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Dachun Yang

Publications and source records attributed to Dachun Yang.

At least 217 records · Page 12Linked to original sources
arXiv2010

Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups

Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure $ρ$, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,ρ)$ admits a Calderón--Zygmund decomposition which involves a particular family of sets, called Calderón--Zygmund sets. In this paper, we first show the existence of a dyadic grid in the group $S$, which has {nice} properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid we shall prove a Fefferman--Stein type inequality, involving the dyadic maximal Hardy--Littlewood function and the dyadic sharp dyadic function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space $H^1$ and the $BMO$ space introduced in [Collect. Math. 60(2009), 277--295].

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arXiv2010

Orlicz-Hardy Spaces Associated with Operators Satisfying Davies-Gaffney Estimates

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 $\mathrm{BMO}_{ρ,L}({\mathcal X})$; as a corollary, the authors obtain the $ρ$-Carleson measure characterization of the space ${\mathop\mathrm{BMO}_{ρ,L}(\mathcal X)}$. Characterizations of $H_{ω,L}({\mathcal X})$, including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let ${\mathcal X}={\mathbb R}^n$ and $ L=-Δ+V$ be a Schrödinger operator, where $V\in L^1_{\mathrm{\,loc\,}}({\mathbb R}^n)$ is a nonnegative potential. As applications, the authors show that the Riesz transform $\nabla L^{-1/2}$ is bounded from $H_{ω,L}({{\mathbb R}^n})$ to $L(ω)$; moreover, if there exist $q_1,\,q_2\in (0,\infty)$ such that $q_1<1<q_2$ and {\normalsize$[ω(t^{q_2})]^{q_1}$} is a convex function on $(0,\infty)$, then several characterizations of the Orlicz-Hardy space $H_{ω,L}({{\mathbb R}^n})$, in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1)$.

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arXiv2010

New Properties of Besov and Triebel-Lizorkin Spaces on RD-Spaces

An RD-space $\mathcal X$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal X$. In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on $\mathcal X$ are independent of the choice of the regularity $ε\in (0,1)$; as a result of this, the Besov and Triebel-Lizorkin spaces on $\mathcal X$ are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and Triebel-Lizorkin spaces by the norms of homogeneous Besov and Triebel-Lizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.

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arXiv2010

Anisotropic Singular Integrals in Product Spaces

Let $A_i$ for $i=1, 2$ be an expansive dilation, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$ and $\vec A\equiv(A_1, A_2)$. Denote by ${\mathcal A}_\infty(\rnm; \vec A)$ the class of Muckenhoupt weights associated with $\vec A$. The authors introduce a class of anisotropic singular integrals on $\mathbb R^n\times\mathbb R^m$, whose kernels are adapted to $\vec A$ in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on $L^q_w(\mathbb R^n\times\mathbb R^m)$ with $q\in(1, \infty)$ and $w\in\mathcal A_q(\mathbb R^n\times\mathbb R^m; \vec A)$ or on $H^p_w(\mathbb R^n\times\mathbb R^m; \vec A)$ with $p\in(0, 1]$ and $w\in\mathcal A_\infty(\mathbb R^n \times\mathbb R^m; \vec A)$. These results are also new even when $w=1$.

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arXiv2010

Pointwise Characterizations of Besov and Triebel-Lizorkin Spaces and Quasiconformal Mappings

In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces $\dot B^s_{p,\,q}$ and Triebel-Lizorkin spaces $\dot F^s_{p,\,q}$ for all $s\in(0,\,1)$ and $p,\,q\in(n/(n+s),\,\infty],$ both in ${\mathbb R}^n$ and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve $\dot F^s_{n/s,\,q}$ on $\rn$ for all $s\in(0,\,1)$ and $q\in(n/(n+s),\,\infty]$. A metric measure space version of the above morphism property is also established.

