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Subgaussian estimates in probability and harmonic analysis

Subgaussian estimates in probability and harmonic analysis We will discuss subgaussian estimates in harmonic analysis involving the non-tangential maximal function $$N_{\alpha }f$$ N α f and the area function $$A_{\beta } f$$ A β f of a function f on $${\mathbb {R}}^n.$$ R n . We will first introduce subgaussian estimates in the setting of martingales; these then lead to analogous estimates for harmonic functions. Among the consequences of these are sharp $$L^p$$ L p inequalities $$\Vert N_{\alpha }f\Vert _p \le C_p \Vert A_{\beta }f\Vert _p$$ ‖ N α f ‖ p ≤ C p ‖ A β f ‖ p ; here $$C_p= O(\sqrt{p})$$ C p = O ( p ) as $$p \rightarrow \infty $$ p → ∞ and this order is sharp. The subgaussian estimates produce Laws of the Iterated Logarithm (LILs) involving the non-tangential maximal function and area function. These ideas are also applied to lacunary series of more general functions to yield LILs. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

Subgaussian estimates in probability and harmonic analysis

The Journal of Analysis , Volume 26 (2) – Nov 9, 2018

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References (18)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Forum D'Analystes, Chennai
Subject
Mathematics; Analysis; Functional Analysis; Abstract Harmonic Analysis; Special Functions; Fourier Analysis; Measure and Integration
ISSN
0971-3611
eISSN
2367-2501
DOI
10.1007/s41478-018-0136-z
Publisher site
See Article on Publisher Site

Abstract

We will discuss subgaussian estimates in harmonic analysis involving the non-tangential maximal function $$N_{\alpha }f$$ N α f and the area function $$A_{\beta } f$$ A β f of a function f on $${\mathbb {R}}^n.$$ R n . We will first introduce subgaussian estimates in the setting of martingales; these then lead to analogous estimates for harmonic functions. Among the consequences of these are sharp $$L^p$$ L p inequalities $$\Vert N_{\alpha }f\Vert _p \le C_p \Vert A_{\beta }f\Vert _p$$ ‖ N α f ‖ p ≤ C p ‖ A β f ‖ p ; here $$C_p= O(\sqrt{p})$$ C p = O ( p ) as $$p \rightarrow \infty $$ p → ∞ and this order is sharp. The subgaussian estimates produce Laws of the Iterated Logarithm (LILs) involving the non-tangential maximal function and area function. These ideas are also applied to lacunary series of more general functions to yield LILs.

Journal

The Journal of AnalysisSpringer Journals

Published: Nov 9, 2018

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