Gradient perturbations of the sum of two fractional Laplacians (Q2905809)

In this paper, the author studies the gradient perturbation of \(\Delta^{\alpha/2}+\Delta^{\beta/2}\), where \(0<\beta<\alpha<2\) and \(\beta<1\). The gradient perturbation leads to the perturbation series corresponding to the transition densities of the corresponding semigroup. This builds upon the results from [\textit{T.~Jakubowski} and \textit{K.~Szczypkowski}, ``Estimates of gradient perturbation series'', J. Math. Anal. Appl. 389, No. 1, 452--460 (2012; Zbl 1248.47043)], where it was shown (in a more general setting) that, if the drift \(b\) belongs to a certain class \(\mathcal N\), then the perturbation series converges. In that paper, the authors checked that, if \(1< \beta <\alpha <2\) and the drift belongs to the Kato class \(K_d^{\beta-1}\), then it is in \(\mathcal N\). The novelty in the paper under review is that a similar result is proved without the restriction that \(\beta >1\). More precisely, it is shown that the Kato class \(K_d^{\alpha-1}\) is contained in \(\mathcal N\).