Matrix-weighted inequalities for fractional type integrals associated to operators with new classes of weights

Let e−tL be the analytic semigroup generated by −L, where L is a non-negative self-adjoint operator on L2(Rd). Assume that the kernels of e−tL, denoted by pt(x,y), only satisfy the upper bound: for all N>0, there are constants c,C>0 such that |pt(x,y)|≤Ctd/2e−∣∣x−y∣∣2ct(1+t√ρ(x)+t√ρ(y))−N holds for all x,y∈Rd and t>0. We first establish the quantitative matrix-weighted inequalities for fractional type integrals associated to L with new classes of matrix weights, which are a nontrivial extension of the results established by Li, Rahm and Wick (2019). Next, we give new two-weight bump conditions with Young functions satisfying wider conditions for fractional type integrals associated to L, which cover the result obtained by Cruz-Uribe, Isralowitz and Moen (2018). The new classes of matrix weights and bump conditions are larger and weaker than the classical ones. Our results can be applied to the settings of the magnetic Schrödinger operator, Laguerre operators, etc.