On maximal operator of partial sums of Fourier series in Orlicz spaces (Q1271940)

Let \(f\) be an integrable function on \(T=[-\pi, \pi]\) and let \(S_n(f;x)\) be the \(n\)-th partial sum of the trigonometric Fourier series of \(f\). The maximal partial sum operator is \[ S^*(f)(x)= \sup \{| S_n(f;x)| :n \geq 0\},\quad x \in T. \] The Hunt-Carleson theorem, that the operator is bounded in \(L^p(T)\) and on weak \(L^1\), has been extended by the author to Orlicz spaces for the function \(\varphi_{\gamma}(t)= \exp(t^{\gamma})- t^{\gamma}- 1\). He proved formerly that if there is an \(\varepsilon_0> 0\) such that \[ \int_{-\pi}^{\pi} \varphi_{\gamma}(\varepsilon_0| f(x)|) dx < \infty, \] then there is an \(\varepsilon_1>0\) such that \[ \int_{-\pi}^{\pi} \varphi_{\gamma/(\gamma +1)}(\varepsilon_1| S^*f(x)|) dx < \infty, \] and that there is a function \(f\) that satisfies the above condition such that \[ \int_{-\pi}^{\pi} \varphi_{\gamma/(\gamma +1)}(\alpha_0| S^*f(x)|) dx= \infty, \text{ for some } \alpha_0> 0, \] for which \[ \limsup_{n \rightarrow \infty} \int_{-\pi}^{\pi} \varphi_{\gamma/(\gamma +1)}(\alpha_0| S_n(f;x)- f(x)|) dx= \infty. \] In this paper, the author investigates these questions for general \(\varphi\) that are strictly increasing continuous functions on \([0,\infty)\) such that \( \varphi(0)= 0\), \(\varphi(t)>0\) for \(t>0\), and \(\varphi(t) \uparrow \infty\) as \(t\uparrow \infty\). Denote by \(\varphi \in \Phi\) if \(\varphi\) satisfies these conditions. Let \( L^{\varphi}= \bigcup_{\varepsilon>0} \varphi(\varepsilon L) \) and \( M^{\varphi}= \bigcap_{\alpha>0} \varphi(\alpha L). \) A function \(\varphi\) is said to satisfy the \(\Delta^2\) condition if \( \varphi(t)^2 \leq \varphi(C_1t),\;t \geq t_1. \) After giving some properties of functions satisfying the \(\Delta^2\) condition, the author shows that for any \(\varphi \in \Phi\) satisfying the \(\Delta^2\) condition, there is an \(f_0\) and a sequence \(\{N_k\}\) such that \(S_{N_k}(f_0;x) \geq C \varphi^{-1}(N_k) \log N_k\), for all \(0 \leq x \leq \frac{\pi}{3N_k}\), and if \(\psi \in \Phi\) satisfies \(\psi^{-1}(s) \leq C_3 \varphi^{-1}(s) \log s\), \(s \geq s_3\), then there is a function \(f_0\) in \(L^{\varphi}\) for which \(S^*(f_0) \notin M^{\psi}\) and for sufficiently large \(\alpha_0\), we have \[ \limsup_{n \rightarrow \infty} \int_{-\pi}^{\pi} \psi (\alpha_0| S_n(f;x)- f(x)|) dx= \infty. \] Functions having a power series expansion are considered, as well.