Weak type (p, p) bounds for Schrodinger groups via generalized Gaussian estimates

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:495

Article

Fan, Zhijie¹ 

[1] Sun Yat Sen Zhongshan Univ, Dept Math, Guangzhou 510275, Peoples R China

关键词: Weak type (p, p) bounds;    Schrodinger group;    Generalized Gaussian estimates;    Off-diagonal estimates;    Space of homogeneous type;   

DOI  :  10.1016/j.jmaa.2020.124766

来源: Elsevier

【 摘 要 】

Let L be a non-negative self-adjoint operator acting on L-2 (X), where X is a space of homogeneous type with a dimension n. Suppose that the heat operator e(-tL) satisfies the generalized Gaussian (p(0) , p'(0))-estimates of order m for some 1 < p(0) < 2. It is known that the operator (I+L) _seitL is bounded on L-p (X) for s >= n vertical bar 1/2-1/p vertical bar and p is an element of (p(0),p'(0)) (see for example, [5,7,9,10,13,2E1 ). In this paper we study the endpoint case p = p(0) and show that for s(0) = n vertical bar 1/2 - 1/p(0)vertical bar, the operator (I + L) -s(0) e(itL)( )is of weak type (p(0),p(0)), that is, there is a constant C > 0, independent of t and f so that mu ({x: vertical bar(I + L) -s(0) e(itL) f(X)vertical bar> alpha}) <= C(1 + vertical bar t vertical bar)(n(1-p)(/2) )(0)(vertical bar vertical bar f vertical bar vertical bar p(0)/alpha)(p0), t is an element of R for a > 0 when mu(X) =infinity, and a > (vertical bar vertical bar f vertical bar vertical bar p(0)/mu(X))(P0) when mu(X) < infinity. Our results can be applied to Schriidinger operators with rough potentials and higher order elliptic operators with bounded measurable coefficients although in general, their semigroups fail to satisfy Gaussian upper bounds. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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