Weighted inequalities for holomorphic functional calculi of operators with heat kernel bounds

Let script X sign be a space of homogeneous type. Assume that L has a bounded holomorphic functional calculus on L ²(Ω) and L generates a semigroup with suitable upper bounds on its heat kernels where Ω is a measurable subset of script X sign. For appropriate bounded holomorphic functions b, we can define the operators b(L) on L ^p(Ω), 1 ≤ p ≤ ∞. We establish conditions on positive weight functions u, v such that for each p, l < p < ∞, there exists a constant c _p such that ∫ _Ω|b(L)f(x) | ^pu(x)dμ(x) ≤ c _p||b|| _∞ ^p ∫ _Ω |f(x)| ^pv(x)dμ(x) for all f ∈ L ^P(vdμ). Applications include two-weight L ^p inequalities for Schrödinger operators with non-negative potentials on R ⁿ and divergence form operators on irregular domains of R ⁿ.