Hölder and \(L^ p\) estimates for the solutions of the \({\bar \partial}\)-equation in non-smooth strictly pseudoconvex domains
This paper is a continuation of the author's previous results in Duke Math. J. 55, 539-596 (1987; Zbl 0645.32009)]. In that paper it was proven that \(Tf=\int_{D}K(\xi,z)\wedge f(\xi)\) mapping \(L^{loc}_{(\alpha,\beta)}\) to \(L^{loc}_{(\alpha,\beta -1)}\) such that \({\bar \partial}Tf=f\) if \({\bar \partial}f=0\) with some additional technical assumptions. This paper improves several previous estimates and extends them to the non-smooth case. The results are within the framework of \(L^ p\) spaces with respect to the Lebesgue measure, dm. The functions satisfy Lip(s,D) which are continuous functions on D having modulus of continuity \(0(\delta^ s)\). The \(0(\delta^ s)\) denotes the standard Bachmann-Landau 0 notation introduced in asymptotic methods. The principle theorems are very well proven and are as follows: The operator T satisfies the following \(L^ p\)-estimates: a) \(\| Tf\|_ q\leq c\| f\|_ p\), \(1<p<2n+2\), \(1/q=1/p- 1/(2n+2)\) and furthermore if \(p>2n+2,\) T maps \(L^ p_{(0,1)}\) continuously into \(Lip(1/2-(n+1)/p,D)\) and moreover for \(2n+2<p<\infty,\) T maps \(L^ p_{(0,1)}\) into \(Lip(1/2-(n+1)/p).\) b) \(\| Tf\|_ q<c_ q\| f\|_ 1\), \(\forall q<2n+2\) and furthermore if \(f\in L^ p_{(\alpha,\beta)}\), \(p>2n+2\) and \(\beta >1\), Tf is continuous on \(\bar D\) and moreover if \(f\in L^{\infty}_{(0,1)}\) then \(| Tf(z)-Tf(w)| \leq\)C\(\| f\|_{\infty}p(z,w)^{1/2}| \log p(z,w)|.\) c) For \(p=2n+2\), \(| Tf|\) satisfies the estimate, \(\int_{D}\exp \{c| Tf(z)|^{(2n+2)/(2n+1)}\}dm(s)<\infty\) for some constant, c, depending on \(\| f\|_{2n+2}\) and furthermore if the function r defining, D, is of class, \(c^{2+\gamma}\), \(0<\gamma \leq 1/2\) then for \(p\geq (2n+2)/(1-2\gamma)\), T maps \(L^ p_{(\alpha,\beta)}\) into \(Lip(\min \{\gamma,1/2-(n+1)/p\},D)\), for all \(\gamma\),\(\beta\) and moreover an additional result if \(r\in c^{2+\gamma}\), \(0<\gamma \leq 1/2.\) The paper revisits the construction of the operator T and elegantly proves estimates for the kernel, K.
scientific article; zbMATH DE number 4171333