Weighted Norm Inequalities of (1, q)-Type for Integral and Fractional Maximal Operators

Abstract

We study weighted norm inequalities of (1, q)- type for 0 < q < 1,

$$\displaystyle{\Vert \mathbf{G}\nu \Vert _{L^{q}(\Omega,d\sigma )} \leq C\,\Vert \nu \Vert,\quad \text{for all positive measures}\,\nu \text{ in }\Omega,}$$

along with their weak-type counterparts, where $\Vert \nu \Vert =\nu (\Omega )$, and G is an integral operator with nonnegative kernel,

$$\displaystyle{\mathbf{G}\nu (x) =\int _{\Omega }G(x,y)d\nu (y).}$$

These problems are motivated by sublinear elliptic equations in a domain $\Omega \subset \mathbb{R}^{n}$ with non-trivial Green’s function G(x, y) associated with the Laplacian, fractional Laplacian, or more general elliptic operator. We also treat fractional maximal operators M _α (0 ≤ α < n) on $\mathbb{R}^{n}$, and characterize strong- and weak-type (1, q)-inequalities for M _α and more general maximal operators, as well as (1, q)-Carleson measure inequalities for Poisson integrals.

Dedicated to Richard L. Wheeden