Exponential-square integrability, weighted inequalities for the square functions associated to operators and applications (1609.01516v1)
Abstract: 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 H\"older's continuity in $x$ but we do not require the semigroup to satisfy the preservation condition $e{-tL}1 = 1$. 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 ${\mathbb Rn}$. We then apply this result to obtain: (i) estimates of the norm on $Lp$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (ii) weighted norm inequalities for the square functions; and (iii) eigenvalue estimates for Schr\"odinger operators on ${\mathbb R}n$ or Lipschitz domains of ${\mathbb R}n$.