Variable Martingale Hardy Spaces and Their Applications in Fourier Analysis

Let $p(\cdot)$ be a measurable function defined on a probability space satisfying $0<p_-:={\rm ess}\inf_{x\in \Omega}p(x)\leq {\rm ess}\sup_{x\in\Omega}p(x)=:p_+<\infty$. We investigate five types of martingale Hardy spaces $H_{p(\cdot)}$ and $H_{p(\cdot),q}$ and prove their atomic decompositions when each $\sigma$-algebra $\mathcal F_n$ is generated by countably many atoms. Martingale inequalities and the relation of the different martingale Hardy spaces are proved as application of the atomic decomposition. In order to get these results, we introduce the following condition to replace (generalize) the so-called log-H\"{o}lder continuity condition in harmonic analysis: \mathbb P(A)^{p_-(A)-p_+(A)}\leq C_{p(\cdot)} \quad \mbox{ for all atom $A$}. Some applications in Fourier analysis are given by use of the previous results. We generalize the classical results and show that the partial sums of the Walsh-Fourier series converge to the function in norm if $f \in L_{p(\cdot)}$ or $f \in L_{p(\cdot),q}$ and $p_->1$. The boundedness of the maximal Fej{\'e}r operator on $H_{p(\cdot)}$ and $H_{p(\cdot),q}$ is proved whenever $p_->1/2$ and the condition $\frac{1}{p_-}-\frac{1}{p_+} <1$ hold. It is surprising that this last condition does not appear for trigonometric Fourier series. One of the key points of the proof is that we introduce two new dyadic maximal operators and prove their boundedness on $L_{p(\cdot)}$ with $p_->1$. The method we use to prove these results is new even in the classical case. As a consequence, we obtain theorems about almost everywhere and norm convergence of the Fej\'er means.