Muckenhoupt weights and maximal \(L^p\)-regularity (Q1423812)

For \(X\) a Banach space, \(q\in(1,\infty)\), \(f\in L^q(\mathbb R_+,X)\), consider the equation \(u'(t)+Au(t)=f(t)\) with a sectorial operator \(-A\) of angle \(<\pi/2\). The operator \(A\) is of maximal regularity, if for any \(f\in L^q(\mathbb R_+,X)\) both \(u'\) and \(Au\) are also in \(L^q(\mathbb R_+,X)\). This property is independent of \(q\in(1,\infty)\). In the current paper, sufficient conditions for maximal regularity are given in terms of weighted \(L^p\)-estimates (with weight belonging to some Muckenhoupt class) for a sublinear operator dominating \(is(is+A)^{-1}\) for all \(s\in \mathbb R\). The results are applied to elliptic differential operators with continuous top order coefficients.