New martingale inequalities and applications to Fourier analysis

Abstract

Let (Ω, F , P) be a probability space and ϕ : Ω × [0, ∞) → [0, ∞) be a Musielak-Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak-Orlicz space L ϕ (Ω). Using this and extrapolation method, the authors then establish a Fefferman-Stein vector-valued Doob maximal inequality on L ϕ (Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for L ϕ (Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak-Orlicz Hardy spaces H s ϕ (Ω), P ϕ (Ω), Q ϕ (Ω), H S ϕ (Ω) and H M ϕ (Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak-Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on H S ϕ (Ω) and H M ϕ (Ω), the authors obtain the Burkholder-Davis-Gundy inequality associated with Musielak-Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from H ϕ [0, 1) to L ϕ [0, 1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

Atomic Decompositions and John-Nirenberg Theorem of Grand Martingale Hardy Spaces with Variable Exponents

“…The method of atomic characterization plays an useful tool in martingale theory (see for instance [1,4,6,[31][32][33]). We shall construct the atomic characterizations for grand Hardy martingale spaces with variable exponents in this section.…”

Martingale Transforms between Martingale Hardy-amalgam Spaces

“…However, in 1966, Burkholder had already introduced the notion of martingale transforms [9] which became an indispensable tool in the study of some relations between classical martingale Hardy spaces, mostly the predictive spaces P p in the classical settings [6,10]. In the past years, various authors have generalized the classical martingale Hardy spaces of the classical Lebesgue spaces to Lorentz spaces, Orlicz spaces, and Orlicz-Musielak spaces [11][12][13][14][15][16][17].…”

Mixed Martingale Hardy Spaces

“…Moreover, Musielak-Orlicz-Hardy spaces were studied in Yang et al [44]. These results were also investigated for martingale Hardy spaces in Jiao et al [26,27] and Xie et al [42]. The mixed-norm classical Hardy spaces were intensively studied by Huang et al [22,23] and Huang and Yang [24].…”