{
  "id": "1609.01516",
  "version": "v1",
  "published": "2016-09-06T12:39:20.000Z",
  "updated": "2016-09-06T12:39:20.000Z",
  "title": "Exponential-square integrability, weighted inequalities for the square functions associated to operators and applications",
  "authors": [
    "Peng Chen",
    "Xuan Thinh Duong",
    "Liangchuan Wu",
    "Lixin Yan"
  ],
  "comment": "1 figure",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "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 ${\\mathbb R}^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-06T12:39:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "square function",
      "exponential-square integrability",
      "weighted inequalities",
      "satisfy gaussian upper bounds",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}