Let $\mathcal{X}$ be a space of homogenous type and $φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ a growth function such that $φ(\cdot,t)$ is a Muckenhoupt weight uniformly in $t$ and $φ(x,\cdot)$ an Orlicz function of uniformly upper type 1 and lower type $p\in(0,1]$. In this article, the authors introduce a new Musielak-Orlicz BMO-type space $\mathrm{BMO}^φ_A(\mathcal{X})$ associated with the generalized approximation to the identity, give out its basic properties and establish its two equivalent characterizations, respectively, in terms of the spaces $\mathrm{BMO}^φ_{A,\,\mathrm{max}}(\mathcal{X})$ and $\widetilde{\mathrm{BMO}}^φ_A(\mathcal{X})$. Moreover, two variants of the John-Nirenberg inequality on $\mathrm{BMO}^φ_A(\mathcal{X})$ are obtained. As an application, the authors further prove that the space $\mathrm{BMO}^φ_{\sqrtΔ}(\mathbb{R}^n)$, associated with the Poisson semigroup of the Laplace operator $Δ$ on $\mathbb{R}^n$, coincides with the space $\mathrm{BMO}^φ(\mathbb{R}^n)$ introduced by L. D. Ky.
Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their ApplicationsLet $φ: \mathbb R^n\times [0,\infty)\to[0,\infty)$ be a growth function such that $φ(x,\cdot)$ is nondecreasing, $φ(x,0)=0$, $φ(x,t)>0$ when $t>0$, $\lim_{t\to\infty}φ(x,t)=\infty$, and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty(\mathbb{R}^n)$ weight uniformly in $t$. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak-Orlicz Hardy space $H_φ(\mathbb{R}^n)$ introduced by Luong Dang Ky via the grand maximal function. As an applicatio…
Let $φ: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight. The Musielak-Orlicz Hardy space $H^φ(\mathbb R^n)$ is defined to be the space of all $f\in{\mathcal S}'({\mathbb R^n})$ such that the grand maximal function $f^*$ belongs to the Musielak-Orlicz space $L^φ(\mathbb R^n)$. Luong Dang Ky established its atomic characterization. In this paper, the authors establish some new real-variable characterizations of $H^φ(\mathbb R^n)$ in terms of the vertical or the non-tangential maximal f…
Let $φ: \mathbb R^n\times [0,\infty)\to[0,\infty)$ be such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty(\mathbb R^n)$ weight uniformly in $t$. In this article, the authors introduce the Musielak-Orlicz Campanato space ${\mathcal L}_{φ,q,s}({\mathbb R}^n)$ and, as an application, prove that some of them is the dual space of the Musielak-Orlicz Hardy space $H^φ(\mathbb R^n)$, which in the case when $q=1$ and $s=0$ was obtained by L. D. Ky [arXiv: 1105.0486]. The authors also establish a John-Nirenberg inequality for functions in ${\mathcal L}_{φ,1,s}({\math…
Let $(Ω,\mathcal{F},\mathbb{P})$ be a probability space and $φ:\ Ω\times[0,\infty)\to [0,\infty)$ be a Musielak--Orlicz function. In this article, the authors establish the atomic characterizations of weak martingale Musielak--Orlicz Hardy spaces $WH_φ^s(Ω)$, $WH_φ^M(Ω)$, $WH_φ^S(Ω)$, $WP_φ(Ω)$ and $WQ_φ(Ω)$. Using these atomic characterizations, the authors then obtain the boundedness of sublinear operators from weak martingale Musielak--Orlicz Hardy spaces to weak Musielak--Orlicz spaces, and some martingale inequalities which further clarify the relationships among $WH_φ^s(Ω)$, $WH_φ^M(Ω)$,…
We introduce a new class of Hardy spaces $H^{φ(\cdot,\cdot)}(\mathbb R^n)$, called Hardy spaces of Musielak-Orlicz type, which generalize the Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of García-Cuerva, Strömberg, and Torchinsky. Here, $φ: \mathbb R^n\times [0,\infty)\to [0,\infty)$ is a function such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight. A function $f$ belongs to $H^{φ(\cdot,\cdot)}(\mathbb R^n)$ if and only if its maximal function $f^*$ is so that $x\mapsto φ(x,|f^*(x)|)$ is integrable. Such a space arises naturally for …
Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $φ:\,\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{φ,\,L}(\mathbb{R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the spac…
Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a nonnegative self-adjoint operator in $L^2(\mathcal{X})$ satisfying the Davies-Gaffney estimates. Let $φ:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that $φ(x,\cdot)$ is an Orlicz function, $φ(\cdot,t)\in A_{\infty}(\mathcal{X})$ (the class of Muckenhoupt weights) and its uniformly critical lower type index $i(φ)\in(0,1]$. In this paper, the authors introduce a Musielak-Orlicz Hardy space $H_{φ,\,L}(\mathcal{X})$ by the Lusin area function associated with the heat semigroup generated by $L$, and a Musielak-Or…
In Musilak-Orlicz type spaces ${\mathcal S}_{\bf M}$, direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.
