Lusin Area Research Articles

Intrinsic square function and wavelet characterizations of variable Hardy–Lorentz spaces

Let p(⋅):Rn→(0,∞] be a variable exponent function satisfying the globally log-Hölder continuous condition and q∈(0,1]. The goal of this article is to characterize the variable Hardy–Lorentz space Hp(⋅),q(Rn) in terms of various intrinsic square functions including the intrinsic g-function, the intrinsic Lusin area integral and the intrinsic gλ⁎-function. Using the intrinsic square function and the known atomic characterizations of Hp(⋅),q(Rn), and establishing some estimates on a discrete Littlewood–Paley g-function and a Peetre-type maximal function, the authors further give several equivalent characterizations of Hp(⋅),q(Rn) via wavelets. What is different from the existing works is that we merely assume p+=ess supx∈Rnp(x)∈(0,∞). One of the important tools that make it possible is to find a new dual space of Hp(⋅),q(Rn).

Characterizations of Sobolev multiplier spaces and their preduals in terms of Littlewood–Paley and Lusin area functions

We characterize Sobolev multiplier spaces and their preduals in terms of Littlewood–Paley and Lusin area functions. The vector-valued and weighted norm estimates of Hardy–Littlewood maximal operator will be used as the main tools in such characterizations.

A two-weight boundedness criterion and its applications

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, [Formula: see text], on [Formula: see text] in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley [Formula: see text]-functions, Lusin area functions, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on [Formula: see text] in the aforementioned function spaces.

Characterizations of a class of weighted 𝐵𝑀𝑂 spaces via commutators of intrinsic square functions

The intrinsic square functions including the Lusin area function, Littlewood-Paley g g -function and g λ ∗ g^\ast _\lambda -function dominate pointwisely the classical Littlewood-Paley functions and can be used to characterize the weighted Hardy spaces and more general Musielak-Orlicz Hardy spaces etc. This paper shows that for b ∈ B M O ( R n ) b\in \mathrm {BMO(\mathbb {R}^n)} , the commutators generated by these intrinsic square functions with b b are bounded from H ω p ( R n ) H^p_\omega (\mathbb {R}^n) to L ω p ( R n ) L^p_\omega (\mathbb {R}^n) for some 0 > p ≤ 1 0>p\le 1 and ω ∈ A ∞ \omega \in A_\infty if and only if b ∈ B M O ω , p ( R n ) b\in \mathcal {BMO}_{\omega ,p}(\mathbb {R}^n) , which are a class of non-trivial subspaces of B M O ( R n ) \mathrm {BMO(\mathbb {R}^n)} .

Real-variable characterizations and their applications of matrix-weighted Triebel–Lizorkin spaces

Let α∈R, q∈(0,∞], p∈(0,∞), and W be an Ap(Rn,Cm)-matrix weight. In this article, the authors characterize the matrix-weighted Triebel–Lizorkin space F˙pα,q(W) via the Peetre maximal function, the Lusin area function, and the Littlewood–Paley gλ⁎-function. As applications, the authors establish the boundedness of Fourier multipliers on matrix-weighted Triebel–Lizorkin spaces under the generalized Hörmander condition. The main novelty of these results exists in that their proofs need to fully use both the doubling property of matrix weights and the reducing operator associated to matrix weights, which are essentially different from those proofs of the corresponding cases of classical Triebel–Lizorkin spaces that strongly depend on the Fefferman–Stein vector-valued maximal inequality on Lebesgue spaces.

Wavelet characterization of Triebel–Lizorkin spaces for p=∞ on spaces of homogeneous type and its applications

Let ( X , d , μ ) be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, η ∈ ( 0 , 1 ) the Hölder regularity of wavelets on X constructed by P. Auscher and T. Hytönen, s ∈ ( − η , η ) , and q ∈ ( max { ω ω + η , ω ω + η + s } , ∞ ] . In this article, the authors establish the wavelet characterization of the Triebel–Lizorkin space F ̇ ∞ , q s ( X ) . Moreover, the authors introduce almost diagonal operators on the Triebel–Lizorkin sequence space f ̇ ∞ , q s ( X ) and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors establish the molecular characterization of F ̇ ∞ , q s ( X ) . The authors also obtain both the Lusin area function and the Littlewood–Paley g λ ∗ -function characterizations of F ̇ ∞ , q s ( X ) . Besides, the inhomogeneous counterparts are also given. The main novelty of this article is that all these results get rid of their dependence on the reverse doubling property of μ and also the triangle inequality of d , by fully using the geometrical properties of X expressed via its equipped quasi-metric d , dyadic reference points, dyadic cubes, and wavelets.

Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations

Abstract Let ( 𝕏 , d , μ ) {(\mathbb{X},d,\mu)} be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and let X ⁢ ( 𝕏 ) {X(\mathbb{X})} be a ball quasi-Banach function space on 𝕏 {\mathbb{X}} . In this article, the authors introduce the weak Hardy space W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} associated with X ⁢ ( 𝕏 ) {X(\mathbb{X})} via the Lusin area function. Then the authors characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the molecule, the grand maximal function, and the Littlewood–Paley g-function and g λ * {g^{*}_{\lambda}} -function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of 𝕏 {\mathbb{X}} expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm of X ⁢ ( 𝕏 ) {X(\mathbb{X})} and the reverse doubling condition of μ, and that the authors use the tent space on 𝕏 × ℤ {\mathbb{X}\times\mathbb{Z}} to characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the Littlewood–Paley g λ * {g^{*}_{\lambda}} -function, where the range of λ might be best possible in some cases.

