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Maximal operators of Cesáro means with varying parameters of Walsh-Fourier series. (English) Zbl 1449.42045

The Cesaro means of the Walsh-Fourier series of an integrable function \(f\) on \([0,1)\) are \(\frac1{A^{\alpha_n}_{n-1}}\sum_{j=0}^{n-1} A^{\alpha_n}_{n-1-j}\hat f(j)w_j(x)\) where \(A^{\alpha_n}_n=\frac {(1+\alpha_n)\cdots (n+\alpha_n)}{n!}\), \(w_n(x)=(-1)^{\sum_{j=0}^\infty n_jx_j}\) when \(n=\sum_{k=0}^\infty n_k2^k\), \(x=\sum_{j=0}^\infty x_j2^{-j-1}\) for \(x\in [0,1)\) and \(\hat f(j)=\int_0^1 f(x)w_j(x)\, dx\). These means can also be written as convolutions in the form \( f*K^{\alpha_n}_n \) where \(K^{\alpha_n}_n=\frac 1{A^{\alpha_n}_{n-1}}\sum_{j=0}^{n-1} A^{\alpha_n}_{n-1-j}w_j\) when the group operation used in the definition of the convolution is \(x\dot + y=\sum_{j=0}^\infty \vert x_j-y_j\vert 2^{-j-1}\). The main result of the paper is that for every \(p>0\) there is a constant \(c_p<\infty\) such that \(\Vert \sup_{N\in \mathbb{N}} \vert f*\vert K^{\alpha_N}_{2^N}\vert \Vert_{L_p}\leq c_p\Vert f^*\Vert_{L_p}\) when \(0<\alpha_N<1\) and where \(f^*(x)= \sup_{n\in \mathbb{N}} \frac 1{\vert I_n(x)\vert} \int_{I_n(x)}f(t)\, dt\vert \) where \(I_n(x)\) is the dyadic interval \([\frac {j}{2^n},\frac {j+1}{2^n})\) containing \(x\).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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