Shen, Zhongwei Estimates in \(L^ p\) for magnetic Schrödinger operators. (English) Zbl 0880.35034 Indiana Univ. Math. J. 45, No. 3, 817-841 (1996). Let \(H({\mathbf a},V)\) be a magnetic Schrödinger operator \[ \sum_{j=1}^n\Biggl({1 \over i} {\partial \over \partial x_j } - a_j(x)\Biggr)^2 + V(x) \] in \({\mathbb{R}}^n, n \geq 3\) and \(L_j = {1 \over i} {\partial \over \partial x_j } - a_j\). Under certain conditions on \(\text{curl}{\mathbf a}\) and \(V\), given in terms of the reverse Hölder inequality (in particular, they are valid for polynomial coefficients), the estimates of the following type \[ \sum_{1\leq j,k \leq n}|L_jL_k f|\leq C_p|H({\mathbf a},V)f|, \qquad 1<p<\infty \] are proved. There are also similar weak-type (1,1) inequalities. Reviewer: M.Perelmuter (Kiev) Cited in 2 ReviewsCited in 31 Documents MSC: 35J10 Schrödinger operator, Schrödinger equation 35B45 A priori estimates in context of PDEs 35Q40 PDEs in connection with quantum mechanics Keywords:weak-type inequalities; reverse Hölder inequality PDFBibTeX XMLCite \textit{Z. Shen}, Indiana Univ. Math. J. 45, No. 3, 817--841 (1996; Zbl 0880.35034) Full Text: DOI