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

Abstract

Let 𝒳 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 𝒳 . 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 , 1 < p < ∞ , there exists a constant c p such that ∫ Ω | b ( L ) f ( x ) | p u ( x ) d μ ( x ) ≤ c p | | b | | ∞ p ∫ Ω | f ( x ) | p v ( x ) d μ ( x ) for all f ∈ L p ( v d μ ) . Applications include two-weight L p inequalities for Schrodinger operators with non-negative potentials on R n and divergence form operators on irregular domains of R n .