Carleson measures on spaces of homogeneous type

Abstract:

Let $X$ be a space of homogeneous type in the sense of Coifman and Weiss $[{\text {CW}}2]$ and let ${X^ + } = X \times {{\mathbf {R}}^ + }$. A positive function $F$ on ${X^ + }$ is said to have horizontal bounded ratio $({\text {HBR}})$ on ${X^ + }$ if there is a constant ${A_F}$ so that $F(x,t) \leq {A_F}F(y,t)$ whenever $\rho (x,y) < t$. (By Harnack’s inequality, a well-known example is any positive harmonic function in the upper half plane.) ${\text {HBR}}$ is a rich class that is closed under a wide variety of operations and it provides useful majorants for many classes of functions that are encountered in harmonic analysis. We are able to prove theorems in spaces of homogeneous type for functions in ${\text {HBR}}$ which are analogous to the classical Carleson measure theorems and to extend these results to the functions which they majorize. These results may be applied to obtain generalizations of the original Carleson measure theorem, and of results of Flett’s which contain the Hardy-Littlewood theorems on intermediate spaces of analytic functions. Hörmander’s generalization of Carleson’s theorem is included and it is possible to extend those results to the atomic ${H^p}$ spaces of Coifman and Weiss.