Classes of weights related to Schrödinger operators
The authors consider the Schrödinger operator on \(\mathbb R^d\) for \(d\geq 3\), \(\mathcal L=-\Delta + V\), where \(V\) is a non-negative potential satisfying the reverse Hölder inequality \((\frac{1}{|B|}\int_B V^q)^{1/q}\leq C \frac{1}{|B|}\int_B V\) for some \(q>d/2\). The paper concerns the study of weighted \(L^p\)-estimates for several operators associated to \(\mathcal L\), namely the maximal operator of the diffusion semigroup \(\mathcal T^*f(x)=\sup_{t>0} e^{-t\mathcal L}f(x)\), the \(\mathcal L\)-fractional operator \(\mathcal L^{-\alpha/2}\), the \(\mathcal L\)-Riesz transform \(\mathcal R=\nabla \mathcal L^{-1/2}\) or the \(\mathcal L\)-square function. While in the case \(V=0\) the weighted theory depends on the class of \(A_p\)-weights, they show that there is a new class appearing in this context which fits with the boundedness of those operators. They introduce the class \(A^{\rho, \theta}_p\) of those weights \(w\) such that \[ (\int_B w)^{1/p}(\int_B w^{-1/(p-1)})^{1/p'}\leq C |B|(1+\frac{r}{\rho(x)})^\theta \] for every ball \(B=B(x,r)\), where \(\rho(x)\) is the critical radius introduced by \textit{Z. Shen} [Ann. Inst. Fourier 45, No. 2, 513--546 (1995; Zbl 0818.35021)] given by \(\rho(x)= \sup\{ r: \frac{1}{r^{d-2}}\int_{B(x,r)} V\leq 1\} \). They show that if \(w\in A_p^{\rho,\infty}=\cup_{\theta \geq 0}A^{\rho, \theta}_p\) for \(1<p<\infty\) then the maximal operator \(\mathcal T^*\) and the \(\mathcal L\)-square function operator are bounded in \(L^p(w)\). Also the weak-type result holds in the case \(w\in A_1^{\rho,\infty}\). The boundedness for the \(\mathcal L\)-Riesz transform is shown under the same assumptions on the weight \(w\), but only for \(V\) satisfying the revere Hölder inequality for \(q\geq d\). Finally, the expected result on the \(\mathcal L\)-fractional operator is also achieved.
scientific article; zbMATH DE number 5804530