On an interpolation theorem of Hunt-Sjölin
For a real-valued integrable function \(f\) on \([0,2 \pi]\), let \(S_n (f;x)\) denote the \(n\)th partial sum of the trigonometric Fourier series of \(f\) and let \(S^*f(x) =\sup \{|S_n (f;x) |: x\geq 0\}\), \(x\in [0,2\pi]\), denote the majorant function. Let \(\psi (t)\) be a concave function on \([0,\infty)\) satisfying: \(\psi(t) =\begin{cases} t, \text{ if } 0\leq t\leq e^2 \\ \text{concave, if } t>e^2,\;\psi(t) \uparrow+ \infty \text{ as } t\to \infty; \end{cases}\) \(\psi '(t)\) is non-increasing on \([0,\infty)\) with \(\psi' (e^2)=1\) and \(\lim_{t\to \infty} \int^t_{e^2} {\psi'(s) \over s} ds = +\infty\); \(\psi''(t)\) is piecewise continuous on \([0,\infty)\) with \(-\infty< \int^\infty_{e^2} {\psi'' (t) \over t} dt \leq 0\). Let \(\varphi(t) =\begin{cases} t(\log t) \int^t_{e^2} {\psi'(s) \over s} ds, \quad & \text{if } t\geq e^2 \\ 0 \quad & \text{if } 0\leq t< e^2. \end{cases}\) The main result of this paper is the following: Suppose \(\int^t_{e^2} {\psi'(s) \over s} ds \geq a_0 (\log t)^\varepsilon\) for some constants \(a_0>0\) and \(0< \varepsilon <1\). Then \(\int^{2\pi}_0 \psi (S^*(f)(x)) dx\leq C (\varepsilon) \int^{2\pi}_0 \varphi (|f(x) vert)dx +C (\varepsilon)\) for some constant \(C (\varepsilon) >0\) depending on \(\varepsilon\).
scientific article; zbMATH DE number 755473