Let $p\in(0,1)$, $α:=1/p-1$ and, for any $τ\in [0,\infty)$, $Φ_{p}(τ):=τ/(1+τ^{1-p})$. Let $H^p(\mathbb R^n)$, $h^p(\mathbb R^n)$ and $Λ_{nα}(\mathbb{R}^n)$ be, respectively, the Hardy space, the local Hardy space and the inhomogeneous Lipschitz space on $\mathbb{R}^n$. In this article, applying the inhomogeneous renormalization of wavelets, the authors establish a bilinear decomposition for multiplications of elements in $h^p(\mathbb R^n)$ [or $H^p(\mathbb R^n)$] and $Λ_{nα}(\mathbb{R}^n)$, and prove that these bilinear decompositions are sharp in some sense. As applications, the authors also obtain some estimates of the product of elements in the local Hardy space $h^p(\mathbb R^n)$ with $p\in(0,1]$ and its dual space, respectively, with zero $\lfloor nα\rfloor$-inhomogeneous curl and zero divergence, where $\lfloor nα\rfloor$ denotes the largest integer not greater than $nα$. Moreover, the authors find new structures of $h^{Φ_p}(\mathbb R^n)$ and $H^{Φ_p}(\mathbb R^n)$ by showing that $h^{Φ_p}(\mathbb R^n)=h^1(\mathbb R^n)+h^p(\mathbb R^n)$ and $H^{Φ_p}(\mathbb R^n)=H^1(\mathbb R^n)+H^p(\mathbb R^n)$ with equivalent quasi-norms, and also prove that the dual spaces of both $h^{Φ_p}(\mathbb R^n)$ and $h^p(\mathbb R^n)$ coincide. These results give a complete picture on the multiplication between the local Hardy space and its dual space.
We give a div-curl type lemma for the wedge product of closed differential forms on R^n when they have coefficients respectively in a Hardy space and L^infinity or BMO. In this last case, the wedge product belongs to an appropriate Hardy-Orlicz space.
The main aim of this paper to provide several scales of equivalent conditions for the bilinear Hardy inequalities in the case $1< q, p_1, p_2<\infty$ with $q \geq \max(p_1,p_2)$.
This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H^1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear Calderon-Zygmund operators in a far more abs…
Recently, both the bilinear decompositions $h^1(\mathbb{R}^n)\times \mathrm{\,bmo}(\mathbb{R}^n) \subset L^1 (\mathbb{R}^n)+h_\ast^Φ(\mathbb{R}^n)$ and $h^1(\mathbb{R}^n) \times \mathrm{bmo}(\mathbb{R}^n) \subset L^1 (\mathbb{R}^n) + h^{\log}(\mathbb{R}^n)$ were established. In this article, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains the variant $h_\ast^Φ(\mathbb{R}^n)$ of the local Orlicz Hardy space introduced by A. Bonami and J. Feuto as a special case, and obta…
The aim of this article is to give a complete solution to the problem of the bilinear decompositions of the products of some Hardy spaces $H^p(\mathbb{R}^n)$ and their duals in the case when $p<1$ and near to $1$, via wavelets, paraproducts and the theory of bilinear Calderón-Zygmund operators. Precisely, the authors establish the bilinear decompositions of the product spaces $H^p(\mathbb{R}^n)\times\dotΛ_α (\mathbb{R}^n)$ and $H^p(\mathbb{R}^n)\timesΛ_α(\mathbb{R}^n)$, where, for all $p\in(\frac{n}{n+1},\,1)$ and $α:=n(\frac{1}{p}-1)$, $H^p(\mathbb{R}^n)$ denotes the classical real Hardy spac…
More work needs to be done to move from the tail to the averages themselves. So at this time we prefer to withdraw the paper about the averages. However a previous version of the paper which deals with the tail has been checked and we believe it to be complete and correct.
Let $(X,\mathcal{B}, μ, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $\dis R^*:(f, g) \in L^1\times L^1 \to R^*(f, g)(x) = \sup_{n} \frac{f(T^nx)g(T^{2n}x)}{n}.$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy--Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times L^1.$ The Furstenberg averages do not converge for all pairs of $(L^{1},L^{1})$ functions, while by a result of J. Bourgain these averages conve…
This article is devoted to a study of the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space $H^1$ and a function in $BMO$ to distributions that belong to $H^{\log}$ based on dyadic paraproducts. We also point out analogues of classical results of Hardy-Littlewood, Zygmund, and Stein for $H^{\log}$ and related Musielak-Orlicz spaces.
