Convexity and Cauchy-Riemann equations with \(L^p\) estimations (Q1580328)

The author solves the equation \(\overline\partial u=f\), for a \((0,q)\) form \(f\), with \(q\geq 1\), on a bounded convex domain of \(\mathbb C^n\), with a \(\mathcal C^3\) smooth boundary, but not satisfying any finite type assumption. The solution is of the form \(u=T_q(f)\) for an operator \(T_q\), which only involves integration on \(D\). In this way some \(L^p\) estimates are obtained, in the spirit of previous results by \textit{R. M. Range} [J. Geom. Anal. 2, No. 6, 575-584 (1992; Zbl 0768.32013)], \textit{John C. Polking} [Proc. Summer Res. Inst., Santa Cruz/CA (USA) 1989, Proc. Symp. Pure Math. 52, Part 3, 309-322 (1991; Zbl 0742.32013)] and \textit{J. Chaumat} and \textit{A.-M. Chollet} [Math. Z. 207, No. 4, 501-534 (1991; Zbl 0748.32008)]. Namely the author shows that there is a solution \(u\in L^p_{(0,q-1)}(D)\) (with \(1<p\leq +\infty\)) when \(f\in L^p_{(0,q)}(D)\) satisfies the extra regularity assumption: \(\frac{\overline\partial\rho\wedge f}{(-\rho)^{1-\varepsilon}}\in L^p_{(0,q+1)}(D)\) for an \(\varepsilon\) with \(0<\varepsilon<\frac{1-\frac{1}{p}}{n-q-\frac{1}{p}}\). The function \(\rho\) is a defining function for \(D\), smooth of class \(\mathcal C^3\).