Let $φ:\mathbb{R}^n\times[0,\,\infty) \rightarrow [0,\,\infty)$ satisfy that $φ(x,\,\cdot)$, for any given $x\in\mathbb{R}^n$, is an Orlicz function and $φ(\cdot\,,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t\in(0,\,\infty)$. The Musielak-Orlicz Hardy space $H^φ(\mathbb{R}^n)$ generalizes both of the weighted Hardy space and the Orlicz Hardy space and hence has a wide generality. In this paper, the authors first prove the completeness of both of the Musielak-Orlicz space $L^φ(\mathbb{R}^n)$ and the weak Musielak-Orlicz space $WL^φ(\mathbb{R}^n)$. Then the authors obtain two boundedne…
In this article, the authors establish a new characterization of the Musielak--Orlicz--Sobolev space on $\mathbb{R}^n$, which includes the classical Orlicz--Sobolev space, the weighted Sobolev space and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hästö and A. M. Ribeiro [Commun. Contemp. Math. 19 (2017), 1650022, 13 pp] via weakening the assumption $f\in L^1(\mathbb R^n)$ into $f\in L_{\ma…
Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a one-to-one operator of type $ω$ having a bounded $H_\infty$-functional calculus in $L^2(\mathcal{X})$ satisfying the reinforced $(p_L, q_L)$ off-diagonal estimates on balls, where $p_L\in[1,2)$ and $q_L\in(2,\infty]$. Let $φ:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that $φ(x,\cdot)$ is an Orlicz function, $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index $I(φ)\in(0,1]$ and $φ(\cdot,t)$ satisfies the uniformly reverse Höl…
Let $ϕ: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz)$ be a function such that $ϕ(x,\cdot)$ is an Orlicz function and $ϕ(\cdot,t)\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$ (the class of local weights introduced by V. S. Rychkov). In this paper, the authors introduce a local Hardy space $h_ϕ(\mathbb{R}^n)$ of Musielak-Orlicz type by the local grand maximal function, and a local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_ϕ(\mathbb{R}^n)$ which is further proved to be the dual space of $h_ϕ(\mathbb{R}^n)$. As an application, the authors prove that the class of pointwise multipli…
Let $φ$ be a Musielak-Orlicz function satisfying that, for any $(x,\,t)\in\mathbb{R}^n\times(0,\,\infty)$, $φ(\cdot,\,t)$ belongs to the Muckenhoupt weight class $A_\infty (\mathbb{R}^n)$ with the critical weight exponent $q(φ)\in[1,\,\infty)$ and $φ(x,\,\cdot)$ is an Orlicz function with $0<i(φ)\le I(φ)\le 1$ which are, respectively, its critical lower type and upper type. In this article, the authors establish the Riesz transform characterizations of the Musielak-Orlicz-Hardy spaces $H_φ(\mathbb{R}^n)$ which are generalizations of weighted Hardy spaces and Orlicz-Hardy spaces. Precisely, the…
Let $φ: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $\vz(x,\cdot)$ is nondecreasing, $φ(x,0)=0$, $φ(x,t)>0$ when $t>0$, $\lim_{t\to\infty}φ(x,t)=\infty$ and $\vz(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight uniformly in $t$. Let $ϕ: [0,\infty)\to[0,\infty)$ be nondecreasing. In this article, the authors introduce the Musielak-Orlicz Morrey space $\mathcal M^{φ,ϕ}(\mathbb R^n)$ and obtain the boundedness on $\mathcal M^{φ,ϕ}(\mathbb R^n)$ of the intrinsic Lusin area function $S_α$, the intrinsic $g$-function $g_α$, the intrinsic $g_λ^*$-function $g^\ast_{λ, α}$ and …
In this paper, the author establishes the boundedness of parametric Littlewood-Paley operators from Musielak-Orlicz Hardy space to Musielak-Orlicz space, or to weak Musielak-Orlicz space at the critical index. Part of these results are new even for classical Hardy space of Fefferman and Stein.
In the Musielak-Orlicz type spaces ${\mathcal S}_{\bf M}$, exact Jackson-type inequalities are obtained in terms of best approximations of functions and the averaged values of their generalized moduli of smoothness. The values of Kolmogorov, Bernstein, linear, and projective widths in ${\mathcal S}_{\bf M}$ are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.
