\(L^ p\)-integrability of the second order derivatives of Green potentials in convex domains (Q811582)

We give estimates in \(L^ p\), \(1<p\leq2\), of the second order derivatives of the Green potential of \(f\in L^ p\), for convex domains. This is done by interpolating between estimates in \(L^ 1\) and \(L^ 2\) of functions in atomic \(H^ 1\) and \(L^ 2\), respectively. The crucial step is obtaining the atomic estimate which is done by adapting to the present situation, a technique introduced by \textit{B. E. J. Dahlberg} and \textit{C. Kenig} [e.g.: Lect. Notes Pure Appl. Math. 122, 631-634 (1990; Zbl 0702.35076)].