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Exponential-square integrability, weighted inequalities for the square functions associated to operators and applications | | Peng Chen
; Xuan Thinh Duong
; Liangchuan Wu
; Lixin Yan
; | | Date: |
6 Sep 2016 | | Abstract: | 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 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 ${mathbb R^n}$.
We then apply this result to obtain: (i) estimates of the norm on $L^p$ 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 ${mathbb R}^n$. | | Source: | arXiv, 1609.1516 | | Services: | Forum | Review | PDF | Favorites | |
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