Higher rank Haar wavelet bases in spaces

2011 ◽  
Vol 18 (3) ◽  
pp. 517-532
Author(s):  
Tengiz Kopaliani
Keyword(s):  
Unconditional Basis ◽  
Function Method ◽  
Haar Wavelet ◽  
Bellman Function ◽  
Wavelet Bases ◽  
Wavelet System ◽  
Higher Rank

Abstract Using the Bellman function method, we prove that a Haar wavelet system of rank N (N ∈ ℕ, N ≥ 2) is an unconditional basis in , 1 < p < ∞, if and only if .

scholarly journals A Remark on Wavelet Bases in Weighted Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Agnieszka Wojciechowska
Keyword(s):  
Unconditional Basis ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weighted Spaces ◽  
Wavelet Bases ◽  
Wavelet System ◽  
Necessary And Sufficient

The paper deals with unconditional wavelet bases in weighted spaces. Inhomogeneous wavelets of Daubechies type are considered. Necessary and sufficient conditions for weights are found for which the wavelet system is an unconditional basis in weighted spaces in dependence on .

A weighted weak-type bound for Haar multipliers

2018 ◽  
Vol 148 (3) ◽  
pp. 643-658
Author(s):  
Adam Osȩkowski
Keyword(s):  
Maximal Operator ◽  
Function Method ◽  
Dual Problem ◽  
Weak Type ◽  
Analogous Result ◽  
Type Inequality ◽  
Bellman Function ◽  
Weak Type Inequality ◽  
Image Position

We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [−1, 1], and is the r-maximal operator, then for any weight w and any function f on [0, 1) we havefor some constant Cr depending only on r. We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md. The proof rests on the Bellman function method; using this technique we establish related weighted Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.

On the Bellman Function Method for Operators on Martingales

2021 ◽  
Vol 103 (3) ◽  
pp. 118-121
Author(s):  
V. A. Borovitskiy ◽  
N. N. Osipov ◽  
A. S. Tselishchev
Keyword(s):  
Function Method ◽  
Bellman Function

Some Sharp Estimates for the Haar System and Other Bases In $L^1(0,1)$

2014 ◽  
Vol 115 (1) ◽  
pp. 123
Author(s):  
Adam Osȩkowski
Keyword(s):  
Nonnegative Integer ◽  
Unconditional Basis ◽  
Haar System ◽  
Bellman Function ◽  
Real Numbers ◽  
Best Constant ◽  
Sharp Estimates

Let $h=(h_k)_{k\geq 0}$ denote the Haar system of functions on $[0,1]$. It is well known that $h$ forms an unconditional basis of $L^p(0,1)$ if and only if $1<p<\infty$, and the purpose of this paper is to study a substitute for this property in the case $p=1$. Precisely, for any $\lambda>0$ we identify the best constant $\beta=\beta_h(\lambda)\in [0,1]$ such that the following holds. If $n$ is an arbitrary nonnegative integer and $a_0$, $a_1$, $a_2$, $\ldots$, $a_n$ are real numbers such that $\bigl\|\sum_{k=0}^n a_kh_k\bigr\|_1\leq 1$, then \[ \Bigl|\Bigl\{x\in [0,1]:\Bigl|\sum_{k=0}^n \varepsilon_ka_kh_k(x)\Bigr|\geq \lambda\Bigr\}\Bigr|\leq \beta, \] for any sequence $\varepsilon_0, \varepsilon_1, \varepsilon_2,\ldots, \varepsilon_n$ of signs. A related bound for an arbitrary basis of $L^1(0,1)$ is also established. The proof rests on the construction of the Bellman function corresponding to the problem.

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