Martingale Hardy Spaces Research Articles

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

Let θ ≥ 0 and p · be a variable exponent, and we introduce a new class of function spaces L p · , θ in a probabilistic setting which unifies and generalizes the variable Lebesgue spaces with θ = 0 and grand Lebesgue spaces with p · ≡ p and θ = 1 . Based on the new spaces, we introduce a kind of Hardy-type spaces, grand martingale Hardy spaces with variable exponents, via the martingale operators. The atomic decompositions and John-Nirenberg theorem shall be discussed in these new Hardy spaces.

Modulus of continuity and convergence of subsequences of Vilenkin–Fejér means in martingale Hardy spaces

Abstract In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 < p < 1 2 {0<p<\frac{1}{2}} .

Interpolation between noncommutative martingale Hardy and BMO spaces: the case

AbstractLet$\mathcal {M}$be a semifinite von Nemann algebra equipped with an increasing filtration$(\mathcal {M}_n)_{n\geq 1}$of (semifinite) von Neumann subalgebras of$\mathcal {M}$. For$0<p <\infty $, let$\mathsf {h}_p^c(\mathcal {M})$denote the noncommutative column conditioned martingale Hardy space and$\mathsf {bmo}^c(\mathcal {M})$denote the column “little” martingale BMO space associated with the filtration$(\mathcal {M}_n)_{n\geq 1}$.We prove the following real interpolation identity: if$0<p <\infty $and$0<\theta <1$, then for$1/r=(1-\theta )/p$,$$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$with equivalent quasi norms.For the case of complex interpolation, we obtain that if$0<p<q<\infty $and$0<\theta <1$, then for$1/r =(1-\theta )/p +\theta /q$,$$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$with equivalent quasi norms.These extend previously known results from$p\geq 1$to the full range$0<p<\infty $. Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned$L_p$-spaces are also shown to form interpolation scale for the full range$0<p<\infty $when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned$L_p$-spaces.We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

New variable martingale Hardy spaces

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces$\mathcal {L}_{p(\cdot )}$defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space$\mathcal {H}_{p(\cdot )}^{s}$with$0<p_{-}\leq p_{+}\leq 1$can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

Martingale Musielak–Orlicz–Lorentz Hardy Spaces with Applications to Dyadic Fourier Analysis

Let $$(\Omega ,{\mathcal {F}},{\mathbb {P}})$$ be a probability space, $$\varphi :\ \Omega \times [0,\infty )\rightarrow [0,\infty )$$ a Musielak–Orlicz function, and $$q\in (0,\infty ]$$ . In this article, the authors introduce five martingale Musielak–Orlicz–Lorentz Hardy spaces and prove that these new spaces have some important features such as atomic characterizations, the boundedness of $$\sigma $$ -sublinear operators, and martingale inequalities. This new scale of martingale Hardy spaces requires the introduction of the Musielak–Orlicz–Lorentz space $$L^{\varphi ,q}(\Omega )$$ . In particular, the authors show that this Lorentz type space has some fundamental properties including the completeness, the convergence, real interpolations, and the Fefferman–Stein vector-valued inequality for the Doob maximal operator. As applications, the authors prove that the maximal Fejer operator is bounded from the martingale Musielak–Orlicz–Lorentz Hardy space $$H_{\varphi ,q}[0,1)$$ to $$L^{\varphi ,q}[0,1)$$ , which further implies some convergence results of the Fejer means. Moreover, all the above results are new even for Musielak–Orlicz functions with particular structure such as weight, weight Orlicz, and double-phase growth. The main approach used in this article can be viewed as a combination of the stopping time argument in probability theory and the real-variable technique of function spaces in harmonic analysis.

Martingale inequalities and fractional integral operator in variable Hardy-Lorentz spaces

Let (Ω,F,P) be a complete probability space. We introduce variable Lorentz space Lp(⋅),q(Ω) defined by rearrangement functions and its related properties. Then, we establish martingale inequalities among these martingale Hardy-Lorentz spaces Hp(⋅),q(Ω) by applying the interpolation theorem. Furthermore, we study the boundedness of the fractional integral operator in variable martingale Hardy spaces Hp(⋅)M(Ω) and Hp(⋅),qM(Ω).

