Littlewood-Paley decompositions and Besov spaces related to symmetric cones

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.