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, $B_{\nu }^{p,q}$, 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 $B_{\nu }^{p,q}$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A_{\nu }^{p,q}(T_{\Omega })$, at least in a ``good range'' of indices $1\le q<q_{\nu ,p}$. We obtain the sharp $q_{\nu ,p}$ when $p\le 2$, and conjecture a critical index for $p>2$. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors $P_{\nu }:L_{\nu }^{p,q}\to A_{\nu }^{p,q}$, for which our result implies a positive answer when $q_{\nu ,p}^{\prime }<q<q_{\nu ,p}$. 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.

48 pages, 1 figure