Littlewood-Paley decompositions and Besov spaces related to symmetric cones

Simple item page
dc.creatorBekolle, D.
dc.creatorBonami, A.
dc.creatorGarrigos, G.
dc.creatorRicci, F.
dc.date2003-05-05
dc.date.accessioned2026-07-07T04:57:45Z
dc.date.available2026-07-07T04:57:45Z
dc.descriptionStarting from a Whitney decomposition of a symmetric cone $Ω$, 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 $Ω$. In particular, we define a natural class of Besov spaces of such functions, $B^{p,q}_ν$, where the role of usual derivation is now played by the generalized wave operator of the cone $Δ(\frac{\partial}{\partial x})$. Our main result shows that $B^{p,q}_ν$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p,q}_ν(T_Ω)$, at least in a ``good range'' of indices $1\leq q<q_{ν,p}$. We obtain the sharp $q_{ν,p}$ 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 $P_ν\colon L^{p,q}_ν\to A^{p,q}_ν$, for which our result implies a positive answer when $q_{ν,p}'<q<q_{ν,p}$. 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.
dc.description48 pages, 1 figure
dc.identifierhttps://arxiv.org/abs/math/0305072
dc.identifierhttp://arxiv.org/abs/math/0305072
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/67371
dc.subjectClassical Analysis and ODEs
dc.subjectFunctional Analysis
dc.subject42B35; 32M15
dc.titleLittlewood-Paley decompositions and Besov spaces related to symmetric cones
dc.typetext

Files

Collections

arXiv