We study the weighted norm inequality of $(1,q)$-type, \[ \Vert \mathbf{ G}\nu \Vert_{ L^q(\Omega, d\sigma)} \le C \Vert \nu \Vert, \quad \text{ for all } \nu \in \mathcal{ M}^+(\Omega), \] along with its weak-type analogue, for $0 < q < 1$, where $\mathbf{ G}$ is an integral operator associated with the nonnegative kernel $G(x,y)$. Here $\mathcal{ M}^+(\Omega)$ denotes the class of positive Radon measures in $\Omega$; $\sigma, \nu \in \mathcal{ M}^+(\Omega)$, 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 $u \ge \mathbf{ G}(u^q \sigma)$, i.e., a supersolution to the integral equation \[ u - \mathbf{ G}(u^q \sigma) = 0, \quad u \in L^q_{ \rm loc} (\Omega, \sigma). \] This study is motivated by solving sublinear equations involving the fractional Laplacian, \[ (-\Delta)^{ \frac{ \alpha}{ 2}} u - u^q \sigma = 0\] in domains $\Omega \subseteq \mathbf{ R}^n$ which have a positive Green function $G$, for $0 < \alpha < n$.
Categories: math.AP math.FA