Projects per year
Abstract
Let X be a metric space with a doubling measure. Let L be a nonnegative selfadjoint 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 exponentialsquare 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 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 R^{n} or Lipschitz domains of R^{n}.
Original language  English 

Pages (fromto)  18057–18117 
Number of pages  61 
Journal  International Mathematics Research Notices 
Volume  2021 
Issue number  23 
DOIs  
Publication status  Published  Dec 2021 
Fingerprint
Dive into the research topics of 'Exponentialsquare integrability, weighted inequalities for the square functions associated to operators, and applications'. Together they form a unique fingerprint.Projects
 1 Finished

Multiparameter Harmonic Analysis: Weighted Estimates for Singular Integrals
Duong, X., Ward, L., Li, J., Lacey, M., Pipher, J. & MQRES, M.
16/02/16 → 30/06/20
Project: Research