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 …
Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ satisfies that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights). Let $H_{φ,\,L}(\mathcal{X})$ be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors characteri…
Let $(Ω,\mathcal{F},\mathbb{P})$ be a probability space and $φ:\ Ω\times[0,\infty)\to[0,\infty)$ 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 characterization…
BMO-Type characterizations and endpoint estimates for commutators of intrinsic square functions on Orlicz-Hardy spaces
School of Information Network Security People’s Public Security University of China（中国人民公安大学信息网络安全学院）；College of Mathematics and System Science Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；School of Mathematics and Statistics Minnan Normal University（闽南师范大学数学与统计学院）；School of Mathematical Sciences Xiamen University（厦门大学数学科学学院）
Let ${\mathcal X}$ be a metric space with doubling measure, $L$ a nonnegative self-adjoint operator in $L^2({\mathcal X})$ satisfying the Davies-Gaffney estimate, $ω$ a concave function on $(0,\infty)$ of strictly lower type $p_ω\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for all $t\in (0,\infty).$ The authors introduce the Orlicz-Hardy space $H_{ω,L}({\mathcal X})$ via the Lusin area function associated to the heat semigroup, and the BMO-type space ${\mathop\mathrm{BMO}_{ρ,L}(\mathcal X)}$. The authors then establish the duality between $H_{ω,L}({\mathcal X})$ and $\mathrm{BMO}_{ρ,L}({\mat…
Let $Φ$ be a concave function on $(0,\infty)$ of strictly lower type $p_Φ\in(0,1]$ and $ω\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$. We introduce the weighted local Orlicz-Hardy space $h^Φ_ω(\mathbb{R}^n)$ via the local grand maximal function. Let $ρ(t)\equiv t^{-1}/Φ^{-1}(t^{-1})$ for all $t\in(0,\infty)$. We also introduce the $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_{ρ,\,ω}(\mathbb{R}^n)$ and establish the duality between $h^Φ_ω(\mathbb{R}^n)$ and $\mathop\mathrm{bmo}_{ρ,\,ω}(\mathbb{R}^n)$. Several real-varaiable characterizations of $h^Φ_ω(\mathbb{R}^n)$ are presente…
Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition and may not be doubling, we define the product of functions in the regular $BMO$ and the atomic block $\H^{1}$ in the sense of distribution, and show that this product may be split into two parts, one in $L^{1}$ and the other in some Hardy-Orlicz space.
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.
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…
Let $L$ be the divergence form elliptic operator with complex bounded measurable coefficients, $ω$ the positive concave function on $(0,\infty)$ of strictly critical lower type $p_\oz\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors study the Orlicz-Hardy space $H_{ω,L}({\mathbb R}^n)$ and its dual space $\mathrm{BMO}_{ρ,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$. Several characterizations of $H_{ω,L}({\mathbb R}^n)$, including the molecular characterization, the Lusin-area function characte…
We define as a distribution the product of a function (or distribution) h in some Hardy space Hp with a function b in the dual space of Hp. Moreover, we prove that the product bxh may be written as the sum of an integrable function with a distribution that belongs to some Hardy-Orlicz space, or to the same Hardy space Hp, depending on the values of p.
Let $n\ge2$ and $Ω\subset\mathbb{R}^n$ be a bounded NTA domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second order elliptic equations of divergence form with an elliptic symmetric part and a BMO anti-symmetric part in $Ω$. More precisely, for any given $p\in(2,\infty)$, the authors prove that a weak reverse Hölder inequality with exponent $p$ implies the global $W^{1,p}$ estimate and the global weighted $W^{1,q}$ estimate, with $q\in[2,p]$ and some Muckenhoupt weights, of solutions to Dirichlet boundary value prob…
We study the holomorphic Hardy-Orlicz spaces H^Φ(Ω), where Ωis the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in Cn . The function Φis in particular such that H ^1(Ω) \subset H^Φ(Ω) \subset H ^p (Ω) for some p > 0. We develop for them maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Omega). As a consequence, we characterize those Hankel operators which are bounded from H ^Φ(Ω) into H^1 (Ω).
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 …
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_{-…
We prove that a generalized Fefferman-Phong type condition on a pair of weights $u$ and $v$ is sufficient for the boundedness of the commutators of potential type operators from $L^{p(\cdot)}_v$ into $L^{q(\cdot)}_u$. We also give an improvement of this result in the sense that we not only consider a variable version of power bump conditions, but also weaker norms related to Musielak-Orlicz functions. We consider a wider class of symbols including Lipschitz symbols and some generalizations.
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 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.
We provide an abstract estimate of the form \[ \|f-f_{Q,μ}\|_{X \left(Q,\frac{\mathrm{d} μ}{Y(Q)}\right)}\leq c(μ,Y)ψ(X)\|f\|_{\mathrm{BMO}(\mathrm{d}μ)} \] for all cubes $Q$ in $\mathbb{R}^n$ and every function $f\in \mathrm{BMO}(\mathrm{d}μ)$, where $μ$ is a doubling measure in $\mathbb{R}^n$, $Y$ is some positive functional defined on cubes, $\|\cdot \|_{X \left(Q,\frac{\mathrm{d} w}{w(Q)}\right)}$ is a sufficiently good quasi-norm and $c(μ,Y)$ and $ψ(X)$ are positive constants depending on $μ$ and $Y$, and $X$, respectively. That abstract scheme allows us to recover the sharp estimate \[ \…
