International Mathematics Research Notices ( IF 1.452 ) Pub Date : 2020-01-08 , DOI: 10.1093/imrn/rnz326
Chen P, Duong X, Wu L, et al.

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 Hölder 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 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 \${\mathbb R}^n\$ or Lipschitz domains of \${\mathbb R}^n\$.

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