Lipschitz estimates for rough fractional multilinear integral operators on local generalized Morrey spaces

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 $r$.