Weighted and Sobolev Estimates for Fractional Schrödinger Operators with Reverse Hölder Potentials

We study weighted and Sobolev estimates for two fractional Schrödinger operators on ${\mathbb{R}}^n$, where $n\ge 3$ and $0<s<1$. For the additive operator $L=(-\mathrm{\Delta }{)}^s+V$, with a nonnegative, nontrivial potential $V$ belonging to a suitable reverse Hölder class, $R{H}_q$, 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 $L^{-\alpha \mathrm{/}2}$, $0<\alpha s<n$, 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 $Q=(-\mathrm{\Delta }+V{)}^s$. Assuming $V\in R{H}_{q_1}$ with $n\mathrm{/}(1+s)\le q_1<n$ and classical critical radius ${\rho }_1(x)\ge 1$, we prove that $\mathrm{\nabla }{Q}^{-1\mathrm{/}2}$ maps $H^{1-s,p}({\mathbb{R}}^n)$ into $L^p({\mathbb{R}}^n)$ for $1<p\le p_0$, where $1\mathrm{/}{p}_0=1\mathrm{/}{q}_1-1\mathrm{/}n$. By duality, its adjoint maps $L^{p^{{}^{\mathrm{\prime }}}}({\mathbb{R}}^n)$ into $H^{s-1,p^{{}^{\mathrm{\prime }}}}({\mathbb{R}}^n)$ for $p_0^{{}^{\mathrm{\prime }}}\le p^{{}^{\mathrm{\prime }}}<\mathrm{\infty }$.