Littlewood-Paley decompositions and Besov spaces related to symmetric cones

Details Littlewood-Paley decompositions and Besov spaces related to symmetric cones Littlewood-Paley decompositions and Besov spaces related to symmetric cones

2003-05-05

arxiv.org/abs/math/0305072v1

Mathematics

Classical Analysis and ODEs

48 pages, 1 figure

Scientific paper

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 $1\leq q2$. 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}'

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