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arXiv2010

Decompositions of Besov-Hausdorff and Triebel-Lizorkin-Hausdorff Spaces and Their Applications

Let $p\in(1,\infty)$, $q\in[1,\infty)$, $s\in\mathbb{R}$ and $τ\in[0, 1-\frac{1}{\max\{p,q\}}]$. In this paper, the authors establish the $φ$-transform characterizations of Besov-Hausdorff spaces $B{\dot H}_{p,q}^{s,τ}(\mathbb{R}^n)$ and Triebel-Lizorkin-Hausdorff spaces $F{\dot H}_{p,q}^{s,τ}(\mathbb{R}^n)$ ($q>1$); as applications, the authors then establish their embedding properties (which on $B{\dot H}_{p,q}^{s,τ}(\mathbb{R}^n)$ is also sharp), smooth atomic and molecular decomposition characterizations for suitable $τ$. Moreover, using their atomic and molecular decomposition characterizations, the authors investigate the trace properties and the boundedness of pseudo-differential operators with homogeneous symbols in $B{\dot H}_{p,q}^{s,τ}(\mathbb{R}^n)$ and $F{\dot H}_{p,q}^{s,τ}(\mathbb{R}^n)$ ($q>1$), which generalize the corresponding classical results on homogeneous Besov and Triebel-Lizorkin spaces when $p\in(1,\infty)$ and $q\in[1,\infty)$ by taking $τ=0$.

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arXiv2010

Localized Hardy Spaces $H^1$ Related to Admissible Functions on RD-Spaces and Applications to Schrödinger Operators

Let ${\mathcal X}$ be an RD-space, which means that ${\mathcal X}$ is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in ${\mathcal X}$. In this paper, the authors first introduce the notion of admissible functions $ρ$ and then develop a theory of localized Hardy spaces $H^1_ρ({\mathcal X})$ associated with $ρ$, which includes several maximal function characterizations of $H^1_ρ({\mathcal X})$, the relations between $H^1_ρ({\mathcal X})$ and the classical Hardy space $H^1({\mathcal X})$ via constructing a kernel function related to $ρ$, the atomic decomposition characterization of $H^1_ρ({\mathcal X})$, and the boundedness of certain localized singular integrals in $H^1_ρ({\mathcal X})$ via a finite atomic decomposition characterization of some dense subspace of $H^1_ρ({\mathcal X})$. This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on $\rn$, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality.

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arXiv2010

Boundedness of Lusin-area and $g_λ^\ast$ Functions on Localized BMO Spaces over Doubling Metric Measure Spaces

Let ${\mathcal X}$ be a doubling metric measure space. If ${\mathcal X}$ has the $δ$-annular decay property for some $δ\in (0,\,1]$, the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from localized spaces ${\rm BMO}_ρ({\mathcal X})$ to ${\rm BLO}_ρ({\mathcal X})$ without invoking any regularity of considered kernels. The same is true for the $g^\ast_\labda$ function and unlike the Lusin-area function, in this case, ${\mathcal X}$ is not necessary to have the $δ$-annular decay property. Moreover, for any metric space, the authors introduce the weak geodesic property and the monotone geodesic property, which are proved to be respectively equivalent to the chain ball property of Buckley. Recall that Buckley proved that any length space has the chain ball property and, for any metric space equipped with a doubling measure, the chain ball property implies the $δ$-annular decay property for some $δ\in (0,1]$. Moreover, using some results on pointwise multipliers of ${\rm bmo}({\mathbb R})$, the authors construct a counterexample to show that there exists a nonnegative function which is in ${\rm bmo}({\mathbb R})$, but not in ${\rm blo}({\mathbb R})$; this further indicates that the above boundedness of the Lusin-area and $g^\ast_λ$ functions even in ${\mathbb R}^d$ with the Lebesgue measure or the Heisenberg group also improves the existing results.

