Exponential-Square Integrability, Weighted Inequalities for the Square Functions Associated to Operators, and Applications

Let X be a metric space with a doubling measure. Let L be a nonnegative self-adjoint operator acting on L-2(X), hence L generates an analytic semigroup e(-tL) . Assume that the kernels p(t) (x,y) of e(-tL) satisfy Gaussian upper bounds and Holder continuity in x, but we do not require the semigroup to satisfy the preservation condition e(-tL) = 1. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator L is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces R-n. We then apply this result to obtain: (1) estimates of the norm on L-P as p becomes large for operators such as the square functions or spectral multipliers; (2) weighted norm inequalities for the square functions; and (3) eigenvalue estimates for Schrodinger operators on R-n or Lipschitz domains of R-n.