Carleson embedding theorem on convex finite type domains

Abstract

This paper concerns certain geometric aspects of function theory on smoothly bounded convex domains of finite type in C n . Specifically, we prove the Carleson-Hörmander inequality for this class of domains and provide examples of Carleson measures improving a known result concerning such measures associated to bounded holomorphic functions.

Hp-corona problem and convex domains of finite type

“…We use results of Jasiczak [17] in order to prove (see Subsection 4.1) that ∂(hω) satisfies the hypothesis of Theorem 1.2 below. In this theorem and in the sequel, A B means there exists a constant c > 0 such that A ≤ cB and A B that A B and B A both hold.…”

“…When θ is a smooth p-form on D, Bruna, Charpentier and Dupain [9] define θ(z) k as a smooth function of z by θ(z) k := sup u 1 ,...,up =0 |ω(z)(u 1 ,...,up)| k(z,u 1 )...k(z,up) which is the norm of the form θ(z) with respect to the norm k(z, •). The following theorem is Theorem 1.2 of [17] . The following theorem is Theorem 1.1 of [17].…”

“…The following theorem is Theorem 1.2 of [17] . The following theorem is Theorem 1.1 of [17]. This result, generally referred as Carleson-Hörmander inequality, will be very useful for us.…”

“…except for the estimates |r| ∂h ∧ ∂h k dV W 1 which are not stated in it, but they are immediate consequences of the inequality Pε(z)∩bD |r| ∂h ∧ ∂h k h 2 BM OA(D) σ(P ε (z) ∩ bD) established for all ε > 0 and all z ∈ bD in the proof of Theorem 1.2 of[17].Theorem 4.1. For all h ∈ BM OA(D), |r| ∂h ∧ ∂h dV and |r| ∂h 2 k dV are Carleson measure and |r| ∂h ∧ ∂h k dV W 1…”

Dyadic decomposition of convex domains of finite type and applications

“…By [DFF99, Proposition 3.10] (see, also [J10,Proposition 2.1]), there exists a constant C 2 > 0, which only depends on C ′ , such that…”