Littlewood-Paley decompositions and Besov spaces related to symmetric cones

Starting from a Whitney decomposition of a symmetric cone $\mathrm{\Omega }$, 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 $\mathrm{\Omega }$. 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 $\Delta(\frac{\partial}{\partial x})$\mathrm{.}Ourmainresultshowsthat$B^{p,q}\nu$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A_{\nu }^{p,q}(T_{\mathrm{\Omega }})$, at least in a ``good range’’ of indices $1\leq q<q{\nu,p}$\mathrm{.}Weobtainthesharp$q_{\nu,p}$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 $P_\nu\colon L^{p,q}\nu\to A^{p,q}\nu$,forwhichourresultimpliesapositiveanswerwhen$q_{\nu,p}'<q<q_{\nu,p}$. 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.