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

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 {align}{upper bound} |p_t(x,y)|≤{C}{td/2}e-|x-y|^2/ct(1+{√t}{ρ(x)}+ {√t}{ρ(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\"{o}dinger operator, Laguerre operators, etc.