Variable martingale Hardy-Lorentz-Karamata spaces and their applications in Fourier Analysis

In this paper, we introduce a new class of function spaces, which unify and generalize Lorentz-Karamata spaces, variable Lorentz spaces and other several classical function spaces. Based on the new spaces, we develop the theory of variable martingale Hardy-Lorentz-Karamata spaces and apply it to Fourier Analysis. To be precise, we discuss the basic properties of Lorentz-Karamata spaces with variable exponents. We introduce five variable martingale Hardy-Lorentz-Karamata spaces and characterize them via simple atoms as well as via atoms. As applications of the atomic decompositions, dual theorems and the generalized John-Nirenberg theorem for the new framework are presented. Moreover, we obtain the boundedness of $\sigma$-sublinear operator defined on variable martingale Hardy-Lorentz-Karamata spaces, which leads to martingale inequalities and the relation of the five variable martingale Hardy-Lorentz-Karamata spaces. Also, we investigate the boundedness of fractional integral operators in this new framework. Finally, we deal with the applications of variable martingale Hardy-Lorentz-Karamata spaces in Fourier analysis by using the previous results. More precisely, we show that the partial sums of the Walsh-Fourier series converge to the function in norm if $f\in L_{p(\cdot),q,b}$ with $1<p_-\le p_+<\infty$. The Fej\'{e}r summability method is also studied and it is proved that the maximal Fej\'{e}r operator is bounded from variable martingale Hardy-Lorentz-Karamata spaces to variable Lorentz-Karamata spaces. As a consequence, we get conclusions about almost everywhere and norm convergence of Fej\'{e}r means. The results obtained in this paper generalize the results for martingale Hardy-Lorentz-Karamata spaces and variable martingale Hardy-Lorentz spaces. Especially, we remove the condition that $b$ is nondecreasing in previous literature.