Interpolation of martingale Orlicz–Hardy spaces

We investigate the real interpolation spaces between martingale Orlicz–Hardy spaces $$H_{\Phi}^s$$ and martingale Hardy spaces $$H_{\infty}^s$$ via atomic decomposition. Moreover, its corresponding weak version is also established.

Real interpolations of martingale Hardy-Lorentz-Karamata spaces

In this paper, we investigate real interpolations in martingale Hardy-Lorentz-Karamata spaces. By introducing new type of martingale Hardy spaces, we find equivalent characterizations for the limiting reiteration of the real interpolations. As an application, we obtain the boundedness of fractional integrals in martingale Hardy-Lorentz-Karamata spaces.

Hoeffding decomposition in H^1 spaces

The well known result of Bourgain and Kwapień states that the projection P_{le m} onto the subspace of the Hilbert space L^2left( Omega ^infty right) spanned by functions dependent on at most m variables is bounded in L^p with norm le c_p^m for 1<p<infty . We will be concerned with two kinds of endpoint estimates. We prove that P_{le m} is bounded on the space H^1left( {mathbb {D}}^infty right) of functions in L^1left( {mathbb {T}}^infty right) analytic in each variable. We also prove that P_{le 2} is bounded on the martingale Hardy space associated with a natural double-indexed filtration and, more generally, we exhibit a multiple indexed martingale Hardy space which contains H^1left( {mathbb {D}}^infty right) as a subspace and P_{le m} is bounded on it.

Applications of mixed martingale Hardy spaces in Fourier analysis

In this paper we consider the martingale Hardy spaces defined with the help of the mixed Lp→-norm. Using atomic decomposition and Doob's inequality, some applications will be shown in Fourier-analysis, such as the uniform boundedness of the partial sums of the Walsh-Fourier series on Lp→(1<p→<∞) and the boundedness of the Fejér maximal operator from Hp→ to Lp→(1/2<p→<∞). As a consequence of the boundedness, we get some almost everywhere and norm convergence results.

Real interpolation for variable martingale Hardy spaces

In this paper, we consider the real interpolation for variable martingale Hardy spaces. Let 0<q≤∞ and 0<θ<1. Then (Hp(⋅)s,H∞s)θ,q=Hp(⋅)/(1−θ),qs, where the variable exponent p(⋅) is a measurable function defined on a probability space. The variable exponent does not need any restriction.

Mixed Martingale Hardy Spaces

In this paper, we consider the martingale Hardy spaces defined with the help of the mixed L_{overrightarrow{p}}-norm. Five mixed martingale Hardy spaces will be investigated: H_{overrightarrow{p}}^{s}, H_{overrightarrow{p}}^S, H_{overrightarrow{p}}^M, mathcal {P}_{overrightarrow{p}}, and mathcal {Q}_{overrightarrow{p}}. Several results are proved for these spaces, like atomic decompositions, Doob’s inequality, boundedness, martingale inequalities, and the generalization of the well-known Burkholder–Davis–Gundy inequality.

On the boundedness of subsequences of Vilenkin-Fejér means on the martingale Hardy spaces

In this paper we characterize subsequences of Fejer means with respect to Vilenkin systems, which are bounded from the Hardy space H-p to the Lebesgue space L-p, for all 0 < p < 1/2. The resu ...

Variable martingale Hardy spaces and their applications in Fourier analysis

Let $p(\cdot)$ be a measurable function defined on a probability space satisfying $$ 0 \lt p_-:={\rm ess}\inf_{x\in \Omega}p(x)\leq {\rm ess}\sup_{x\in\Omega}p(x)=:p_+ \lt \infty. $$ We investigate five types of martingale Hardy spaces $H_{p(\cdot)}$ a

Atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applications

Let (Ω,F,P) be a probability space, and let φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, we establish the atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces WHφs(Ω), WHφM(Ω), WHφS(Ω), WPφ(Ω), and WQφ(Ω). We then use these atomic characterizations to obtain the boundedness of σ-sublinear operators from weak martingale Musielak–Orlicz Hardy spaces to weak Musielak–Orlicz spaces, as well as some martingale inequalities which further clarify the relationships among these weak martingale Musielak–Orlicz Hardy spaces. All these results improve and generalize the corresponding results on weak martingale Orlicz–Hardy spaces. Moreover, we improve all the known results on weak martingale Musielak–Orlicz Hardy spaces. In particular, both the boundedness of σ-sublinear operators and the martingale inequalities, for weak weighted martingale Hardy spaces as well as for weak weighted martingale Orlicz–Hardy spaces, are new.

Orlicz-Lorentz Hardy martingale spaces

Martingale Hardy spaces are widely studied in the field of mathematical physics and probability. In this paper, we develop the theory of Orlicz-Lorentz Hardy martingale spaces, which are much more wider than the classical Lorentz Hardy martingale spaces. More precisely, we first investigate several basic properties of Orlicz-Lorentz spaces, and then construct the atomic decomposition theorems of these martingale function spaces. Also, we establish the dual theorem of Orlicz-Lorentz Hardy spaces for martingales. Furthermore, we study the boundedness of generalized fractional integral operators Iϕ in this new framework, where ϕ is a non-negative concave function. The results partially extend the very recent results Jiao et al. (2017) [21].

A Remark on John–Nirenberg Theorem for Martingales

The paper is mainly devoted to establishing an extension of the John–Nirenberg theorem for martingales. More precisely, let 1 < p < ∞ and 0 < q < ∞. If the stochastic basis (Fn)n≥0 is regular, then BMOp,q = BMO1 with the equivalent norms. Our method is to use a new atomic decomposition construction of the martingale Hardy space.

Square Functions for Noncommutative Differentially Subordinate Martingales

We prove inequalities involving noncommutative differentially subordinate martingales. More precisely, we prove that if $x$ is a self-adjoint noncommutative martingale and $y$ is weakly differentially subordinate to $x$ then $y$ admits a decomposition $dy=a +b +c$ (resp. $dy=z +w$) where $a$, $b$, and $c$ are adapted sequences (resp. $z$ and $w$ are martingale difference sequences) such that: $$ \Big\| (a_n)_{n\geq 1}\Big\|_{L_{1,\infty}({\mathcal M}\overline{\otimes}\ell_\infty)} +\Big\| \Big(\sum_{n\geq 1} \mathcal{E}_{n-1}|b_n|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} + \Big\| \Big(\sum_{n\geq 1} \mathcal{E}_{n-1}|c_n^*|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} \leq C\big\| x \big\|_1 $$ (resp. $$ \Big\| \Big(\sum_{n\geq1} |z_n|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} + \Big\| \Big(\sum_{n\geq 1} |w_n^*|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} \leq C\big\| x \big\|_1). $$ We also prove strong-type $(p,p)$ versions of the above weak-type results for $1<p<2$. In order to provide more insights into the interactions between noncommutative differential subordinations and martingale Hardy spaces when $1\leq p<2$, we also provide several martingale inequalities with sharp constants which are new and of independent interest. As a byproduct of our approach, we obtain new and constructive proofs of both the noncommutative Burkholder-Gundy inequalities and the noncommutative Burkholder/Rosenthal inequalities for $1<p<2$ with the optimal order of the constants when $p \to 1$.

A sharp boundedness result for restricted maximal operators of Vilenkin–Fourier series on martingale Hardy spaces

Abstract The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 &lt; p ≤ 1 {0&lt;p\leq 1} . We also prove that the result is sharp in a particular sense.

On the Convergence of Partial Sums with Respect to Vilenkin System on the Martingale Hardy Spaces

In this paper, we derive characterizations of boundedness of subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces Hp when 0 < p < 1. Moreover, we find necessary and sufficient conditions for the modulus of continuity of martingales f ∈ Hp, which provide convergence of subsequences of partial sums on the martingale Hardy spaces Hp. It is also proved that these results are the best possible in a special sense. As applications, some known and new results are pointed out.