Sublinear equations and Schur's test for integral operators

TL;DR

This paper characterizes weighted norm inequalities for integral operators with quasi-symmetric kernels and explores their connection to solutions of sublinear equations involving fractional Laplacians.

Contribution

It provides necessary and sufficient conditions for weighted inequalities and links these to the existence of solutions for sublinear integral equations involving fractional Laplacians.

Findings

Weighted inequalities hold iff an integral condition on the kernel is satisfied.

Existence of solutions to sublinear equations is characterized by the integral condition.

Counterexamples are provided for the endpoint case p=1.

Abstract

We study weighted norm inequalities of $(p,r)$-type, $ \Vert \mathbf{G} (f \, d \sigma) \Vert_{L^r(\Omega, d\sigma)} \le C \Vert f \Vert_{L^p(\Omega, \sigma)}, \quad \forall \, f \in L^p(\sigma),$ for $0 < r < p$ and $p>1$, where $\mathbf{G}(f d \sigma)(x)=\int_\Omega G(x, y) f(y) d \sigma(y)$ is an integral operator associated with a nonnegative kernel $G$ on $\Omega\times \Omega$, and $\sigma$ is a locally finite positive measure in $\Omega$. We show that this embedding holds if and only if $\int_\Omega (\mathbf{G} \sigma)^{\frac{pr}{p-r}} d \sigma<+\infty,$ provided $G$ is a quasi-symmetric kernel which satisfies the weak maximum principle. In the case $p=\frac{r}{q}$, where $0<q<1$, we prove that this condition characterizes the existence of a non-trivial solution (or supersolution) $u \in L^r(\Omega, \sigma)$, for $r>q$, to the the sublinear integral equation $ u - \mathbf{G}(u^q \, d \sigma) = 0, \quad u \ge 0.$ We also give some counterexamples in the end-point case $p=1$, which corresponds to solutions $u \in L^q (\Omega, \sigma)$ of this integral equation. These problems appear in the investigation of weak solutions to the sublinear equation involving the (fractional) Laplacian, $(-\Delta)^{\alpha} u - \sigma \, u^q = 0, \quad u \ge 0,$ for $0<q<1$ and $0 < \alpha < \frac{n}{2}$ in domains $\Omega \subseteq \mathbb{R}^n$ with a positive Green function.