A sublinear version of Schur’s lemma and elliptic PDE. (English) Zbl 1384.35031

Quinn, Stephen; Verbitsky, Igor E.

Anal. PDE 11, No. 2, 439-466 (2018).

Authors’ abstract: We study the weighted norm inequality of \((1,q)\)-type, \[ \| G\nu \|_{L^q(\Omega, d\sigma)} \leq C \| \nu \|\;\;\text{for all}\; \nu \in \mathscr{M}^+(\Omega), \] along with its weak-type analogue, for \(0 < q < 1\), where \(G\) is an integral operator associated with the nonnegative kernel \(G\) on \(\Omega\times\Omega\). Here \(\mathscr{M}^+(\Omega)\) denotes the class of positive Radon measures in \(\Omega\); \(\sigma, \nu \in \mathscr{M}^+(\Omega)\), and \(\| \nu\|=\nu(\Omega)\). For both weak-type and strong-type inequalities, we provide conditions which characterize the measures \(\sigma\) for which such an embedding holds. The strong-type \((1,q)\)-inequality for \(0< q<1\) is closely connected with existence of a positive function \(u\) such that \(u \geq G(u^q \sigma)\), i.e., a supersolution to the integral equation \[ u - G(u^q \sigma) = 0,\quad u \in L^q_{\text{loc}} (\Omega, \sigma). \] This study is motivated by solving sublinear equations involving the fractional Laplacian, \[ (-\Delta)^{\frac{\alpha}{2}} u - u^q \sigma = 0, \] in domains \(\Omega \subseteq \mathbb{R}^n\) which have a positive Green function \(G\) for \(0 < \alpha < n\).

Reviewer: Michael Perelmuter (Kyïv)

MSC:

35J61	Semilinear elliptic equations
31B15	Potentials and capacities, extremal length and related notions in higher dimensions
30L99	Analysis on metric spaces
42B25	Maximal functions, Littlewood-Paley theory
35R11	Fractional partial differential equations

Keywords:

weighted norm inequalities; sublinear elliptic equations; fractional Laplacian

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