Littlewood-Paley decompositions and Besov spaces related to symmetric cones

TL;DR

This paper develops a Littlewood-Paley theory for functions on symmetric cones, defining Besov spaces and linking them to holomorphic functions in Bergman spaces, with results on boundedness of Bergman projectors and connections to the cone multiplier problem.

Contribution

It introduces an adapted Littlewood-Paley framework for symmetric cones and characterizes Besov spaces as boundary values of holomorphic Bergman functions, extending previous work to general cones.

Findings

Characterization of Besov spaces as boundary values of Bergman space functions.

Sharp bounds for the parameter q when p ≤ 2.

Connections established between the boundedness of Bergman projectors and the cone multiplier problem.

Abstract

Starting from a Whitney decomposition of a symmetric cone $\Omega$, analog to the dyadic partition $[2^j, 2^{j+1})$ of the positive real line, in this paper we develop an adapted Littlewood-Paley theory for functions with spectrum in $\Omega$. In particular, we define a natural class of Besov spaces of such functions, $B^{p,q}_\nu$, where the role of usual derivation is now played by the generalized wave operator of the cone $\Delta(\frac{\partial}{\partial x})$. Our main result shows that $B^{p,q}_\nu$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p,q}_\nu(T_\Omega)$, at least in a ``good range'' of indices $1\leq q<q_{\nu,p}$. We obtain the sharp $q_{\nu,p}$ when $p\leq 2$, and conjecture a critical index for $p>2$. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors $P_\nu\colon L^{p,q}_\nu\to A^{p,q}_\nu$, for which our result implies a positive answer when $q_{\nu,p}'<q<q_{\nu,p}$. This extends to general cones previous work of the authors in the light-cone. Finally, we conclude the paper with a finer analysis in light-cones, for which we establish a link between our conjecture and the cone multiplier problem. Moreover, using recent work by Tao, Vargas and Wolff, we improve in dimension 3 the range of $q$'s for which the Bergman projection is bounded.