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In this note, we consider the regularity of solutions of the nonlinear elliptic systems of$n$-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space$L$^{$n$,∞}. We also obtain the$a priori$global and local estimates for the$L$^{$n$,∞}-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.

In this paper, we generalize the$A$_{∞}extrapolation theorem ($Cruz-Uribe–Martell–Pérez$, Extrapolation from$A$_{∞}weights and applications,$J. Funct. Anal.$$213$(2004), 412–439) and the$A$_{$p$}extrapolation theorem of Rubio de Francia to Schrödinger settings. In addition, we also establish weighted vector-valued inequalities for Schrödinger-type maximal operators by using weights belonging to $A_{p}^{\rho,\infty }$ which includes$A$_{$p$}. As applications, we establish weighted vector-valued inequalities for some Schrödinger-type operators.