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 L²(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 Hölder continuity in x⁠, but we do not require the semigroup to satisfy the preservation condition e^−tL1=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 Schrödinger operators on Rⁿ or Lipschitz domains of Rⁿ⁠.