Weighted Ck estimates for a class of integral operators on non-smooth domains

Abstract

We apply integral representations for (0, q)-forms, q ≥ 1, on nonsmooth strictly pseudoconvex domains, the Henkin-Leiterer domains, to derive weighted C k estimates for a given (0, q)-form, f , in terms of C k norms of∂f , and∂ * f . The weights are powers of the gradient of the defining function of the domain.

Sobolev regularity of the $$\overline{\partial }$$ -equation on the Hartogs triangle

“…In the context of non-smooth domains, it seems that singular weights are a natural device to control the behavior of functions and forms near the singular part of the boundary. Such weights also arise naturally in recent attempts to generalize classical estimates on the ∂ -and ∂ -Neumann problems from smooth to non-smooth strictly pseudoconvex domains (see [9,10,11]. )…”

The Bergman projection and weighted $C^{k}$ estimates for the canonical solution to the $\bar{\partial}$ problem on non-smooth domains

“…Thus, when integrating by parts, special attention has to be paid to these non-smooth terms. We obtain the following theorem from [2]. Theorem 6.5.…”

“…We collect in this section the various mapping properties of our operators. The proofs follow as in [1,2].…”