scholarly journals The weak type inequality for the maximal operator of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series

2008 ◽  
Vol 154 (2) ◽  
pp. 161-180 ◽  
Author(s):  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Maximal Operator ◽  
Weak Type ◽  
Type Inequality ◽  
Two Dimensional ◽  
Fejér Means ◽  
Weak Type Inequality

A weak type inequality for the two-dimensional diagonal Sunouchi operator on the Hardy space

2011 ◽  
Vol 18 (1) ◽  
pp. 67-81
Author(s):  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Hardy Space ◽  
Weak Type ◽  
Type Inequality ◽  
Partial Sums ◽  
Two Dimensional ◽  
Fejér Means ◽  
Weak Type Inequality

Abstract Define the two dimensional diagonal Sunouchi operator where S 2 n , 2 n ƒ and σ 2 n ƒ are the (2 n , 2 n )th cubic-partial sums and 2 n th Marcinkiewicz–Fejér means of a two-dimensional Walsh–Fourier series. The main aim of this paper is to prove that the operator is bounded from the Hardy space H 1/2 to the weak L 1/2 space and is not bounded from the Hardy space H 1/2 to the space L 1/2.

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.

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