Integral representations on nonsmooth domains

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

“…We begin with estimates on the kernels of a certain type. In [1], we proved the Proposition 3.1. Let A j be an operator of type j. Then…”

“…where Φ is a first order differential operator, such that Y can be written (1,2) )). W 1 and W 2 are allowable vector fields while W 2 (ϕA (1,2) ) and W 1 (ϕA (1,2) ) are of the form A ′ 1 where A ′ 1 is of commutator type, proving the theorem for Case 1. Case 2).…”

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

“…and by carefully handling this term we obtain a slight improvement, with tangible benefits, over its treatment in [6] and even in [1], where the additional information has no effect. In our relating boundary integrals to volume integrals, with the use of Stoke's Theorem, we replace ρ 2 with P ǫ , as they are equivalent on ∂D ǫ , in the kernels.…”

“…The proof follows as in [1], up until the examination of the term (4.4) below. By carefully considering such a term, we are able to show that the resulting integral kernels in the representation are of double type (0, 1), whereas the analysis in [1] would only grant us kernels of double type (−1, 1). This information is not needed in the q > 0 case due to the difference in the definitions of the term P ǫ (ζ, z) in the cases q = 0 and q > 0.…”

Principal parts of operators in the $\bar{\partial}$-Neumann problem on strictly pseudoconvex non-smooth domains

“…For ease of notation we will drop here the superscripts ǫ, which were used in [2] to do calculations on the smooth subdomains, D ǫ = {r < −ǫ} noting that the following calculations also hold when the kernels on D × D are replaced with the corresponding kernels on D ǫ × D ǫ . All formulas remain the same and make sense when one looks at the appropriate weighted L p spaces.…”