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arXiv2010

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let $L$ be a linear operator in $L^2({{\mathbb R}^n})$ and generate an analytic semigroup $\{e^{-tL}\}_{t\ge 0}$ with kernels satisfying an upper bound of Poisson type, whose decay is measured by $θ(L)\in (0,\infty].$ Let $ω$ on $(0,\infty)$ be of upper type 1 and of critical lower type $\widetilde p_0(ω)\in (n/(n+θ(L)), 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty)$. In this paper, the authors first introduce the VMO-type space $\mathrm{VMO}_{ρ,L}({\mathbb R}^n)$ and the tent space $T^{\infty}_{ω,\mathrm v}({\mathbb R}^{n+1}_+)$ and characterize the space $\mathrm{VMO}_{ρ,L}({\mathbb R}^n)$ via the space $T^{\infty}_{ω,\mathrm v}({\mathbb R}^{n+1}_+)$. Let $\widetilde{T}_ω ({\mathbb R}^{n+1}_+)$ be the Banach completion of the tent space $T_ω({\mathbb R}^{n+1}_+)$. The authors then prove that $\widetilde{T}_ω({\mathbb R}^{n+1}_+)$ is the dual space of $T^{\infty}_{ω,\mathrm v}({\mathbb R}^{n+1}_+)$. As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{ρ,L^\ast}({\mathbb R}^n)$ is the space $B_{ω,L}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$ and $B_{ω,L}({\mathbb R}^n)$ the Banach completion of the Orlicz-Hardy space $H_{ω,L}({\mathbb R}^n)$. These results generalize the known recent results by particularly taking $ω(t)=t$ for $t\in (0,\infty)$.

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arXiv2010

Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Let $L$ be a divergence form elliptic operator with complex bounded measurable coefficients, $ω$ a positive concave function on $(0,\infty)$ of strictly critical lower type $p_ω\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors introduce the generalized VMO spaces ${\mathop\mathrm{VMO}_ {ρ, L}({\mathbb R}^n)}$ associated with $L$, and characterize them via tent spaces. As applications, the authors show that $(\mathrm{VMO}_{ρ,L} ({\mathbb R}^n))^\ast=B_{ω,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$ and $B_{ω,L^\ast}({\mathbb R}^n)$ the Banach completion of the Orlicz-Hardy space $H_{ω,L^\ast}({\mathbb R}^n)$. Notice that $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1]$ is a typical example of positive concave functions satisfying the assumptions. In particular, when $p=1$, then $ρ(t)\equiv 1$ and $({\mathop\mathrm{VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)$, where $H_{L^\ast}^1({\mathbb R}^n)$ was the Hardy space introduced by Hofmann and Mayboroda.

math.FA↗
arXiv2009

Localized BMO and BLO Spaces on RD-Spaces and Applications to Schrödinger Operators

An RD-space ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in ${\mathcal X}$. Let $ρ$ be an admissible function on RD-space ${\mathcal X}$. The authors first introduce the localized spaces $\mathrm{BMO}_ρ({\mathcal X})$ and $\mathrm{BLO}_ρ({\mathcal X})$ and establish their basic properties, including the John-Nirenberg inequality for $\mathrm{BMO}_ρ({\mathcal X})$, several equivalent characterizations for $\mathrm{BLO}_ρ({\mathcal X})$, and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal functions and their localized versions associated to $ρ$, and the Littlewood-Paley $g$-function associated to $ρ$, where the Littlewood-Paley $g$-function and some of the radial maximal functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on ${\mathbb R}^d$, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

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arXiv2009

Localized Morrey-Campanato Spaces on Metric Measure Spaces and Applications to Schrödinger Operators

