Sobolev and Hölder estimates for homotopy operators of the [formula omitted]-equation on convex domains of finite multitype

Abstract

We construct homotopy formulas for the ∂‾-equation on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain Ω⊂Cn that has q-type mq for 1≤q≤n, our ∂‾ solution operator Hq on (0,q)-forms has (fractional) Sobolev boundedness Hq:Hs,p→Hs+1/mq,p and Hölder–Zygmund boundedness Hq:Cs→Cs+1/mq for all s∈R and 1<p<∞. We also demonstrate the Lp-boundedness Hq:Hs,p→Hs,prq/(rq−p) for all s∈R and 1<p<rq, where rq:=(n−q+1)mq+2q.