On the convergence of Walsh series in spaces close to \(L^ \infty\). (Q1810021)

Let \((G,\Sigma,\mu)\) be a space with finite measure \(\mu\) and \(\mu(G)=1\), \(p>1\), \(\alpha\geq 1\). By \({\widehat L}_{p,\alpha}(G)\) we denote the set of functions for which \(\| f\| _{{\widehat p},\alpha}=(\int^{\infty}_1(\| f\| _x/x^{\alpha})^p\, dx)^{1/p}<\infty\), where \(\| f\| _x\) is the norm in \(L_x(G)\). This set \({\widehat L}_{p,\alpha}(G)\) is a Banach space with norm \(\| f\| _{{\widehat p},\alpha}\). If \(q>0\), \(1/p+1/q=\alpha\), then for any \(\gamma>1\) and \(r>q\) we have \(L^0(\gamma^{| x| ^r})\subset {\widehat L}_{p,\alpha}(G)\subset L^0(\gamma^{| x| ^q})\), where \(L^0(\phi)\) is an Orlicz class of functions \(f\) with the property \(\int_G\phi(f(x))dx<\infty\). The main result is Theorem 3. Suppose that \(p>1\), \(\alpha\geq 1\) and \(S_m(f)\) are partial sums of the Walsh-Fourier series of a function \(f\). Then 1) for any \(f\in{\widehat L}_{p,\alpha}[0,1]\) we have \(\| S_m(f)\| _{{\widehat p},\alpha+1} \leq C_{p,\alpha}\| f\| _{{\widehat p},\alpha}\); 2) for any \(f\in{\widehat L}_{p,\alpha}[0,1]\) we have \(\lim_{m\to\infty}\| S_m(f)-f\| _{{\widehat p},\alpha+1}=0\); 3) there exists a function \(f\in{\widehat L}_{p,\alpha}[0,1]\), such that the norms \(\| S_m(f)\| _{{\widehat p},\alpha+1-\varepsilon}\) are unbounded for all \(\varepsilon>0\).