$L^p$ theory for outer measures and two themes of Lennart Carleson united

Abstract:

We develop a theory of $L^p$ spaces based on outer measures generated through coverings by distinguished sets. The theory includes as a special case the classical $L^p$ theory on Euclidean spaces as well as some previously considered generalizations. The theory is a framework to describe aspects of singular integral theory, such as Carleson embedding theorems, paraproduct estimates, and $T(1)$ theorems. It is particularly useful for generalizations of singular integral theory in time-frequency analysis, the latter originating in Carleson’s investigation of convergence of Fourier series. We formulate and prove a generalized Carleson embedding theorem and give a relatively short reduction of the most basic $L^p$ estimates for the bilinear Hilbert transform to this new Carleson embedding theorem.