Applications of martingale Hardy Orlicz–Lorentz–Karamata theory in Fourier analysis

In this article, we discuss the applications of martingale Hardy Orlicz–Lorentz–Karamata spaces in Fourier analysis. More precisely, we show that the partial sums of the Walsh–Fourier series converge to the function in norm if \(f\in L_{\Phi ,q,b}\) $f\in L_{\mathrm{\Phi },q,b}$ with \(1<p_-\le p_+<\infty \) $1<p_{-}\le p_{+}<\infty$ . The equivalence of maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces is presented. The Fejér summability method is also studied and it is proved that the maximal Fejér operator is bounded from martingale Hardy Orlicz–Lorentz–Karamata spaces to Orlicz–Lorentz–Karamata spaces. As a consequence, we obtain conclusions about almost everywhere and norm convergence of Fejér means.