DECOMPOSITION THEOREMS FOR HARDY SPACES ON CONVEX DOMAINS OF FINITE TYPE

摘要

In this paper we study the holomorphic Hardy space H~p(Ω), where Ω is a smoothly bounded convex domain of finite type in C~n. We show that for 0 < p ≤ 1, H~p(Ω) admits an atomic decomposition. More precisely, we prove that each f ∈ H~p(Ω) can be written as f = P_S(∑_(j=0)~∞v_ja_j) = ∑_(j=0)~∞=v_jP_S(a_j), where P_S is the Szegoe projection, the a_j's are real variable p-atoms on the boundary partial deriv Ω, and the coefficients v_j satisfy the condition ∑_(j=0)~∞ |v_j|~p approx< ||f||_(H~p(Ω))~p. Moreover, we prove the following factorization theorem. Each f ∈ H~p(Ω) can be written as f = ∑_(j=0)~∞f_jg_j, where f_j ∈ H~(2p), g_j ∈ H~(2p), and ∑_(j=0)~∞ ||f_j||_(H~(2p))||g_j||_(H~(2p)) approx< ||f||_(H~p(Ω)). Finally, we extend these theorems to a class of domains of finite type that includes the strongly pseudo-convex domains and the convex domains of finite type.