March 2023 Weak Type (1,1) Bounds for Schrödinger Groups
Peng Chen, Xuan Thinh Duong, Ji Li, Liang Song, Lixin Yan
Michigan Math. J. 73(1): 97-122 (March 2023). DOI: 10.1307/mmj/20205900


Let L be a nonnegative self-adjoint operator acting on L2(X), where X is a space of homogeneous type of dimension n. Suppose that the heat kernel of L satisfies a Gaussian upper bound. It is known that the operator (I+L)seitL is bounded on Lp(X) for s>n|1/21/p| and p(1,) (see, e.g., [7, 22, 33]). The index s=n|1/21/p| was only obtained recently in [9, 10], and this range of s is sharp since it is precisely the range known in the case where L is the Laplace operator Δ on X=Rn [30]. In this paper, we establish that for p=1, the operator (1+L)n/2eitL is of weak type (1,1), that is, there is a constant C, independent of t and f, such that


(for λ>0 when μ(X)= and λ>μ(X)1fL1(X) when μ(X)<). Moreover, we also show that the index n/2 is sharp when L is the Laplacian on Rn by providing an example.

Our results are applicable to Schrödinger groups for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with nonnegative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular nondoubling domains of Euclidean spaces.


Download Citation

Peng Chen. Xuan Thinh Duong. Ji Li. Liang Song. Lixin Yan. "Weak Type (1,1) Bounds for Schrödinger Groups." Michigan Math. J. 73 (1) 97 - 122, March 2023.


Received: 1 April 2020; Revised: 14 February 2021; Published: March 2023
First available in Project Euclid: 14 October 2021

MathSciNet: MR4555222
Digital Object Identifier: 10.1307/mmj/20205900

Primary: 35J10 , 42B37 , 47F05

Rights: Copyright © 2023 The University of Michigan


This article is only available to subscribers.
It is not available for individual sale.

Vol.73 • No. 1 • March 2023
Back to Top