Interior $L^p$-estimates and local $A_p$-weights

Let $ \Omega$ be a nonempty open proper and connected subset of $\mathbb R^ {n}$, $n \geq 3$. Consider the elliptic Schrödinger type operator $L_ {E} u= A_ {E} u+Vu= - \Sigma_{ij} a_ {ij} (x) u_ {x_i x_j} +Vu$ in $\Omega$, and the linear parabolic operator $L_ {P} u=A_ {P} u+Vu = u_ {t} - \Sigma a_ {ij} (x,t)u_ {x_{i}x_{j}} +Vu$ in $\Omega_{T} = \Omega \times (0,T)$, where the coefficients $a_ {ij} \in \mathrm{VMO}$ and the potential $V$ satisfies a reverse Hölder condition. The aim of this paper is to obtain a priori estimates for the operators $L_ {E}$ and $L_ {P}$ in weighted Sobolev spaces involving the distance to the boundary and weights in a local $A_ {p}$ class.

Riesz transform related to a Schrödinger type operator on the Heisenberg group

In this paper, we consider the Schrödinger type operator

Wavelets, Sobolev Multipliers, and Application to Schrödinger Type Operators with Nonsmooth Potentials

We employ Meyer wavelets to characterize multiplier spaceXr,pt(ℝn)without using capacity. Further, we introduce logarithmic Morrey spacesMr,pt,τ(ℝn)to establish the inclusion relation between Morrey spaces and multiplier spaces. By fractal skills, we construct a counterexample to show that the scope of the indexτofMr,pt,τ(ℝn)is sharp. As an application, we consider a Schrödinger type operator with potentials inMr,pt,τ(ℝn).