Let ${\mathcal X}$ be a space of homogeneous type in the sense of Coifman and Weiss and ${\mathcal D}$ a collection of balls in $\cx$. The authors introduce the localized atomic Hardy space $H^{p, q}_{\mathcal D}({\mathcal X})$ with $p\in (0,1]$ and $q\in[1,\infty]\cap(p,\infty]$, the localized Morrey-Campanato space ${\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ and the localized Morrey-Campanato-BLO space $\widetilde{\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ with $\az\in{\mathbb R}$ and $p\in(0, \infty)$ and establish their basic properties including $H^{p, q}_{\mathcal D}({\mathcal X})=H^{p, \infty}_{\mathcal D}({\mathcal X})$ and several equivalent characterizations for ${\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ and $\wz{\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$. Especially, the authors prove that when $p\in(0,1]$, the dual space of $H^{p, \infty}_{\mathcal D}({\mathcal X})$ is ${\mathcal E}^{1/p-1, 1}_{\mathcal D}({\mathcal X})$. Let $ρ$ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces ${\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ and $\widetilde{\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$, respectively, by ${\mathcal E}^{α, p}_ρ({\mathcal X})$ and $\widetilde{\mathcal E}^{α, p}_ρ({\mathcal X})$, when ${\mathcal D}$ is determined by $ρ$. The authors then obtain the boundedness from ${\mathcal E}^{α, p}_ρ({\mathcal X})$ to $\widetilde{\mathcal E}^{α, p}_ρ({\mathcal X})$ of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley $g$-function which are defined via kernels modeled on the semigroup generated by the Schrödinger operator.

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arXiv2009

Weighted Anisotropic Product Hardy Spaces and Boundedness of Sublinear Operators

Let $A_1$ and $A_2$ be expansive dilations, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$. Let $\vec A\equiv(A_1, A_2)$ and $\mathcal A_p(\vec A)$ be the class of product Muckenhoupt weights on ${\mathbb R}^n\times{\mathbb R}^m$ for $p\in(1, \infty]$. When $p\in(1, \infty)$ and $w\in{\mathcal A}_p(\vec A)$, the authors characterize the weighted Lebesgue space $L^p_w({\mathbb R}^n\times{\mathbb R}^m)$ via the anisotropic Lusin-area function associated with $\vec A$. When $p\in(0, 1]$, $w\in {\mathcal A}_\infty(\vec A)$, the authors introduce the weighted anisotropic product Hardy space $H^p_w({\mathbb R}^n\times{\mathbb R}^m; \vec A)$ via the anisotropic Lusin-area function and establish its atomic decomposition. Moreover, the authors prove that finite atomic norm on a dense subspace of $H^p_w({\mathbb R}^n\times{\mathbb R}^m;\vec A)$ is equivalent with the standard infinite atomic decomposition norm. As an application, the authors prove that if $T$ is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space $\mathcal B $, then $T$ uniquely extends to a bounded sublinear operator from $H^p_w({\mathbb R}^n\times{\mathbb R}^m;\vec A)$ to $\mathcal B$. The results of this paper improve the existing results for weighted product Hardy spaces and are new even in the unweighted anisotropic setting.

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arXiv2009

A Characterization of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Grand Littlewood-Paley Functions

In this paper, we establish the equivalence between the Hajłasz-Sobolev spaces or classical Triebel-Lizorkin spaces and a class of grand Triebel-Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when $p\in(n/(n+1),\infty)$, we give a new characterization of the Hajłasz-Sobolev spaces $\dot M^{1, p}({\mathbb R}^n)$ via a grand Littlewood-Paley function.

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arXiv2009

New Orlicz-Hardy Spaces Associated with Divergence Form Elliptic Operators

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 characterization and the maximal function characterization, are established. The $ρ$-Carleson measure characterization and the John-Nirenberg inequality for the space $\mathrm{BMO}_{ρ,L}({\mathbb R}^n)$ are also given. As applications, the authors show that the Riesz transform $\nabla L^{-1/2}$ and the Littlewood-Paley $g$-function $g_L$ map $H_{ω,L}({\mathbb R}^n)$ continuously into $L(ω)$. The authors further show that the Riesz transform $\nabla L^{-1/2}$ maps $H_{ω,L}({\mathbb R}^n)$ into the classical Orlicz-Hardy space $H_ω({\mathbb R}^n)$ for $p_ω\in (\frac{n}{n+1},1]$ and the corresponding fractional integral $L^{-γ}$ for certain $γ>0$ maps $H_{ω,L}({\mathbb R}^n)$ continuously into $H_{\widetildeω,L}({\mathbb R}^n)$, where $\widetildeω$ is determined by $ω$ and $γ$, and satisfies the same property as $ω$. All these results are new even when $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1)$.

