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Lipschitz estimates for rough fractional multilinear integral operators on local generalized Morrey spaces. (English) Zbl 1439.42016
Summary: 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 \(r\).

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35 Function spaces arising in harmonic analysis
47B38 Linear operators on function spaces (general)
47G10 Integral operators
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