Optimal Sobolev estimates for �? on convex domains of finite type

Abstract

We prove that solutions for∂ get 1/M-derivatives more than the data in L p -Sobolev spaces on a bounded convex domain of finite type M by means of the integral kernel method. Also we prove that the Bergman projection is invariant under the L p -Sobolev spaces of fractional orders by different methods from McNeal-Stein's. By using these results, we can get L p -Sobolev estimates of order 1/M for the canonical solution for∂.

Growth Spaces and Growth Norm Estimates for on Convex Domains of Finite Type

“…For more recent results about estimates for∂ on convex domains of finite type by means of integral kernels we refer the reader to [AhCh,Al,Cu,DFF,DiFi,DiFo1,DiFo2,Fi,He1,He2,Wa].…”

Weighted Lipschitz estimates for ∂¯ on convex domains of finite type

“…Similarly as in [2,7], it is enough to consider the case that z and ζ are in a neighborhood of ∂ D and ζ ∈ B(z, γ ).…”

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

“…The first result was obtained by W. Alexandre for C k estimates in [Ale05,Ale06]. For L p -Sobolev estimates, with gain on the Sobolev index, partial results were obtained by Ahn H. and Cho H. in [AC03] and complete results announced by Yao L. in [Yao22].…”