In this paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces $ H^p\ (0<p\leqslant 1) $ and functions in their dual spaces. Our decompositions are based on martingale paraproducts. As a consequence of our work, we also obtain analogous results for dyadic martingales on spaces of homogeneous type equipped with a doubling measure.
We give an atomic decomposition of closed forms on R n , the coefficients of which belong to some Hardy space of Musielak-Orlicz type. These spaces are natural generalizations of weighted Hardy-Orlicz spaces, when the Orlicz function depends on the space variable. One of them, called H log , appears naturally when considering products of functions in the Hardy space H 1 and in BM O. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in H log. Namely, a closed form in H log is the infinite sum of the wedge product between an …
Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón--Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calderón--Zygmund o…
For any $p\in(0,\,1]$, let $H^{Φ_p}(\mathbb{R}^n)$ be the Musielak-Orlicz Hardy space associated with the Musielak-Orlicz growth function $Φ_p$, defined by setting, for any $x\in\mathbb{R}^n$ and $t\in[0,\,\infty)$, $$ Φ_{p}(x,\,t):= \begin{cases} \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}} & \qquad \text{when } n(1/p-1)\notin \mathbb{N} \cup \{0\}; \\ \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}[\log(e+|x|)]^p} & \qquad \text{when } n(1/p-1)\in \mathbb{N}\cup\{0\},\\ \end{cases} $$ which is the sharp target space of the bilinear decomposition of the product of the Hardy space $H^p(\mathbb{R}^n)$ and its …
Fix $λ>0$. Consider the Hardy space $H^1(\mathbb{R}_+,dm_λ)$ in the sense of Coifman and Weiss, where $\mathbb{R_+}:=(0,\infty)$ and $dm_λ:=x^{2λ}dx$ with $dx$ the Lebesgue measure. Also consider the Bessel operators $Δ_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$, and $S_λ:=-\frac{d^2}{dx^2}+\frac{λ^2-λ}{x^2}$ on $\mathbb{R_+}$. The Hardy spaces $H^1_{Δ_λ}$ and $H^1_{S_λ}$ associated with $Δ_λ$ and $S_λ$ are defined via the Riesz transforms $R_{Δ_λ}:=\partial_x (Δ_λ)^{-1/2}$ and $R_{S_λ}:= x^λ\partial_x x^{-λ} (S_λ)^{-1/2}$, respectively. It is known that $H^1_{Δ_λ}$ and $H^1(\mathbb{R}_+,d…
In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may be written as the sum of two continuous bilinear operators, one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$, the other one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of Hardy-Orlicz space denoted by $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the set of distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies $$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathca…
We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 <p,q < \infty$ and $r\geq 1$, boundedly and continuously. The same result holds on $\mathbb{R}^n$ when $r>1$. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.
Let $({\mathcal X},d,μ)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, the authors prove that the product $f\times g$ of $f\in H^1_{\rm at}({\mathcal X})$ and $g\in\mathop\mathrm{BMO}({\mathcal X})$, viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $H^1_{\rm at}({\mathcal X})\times\mathop\mathrm{BMO}({\mathcal X})$ into $L^1…
Let $f:=(f_n)_{n\in \mathbb{Z}_+}$ and $g:=(g_n)_{n\in \mathbb{Z}_+}$ be two martingales related to the probability space $(Ω,\mathcal F,\mathbb P)$ equipped with the filtration $(\mathcal F_n)_{n\in \mathbb{Z}_+}.$ Assume that $f$ is in the martingale Hardy space $H_1$ and $g$ is in its dual space, namely the martingale $\rm BMO.$ Then the semi-martingale $f\cdot g:=(f_ng_n)_{n\in \mathbb{Z}_+}$ may be written as the sum $$f\cdot g=G(f, g)+L( f,g).$$ Here $L( f,g):=(L( f,g)_n)_{n\in\mathbb{Z}_+}$ with $L( f,g)_n:=\sum_{k=0}^n(f_k-f_{k-1})(g_k-g_{k-1)})$ for any $n\in\mathbb{Z}_+$, where $f_{-…
