On maximal regularity of type \(L^p-L^q\) under minimal assumptions for elliptic non-divergence operators
For operators of the form \[ A_{q} = \sum_{\mid \alpha \mid \leq m} a_{\alpha} (x) D^{\alpha}, \;\;\;D(A_{q}) = W_{q}^{m} (\mathbb{R}^{n},\mathbb{C}), \] considered in the space \(L^{q} (\mathbb{R}^{n}, \mathbb{C}),\) \((1 < q < \infty),\) the author gives a positive answer to Brézis' question. Namely, if \(A_{q}\) has maximal regularity of type \(L^{p} - L^{q}\), \(1 < p < \infty,\) whenever \(A_{q}\) generates an analytic semigroup in \(L^{q} (\mathbb{R}^{n}, \mathbb{C})\). The methods applied are new and substantially generalize those previously known for operators with bounded measurable coefficients depending on one coordinate only.
scientific article; zbMATH DE number 5480710
scientific article