Weighted and Sobolev Estimates for Fractional Schr\"odinger Operators with Reverse H\"{o}lder Potentials

Abstract

We study weighted and Sobolev estimates for two fractional Schr\"odinger operators on $\mathbb{R}^n$, where $n\geq3$ and $0<s<1$. For the additive operator $L=(-\Delta)^s+V$, with a nonnegative, nontrivial potential $V$ belonging to a suitable reverse H\"older class, $RH_q$, we establish a critical-radius form inequality and decay estimates for the semigroup mass. These yield a heat-kernel bound combining polynomial off-diagonal decay with a fixed positive exponent of decay relative to the fractional critical radius $\rho_s$. A decomposition at this scale gives weighted strong-type estimates for the semigroup maximal operator and weighted Hardy--Littlewood--Sobolev estimates for $L^{-\alpha/2}$, $0<\alpha s<n$, together with the corresponding endpoint weak-type estimates. The admissible weights satisfy explicit growth restrictions determined by the kernel decay, and the fractional integration order is $\alpha s$. We also consider the spectral operator $Q=(-\Delta+V)^s$. Assuming $V\in RH_{q_1}$ with $n/(1+s)\leq q_1<n$ and classical critical radius $\rho_1(x)\geq1$, we prove that $\nabla Q^{-1/2}$ maps $H^{1-s,p}(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n)$ for $1<p\leq p_0$, where $1/p_0=1/q_1-1/n$. By duality, its adjoint maps $L^{p^{'}}(\mathbb{R}^n)$ into $H^{s-1,p^{'}}(\mathbb{R}^n)$ for $p_0^{'}\leq p^{'}<\infty$.