators $I_{\Omega,\alpha}^{A,m}$ with rough kernels $\Omega\in L_{s}(\mathbb S^{n-1})$, $s>n/(n-\alpha)$ on the local generalized Morrey spaces $LM_{p,\varphi}^{\{x_0\}}$, generalized Morrey spaces $M_{p,\varphi}$ and vanishing generalized Morrey spaces $VM_{p,\varphi}$, where the functions $A$ belong to homogeneous Lipschitz space $\dot{\Lambda}_{\beta}$, $0<\beta<1$. We find the sufficient conditions on the pair $(\varphi_1,\varphi_2)$ which ensures the boundedness of the operators $I_{\Omega,\alpha}^{A,m}$ from $LM_{p,\varphi_1}^{\{x_0\}}$ to $LM_{q,\varphi_2}^{\{x_0\}}$, from $M_{p,\varphi_1}$ to $M_{q,\varphi_2}$ and from $VM_{p,\varphi_1}$ to $VM_{q,\varphi_2}$ for $1<p<q <\infty$ and $1/p-1/q=(\alpha+\beta)/n$. In all cases the conditions for the boundedness of the operator $I_{\Omega,\alpha}^{A,m}$ is given in terms of Zygmund-type integral inequalities on $(\varphi_1,\varphi_2)$, which do not assume any assumption on monotonicity of $\varphi_1(x,r), \varphi_2(x,r)$ in $r$.

Citation

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I. Ekincioglu. C. Keskin. R. V. Guliyev. "Lipschitz estimates for rough fractional multilinear integral operators on local generalized Morrey spaces." Tbilisi Math. J. 13 (1) 47 - 60, January 2020. https://doi.org/10.32513/tbilisi/1585015219

Information

Received: 4 September 2019; Accepted: 4 December 2019; Published: January 2020
First available in Project Euclid: 24 March 2020

zbMATH: 07200151
MathSciNet: MR4079449
Digital Object Identifier: 10.32513/tbilisi/1585015219

Subjects:
Primary: 42B20
Secondary: 42B35, 47B38, 47G10

Rights: Copyright © 2020 Tbilisi Centre for Mathematical Sciences

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Vol.13 • No. 1 • January 2020
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