Sublinear equations and Schur's test for integral operators

Abstract

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