A weak type inequality for the maximal operator of \((C,\alpha)\)-means of Fourier series with respect to the Walsh-Kaczmarz system
The \((C,\alpha)\) means of the martingale \(f\) with respect to the Walsh-Kaczmarz system are \[ \sigma_n^{\alpha,\kappa}(f,x)=\frac{1}{A_{n-1}^\alpha}\sum_{j=0}^{n-1}A_{n-j-1}^\alpha\widehat{f}^\kappa(j)\kappa_j(x). \] Here \[ A_n^\alpha=\frac{(1+\alpha)\cdots(n+\alpha)}{n!}, \] and \[ \widehat{f}^\kappa(j)=\lim_{k\rightarrow\infty}\int_Gf^{(k)}(x)\kappa_j(x) dx \] is the \(j\)th Walsh-Fourier coefficient for the \((\kappa_j(x))_{j=0}^\infty\) Walsh-Kaczmarz system (\(G\) is the Walsh group). For the martingale \(f\) the authors consider the maximal operator \[ \sigma_*^{\alpha,\kappa}f(x)=\sup_n| \sigma_n^{\alpha,\kappa}(f,x)| . \] It was proven by \textit{P. Simon} [J. Approx. Theory 127, No. 1, 39--60 (2004; Zbl 1043.42021)] that this operator is bounded from the Hardy space \(H_p\) into the Lebesgue space \(L_p\) when \(p>1/(1+\alpha)\). In the present paper the authors prove that this boundedness does not hold if \(p\geq1/(1+\alpha)\). In addition, when \(p=1/(1+\alpha)\), the maximal operator \(\sigma_*^{\alpha,\kappa}\) is bounded from the martingale Hardy space \(H_{1/(1+\alpha)}\) to the space weak-\(L_{1/(1+\alpha)}\).
scientific article; zbMATH DE number 5770414
scientific article