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

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 $$0 < \alpha < n$$.

### 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
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