Let e^-tL be a 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 alignupper bound |pₜ (x, y) |t^{d/2}e^-|x-y|²{ct} (1+t (x) + t (y) ) ^-N align holds for all x, yᵈ 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 nontrivial extension of the results established by Li, Rahm and Wick 23. 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 6. We point out that the new classes of matrix weights and bump conditions are larger and weaker than the classical ones given in 17 and 6, respectively. As applications, our results can be applied to settings of magnetic Schr\"odinger operator, Laguerre operators, etc.
Wen et al. (Fri,) studied this question.