Sharp estimates for $\bar \partial $ on convex domains of finite typeon convex domains of finite type

Abstract

Let Q be a bounded convex domain in C n, with smooth boundary of finite type m.The equation c)u~f is solved in f~ with sharp estimates: if f has bounded coefficients, the coefficients of our solution u are in the Lipschitz space A1/m(f~). OptimM estimates are Mso given when data have coefficients belonging to LP([~), p> 1.We solve the c)-equation by means of integral operators whose kernels are not based on the choice of a "good" support function. Weighted kernels are used; in order to reflect the geometry of bf~, we introduce a weight expressed in terms of the Bergman kernel of f/.

Hölder estimates for homotopy operators on strictly pseudoconvex domains with $$C^2$$ C 2 boundary

“…Range [52] obtained a Hölder estimate for ∂ solutions in finite type pseudoconvex domains of C 2 . There are derivative estimates for ∂ solutions in convex domains of D'Angelo finite type m: Diederich-Fornaess-Wiegerlinck [16] obtained the C 1/m estimate for ellipsoids, Diederich-Fischer-Fornaess [15] and Cumenge [13] obtained the C 1/m estimate, and Alexandre [2] achieved the C k+1/m estimate for ∂ solutions.…”

Integral formulas and the ohsawa-takegoshi extension theorem

“…We define the weight factors starting from the Bergman kernel. In this respect the procedure is somewhat similar to [7], but there the construction does not lead up to the canonical solution.…”