Multilinear fractional maximal and integral operators with homogeneous kernels, Hardy--Littlewood--Sobolev and Olsen-type inequalities

TL;DR

This paper establishes boundedness properties of multilinear fractional maximal and integral operators with homogeneous kernels across various Lebesgue and Lorentz spaces, extending classical inequalities and employing innovative proof techniques.

Contribution

It introduces new boundedness results for multilinear operators in Lebesgue, Lorentz, and weak Lebesgue spaces, using ideas from Hedberg and Adams, which were not previously known.

Findings

Boundedness from L^{p_1}×...×L^{p_m} to L^q under certain conditions.

Weak-type boundedness into L^{q,∞} spaces with new parameter ranges.

Boundedness in Morrey-type spaces with specific exponents.

Abstract

Let $m\in \mathbb{N}$ and $0<\alpha<mn$.In this paper, we will use the idea of Hedberg to reprove that the multilinear operators $\mathcal{T}_{\Omega,\alpha;m}$ and $\mathcal{M}_{\Omega,\alpha;m}$ are bounded from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)\times\cdots\times L^{p_m}(\mathbb R^n)$ into $L^q(\mathbb R^n)$ provided that $\vec{\Omega}=(\Omega_1,\Omega_2,\dots,\Omega_m)\in L^s(\mathbf{S}^{n-1})$, $s'<p_1,p_2,\dots,p_m<n/{\alpha}$, \begin{equation*} \frac{\,1\,}{p}=\frac{1}{p_1}+\frac{1}{p_2}+\cdots+\frac{1}{p_m} \quad \mbox{and} \quad \frac{\,1\,}{q}=\frac{\,1\,}{p}-\frac{\alpha}{n}. \qquad (*) \end{equation*} We also prove that under the assumptions that $\vec{\Omega}=(\Omega_1,\Omega_2,\dots,\Omega_m)\in L^s(\mathbf{S}^{n-1})$, $s'\leq p_1,p_2,\dots,p_m<n/{\alpha}$ and $(*)$, the multilinear operators $\mathcal{T}_{\Omega,\alpha;m}$ and $\mathcal{M}_{\Omega,\alpha;m}$ are bounded from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)\times \cdots\times L^{p_m}(\mathbb R^n)$ into $L^{q,\infty}(\mathbb R^n)$, which are completely new. Moreover, we will use the idea of Adams to show that $\mathcal{T}_{\Omega,\alpha;m}$ and $\mathcal{M}_{\Omega,\alpha;m}$ are bounded from $L^{p_1,\kappa}(\mathbb R^n)\times L^{p_2,\kappa}(\mathbb R^n)\times \cdots\times L^{p_m,\kappa}(\mathbb R^n)$ into $L^{q,\kappa}(\mathbb R^n)$ whenever $s'<p_1,p_2,\dots,p_m<n/{\alpha}$, $0<\kappa<1$, \begin{equation*} \frac{\,1\,}{p}=\frac{1}{p_1}+\frac{1}{p_2}+\cdots+\frac{1}{p_m} \quad \mbox{and} \quad \frac{\,1\,}{q}=\frac{\,1\,}{p}-\frac{\alpha}{n(1-\kappa)},\qquad (**) \end{equation*} and also bounded from $L^{p_1,\kappa}(\mathbb R^n)\times L^{p_2,\kappa}(\mathbb R^n)\times \cdots\times L^{p_m,\kappa}(\mathbb R^n)$ into $WL^{q,\kappa}(\mathbb R^n)$ whenever $s'\leq p_1,p_2,\dots,p_m<n/{\alpha}$, $0<\kappa<1$ and $(**)$.