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Superexponential estimates and weighted lower bounds for the square function | Paata Ivanisvili
; Sergei Treil
; | Date: |
19 Nov 2017 | 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 geqlambda }| 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 $0<C<1$ independent of $d$ when
$d o infty$.
For $d=1$ this is a classical result of Chang--Wilson--Wolff [4]; however, in
the case $d>1$ 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]. | Source: | arXiv, 1711.7084 | Services: | Forum | Review | PDF | Favorites |
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