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arXiv2009

Radial Maximal Function Characterizations of Hardy Spaces on RD-Spaces and Their Applications

Let ${\mathcal X}$ be an RD-space with $μ({\mathcal X})=\infty$, which means that ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss and its measure has the reverse doubling property. In this paper, we characterize the atomic Hardy spaces $H^p_{\rm at}(\{\mathcal X})$ of Coifman and Weiss for $p\in(n/(n+1),1]$ via the radial maximal function, where $n$ is the "dimension" of ${\mathcal X}$, and the range of index $p$ is the best possible. This completely answers the question proposed by Ronald R. Coifman and Guido Weiss in 1977 in this setting, and improves on a deep result of Uchiyama in 1980 on an Ahlfors 1-regular space and a recent result of Loukas Grafakos et al in this setting. Moreover, we obtain a maximal function theory of localized Hardy spaces in the sense of Goldberg on RD-spaces by generalizing the above result to localized Hardy spaces and establishing the links between Hardy spaces and localized Hardy spaces. These results have a wide range of applications. In particular, we characterize the Hardy spaces $H^p_{\rm at}(M)$ via the radial maximal function generated by the heat kernel of the Laplace-Beltrami operator $Δ$ on complete noncompact connected manifolds $M$ having a doubling property and supporting a scaled Poincaré inequality for all $p\in(n/(n+α),1]$, where $α$ represents the regularity of the heat kernel. This extends some recent results of Russ and Auscher-McIntosh-Russ.

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arXiv2009

Boundedness of Sublinear Operators on Product Hardy Spaces and Its Application

Let $p\in(0, 1]$. In this paper, the authors prove that a sublinear operator $T$ (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces $H^p({{\mathbb R}^n\times{\mathbb R}^m})$ to some quasi-Banach space ${\mathcal B}$ if and only if $T$ maps all $(p, 2, s_1, s_2)$-atoms into uniformly bounded elements of ${\mathcal B}$. Here $s_1\ge\lfloor n(1/p-1)\rfloor$ and $s_2\ge\lfloor m(1/p-1)\rfloor$. As usual, $\lfloor n(1/p-1)\rfloor$ denotes the maximal integer no more than $n(1/p-1)$. Applying this result, the authors establish the boundedness of the commutators generated by Calderón-Zygmund operators and Lipschitz functions from the Lebesgue space $L^p({{\mathbb R}^n\times{\mathbb R}^m})$ with some $p>1$ or the Hardy space $H^p({{\mathbb R}^n\times{\mathbb R}^m})$ with some $p\le1$ but near 1 to the Lebesgue space $L^q({{\mathbb R}^n\times{\mathbb R}^m})$ with some $q>1$.

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arXiv2009

Boundedness of Linear Operators via Atoms on Hardy Spaces with Non-doubling Measures

Let $μ$ be a non-negative Radon measure on ${\mathbb R}^d$ which only satisfies the polynomial growth condition. Let ${\mathcal Y}$ be a Banach space and $H^1(μ)$ the Hardy space of Tolsa. In this paper, the authors prove that a linear operator $T$ is bounded from $H^1(μ)$ to ${\mathcal Y}$ if and only if $T$ maps all $(p, γ)$-atomic blocks into uniformly bounded elements of ${\mathcal Y}$; moreover, the authors prove that for a sublinear operator $T$ bounded from $L^1(μ)$ to $L^{1, \infty}(μ)$, if $T$ maps all $(p, γ)$-atomic blocks with $p\in(1, \infty)$ and $γ\in{\mathbb N}$ into uniformly bounded elements of $L^1(μ)$, then $T$ extends to a bounded sublinear operator from $H^1(μ)$ to $L^1(μ)$. For the localized atomic Hardy space $h^1(μ)$, corresponding results are also presented. Finally, these results are applied to Calderón-Zygmund operators, Riesz potentials and multilinear commutators generated by Calderón-Zygmund operators or fractional integral operators with Lipschitz functions, to simplify the existing proofs in the corresponding papers.

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