Estimates for Solutions of the $\bar{\partial}$ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type

Abstract

In the late ten years, the resolution of the equation∂ u = f with sharp estimates has been intensively studied for convex domains of finite type in C n by many authors. Generally they used kernels constructed with holomorphic support function satisfying "good" global estimates. In this paper, we consider the case of lineally convex domains. Unfortunately, the method used to obtain global estimates for the support function cannot be carried out in that case. Then we use a kernel that does not gives directly a solution of the∂ -equation but only a representation formula which allows us to end the resolution of the equation using Kohn's L 2 theory.As an application we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.The general scheme of the proof is identical to the one used in the convex case and consists in three steps. First, for the general case of geometrically separated domains, we prove some "Malliavin conditions" on closed positive (1, 1)-currents Θ and then we solve the equation dw = Θ with good estimates. The third step, which solves the∂ -equation for (0, 1)-form with L 1 estimates on the boundary, is only done in the case of lineally convex domains of finite type: Theorem 1.2. Let Ω be a bounded lineally convex domain of finite type in C n with smooth boundary. Let f be a (0, 1)-form in Ω whose coefficients are C 1 (Ω) functions and which is∂ -closed. Then there exists a solution of the equation ∂ u = f , smooth on Ω and continuous on Ω such that u L 1 (∂ Ω) ≤ C||| f ||| k (see Section 2.1 formula (2.3)), the constant C depending only on Ω. In other words, there exists a solution of the equation ∂ b u = f , in the sense of [Sko76], in L 1 (∂ Ω). SOLUTIONS FOR THE∂ -EQUATION FOR LINEALLY CONVEX DOMAINS OF FINITE TYPEFirst of all, we recall the definition of lineally convex domain: Definition 2.1. A domain Ω in C n , with smooth boundary is said to be lineally convex at a point p ∈ ∂ Ω if there exists a neighborhood W of p such that, for all point z ∈ ∂ Ω ∩W ,where T 10 z is the holomorphic tangent space to ∂ Ω at the point z. Furthermore, we always suppose that ∂ Ω is of finite type at every point of ∂ Ω ∩ W . Shrinking W if necessary, we may assume that there exists a C ∞ defining function ρ for Ω and a number η 0 > 0 such that ∇ρ(z) = 0 at every point of W and the level sets {z ∈ W such that ρ(z) = η}, −η 0 ≤ η ≤ η 0 , are lineally convex of finite type.As we want to obtain global results, we need these properties at every boundary point. Thus, in all our work, by "lineally convex domain" we mean a bounded smooth domain having a (global) defining function satisfying the previous hypothesis at every point of ∂ Ω.In Section 2.1 we define a punctual anisotropic norm for forms, . k , related to the geometry of the domain (formula (2.2)). With this notation, the main goal of this Section is to prove the following reformulation of Theorem 1.2 for (0, q)-forms:2000 Mathematics Subject Classification. 32F17, 32T25, 32T40.

Weighted and boundary $$L^{p}$$ L p estimates for solutions of the $$\overline{\partial }$$ ∂ ¯ -equation on lineally convex domains of finite type and applications

“…Precisely (c.f. [CDM14a]) "lineally convex" means that, for all point in the boundary ∂ Ω of Ω, there exists a neighborhood W of p such that, for all point z ∈ ∂ Ω ∩W , Ä z + T 1,0 z ä ∩ (D ∩W ) = / 0, where T 1,0 z is the holomorphic tangent space to ∂ Ω at the point z. Furthermore, we can assume that there exists a a smooth defining function ρ of Ω such that, for δ 0 sufficiently small, the domains Ω t = {ρ(z) < t}, −δ 0 ≤ t ≤ δ 0 , are all lineally convex of finite type ≤ m. • δ Ω denotes the distance to the boundary of Ω.…”

“…So, when r = p = 1 we will write indifferently f (z) or f (z) ,1 . • In the case of strictly pseudo-convex domains it is clear that the integrability of This result can be extended to α = 0 to get an L p (∂ Ω) estimate (for p = 1 this was done in [CDM14a]): For r = 1 this result is weaker than the one which can be obtained using Carleson measure of order β . Before stating our last estimate let us recall these notions.…”

“…P N ) is the component of a kernel K N (formula (2.7) of [CDM14a]) of bi-degree (0, r) in z and (n, n − r − 1) in ζ (resp. (0, r) in z and (n, n − r) in ζ ) constructed with the method of [AB82] using the Diederich-Fornaess support function constructed in [DF03] (see also Theorem 2.2 of [CDM14a]) and the function G(ξ ) = 1 ξ N with a sufficiently large number N (instead of G(ξ ) = 1 ξ in formula (2.7) of [CDM14a]). Then, the form Ω f (ζ ) ∧ P N (z, ζ ) is ∂ -closed and the operator T solving the ∂ -equation in theorems 2.1 and 2.2 is defined on smooth forms by…”

“…For ζ close to ∂ Ω and ε > 0 small, the basic properties of this geometry are (see [Con02] and [CDM14a]):…”

Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

“…There are cases, however, where the Blaschke condition is sufficient. Namely, the sufficiency is known when Ω is 1) a strongly pseudoconvex domain [16, 19, 32]; 2) a pseudoconvex domain in of finite type [5, 30]; 3) a complex or real ellipsoid by [3, 31]; 4) a convex domain of strictly finite type [4, 9, 10] and/or lineally convex and of finite type [6]; 5) an infinite type of domain of the class studied by Ha [17] as well as the example with (see [1]).…”