Superexponential estimates and weighted lower bounds for the square function
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Superexponential estimates and weighted lower bounds for the square function

  • Author(s): Ivanisvili, Paata
  • Treil, Sergei
  • et al.

Published Web Location

https://doi.org/10.1090/tran/7795Creative Commons 'BY' version 4.0 license
Abstract

We prove the following superexponential distribution inequality: for any integrable $g$ on $[0,1)^{d}$ with zero average, and any $\lambda>0$ \[ |\{ x \in [0,1)^{d} \; :\; g \geq\lambda \}| \leq e^{- \lambda^{2}/(2^{d}\|S(g)\|_{\infty}^{2})}, \] where $S(g)$ denotes the classical dyadic square function in $[0,1)^{d}$. The estimate is sharp when dimension $d$ tends to infinity in the sense that the constant $2^{d}$ in the denominator cannot be replaced by $C2^{d}$ with $01$ they work with a special square function $S_\infty$, and their result does not imply the estimates for the classical square function. Using good $\lambda$ inequalities technique we then obtain unweighted and weighted $L^p$ lower bounds for $S$; to get the corresponding good $\lambda$ inequalities we need to modify the classical construction. We also show how to obtain our superexponential distribution inequality (although with worse constants) from the weighted $L^2$ lower bounds for $S$, obtained in [5].

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