Boundedness of Littlewood-Paley Operators in Variable Morrey Spaces

In this paper, the authors prove norm inequalities for the intrinsic square functions and commutators generated by this class operator and BMO function in variable Morrey spaces. This implies that the same norm inequalities for the Lusin area integrals, the Littlewood-Paley operators and the continuous square functions. As application, we get the boundedness for convolution Calderón-Zygmund operators in generalized Morrey spaces.

An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

Identification of anisotropic mixed-norm Hardy spaces and certain homogeneous Triebel–Lizorkin spaces

Let S(Rn) be the Schwartz class on Rn and S∞(Rn)≔ϕ∈S(Rn):∫Rnxαϕ(x)dx=0for any multi-indexα∈({0,1,…})n,and S′(Rn) and S∞′(Rn) be their dual spaces, respectively. Let a→≔(a1,…,an)∈[1,∞)n, p→≔(p1,…,pn)∈(0,1]n, and Ha→p→(Rn)⊂S′(Rn) be the anisotropic mixed-norm Hardy space, associated with an anisotropic quasi-homogeneous norm |⋅|a→, defined via the non-tangential grand maximal function. In this article, via the known atomic characterizations and Lusin area function characterizations of Ha→p→(Rn), the authors first establish a new atomic characterization of Ha→p→(Rn) in terms of S∞′(Rn). Applying this atomic characterization, the authors then obtain the Littlewood–Paley g-function characterization of Ha→p→(Rn) in terms of S∞′(Rn), which further induces the identification of Ha→p→(Rn) and certain anisotropic homogeneous mixed-norm Triebel–Lizorkin spaces. As applications, the authors obtain the dual theorem on some anisotropic homogeneous mixed-norm Triebel–Lizorkin spaces and the boundedness of Fourier multipliers as well as anisotropic pseudo-differential operators on Ha→p→(Rn). All these results are new even for isotropic mixed-norm Hardy spaces on Rn.

Morrey spaces for Schrödinger operators with certain nonnegative potentials, Littlewood–Paley and Lusin functions on the Heisenberg groups

Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on the Heisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_q$ with $q\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. Assume that $\{e^{-s\mathcal L}\}_{s>0}$ is the heat semigroup generated by $\mathcal L$. The Littlewood-Paley function $\mathfrak{g}_{\mathcal L}$ and the Lusin area integral $\mathcal{S}_{\mathcal L}$ associated with the Schr\"odinger operator $\mathcal L$ are defined, respectively, by \begin{equation*} \mathfrak{g}_{\mathcal L}(f)(u) := \bigg(\int_0^{\infty}\bigg|s\frac{d}{ds} e^{-s\mathcal L}f(u) \bigg|^2\frac{ds}{s}\bigg)^{1/2} \end{equation*} and \begin{equation*} \mathcal{S}_{\mathcal L}(f)(u) := \bigg(\iint_{\Gamma(u)} \bigg|s\frac{d}{ds} e^{-s\mathcal L}f(v) \bigg|^2 \frac{dvds}{s^{Q/2+1}}\bigg)^{1/2}, \end{equation*} where \begin{equation*} \Gamma(u) := \big\{(v,s)\in\mathbb H^n\times(0,\infty): |u^{-1}v| < \sqrt{s\,}\big\}. \end{equation*} In this paper the author first introduces a class of Morrey spaces associated with the Schr\"odinger operator $\mathcal L$ on $\mathbb H^n$. Then by using some pointwise estimates of the kernels related to the nonnegative potential $V$, the author establishes the boundedness properties of these two operators $\mathfrak{g}_{\mathcal L}$ and $\mathcal{S}_{\mathcal L}$ acting on the Morrey spaces. It can be shown that the same conclusions also hold for the operators $\mathfrak{g}_{\sqrt{\mathcal L}}$ and $\mathcal{S}_{\sqrt{\mathcal L}}$ with respect to the Poisson semigroup $\{e^{-s\sqrt{\mathcal L}}\}_{s>0}$.

Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

The aim of this paper is to establish the intrinsic square function characterizations in terms of the intrinsic Littlewood–Paleyg-function, the intrinsic Lusin area function, and the intrinsicgλ∗-function of the variable Hardy–Lorentz spaceHp⋅,qℝn, forp⋅being a measurable function onℝnsatisfying0&lt;p−≔ess infx∈ℝnpx≤ess supx∈ℝnpx≕p+&lt;∞and the globally log-Hölder continuity condition andq∈0,∞, via its atomic and Littlewood–Paley function characterizations.

Real-variable characterizations of new anisotropic mixed-norm Hardy spaces

Let $\vec{p}\in(0,\infty)^n$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces $H_A^{\vec{p}}(\mathbb{R}^n)$ associated with $A$ and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize $H_A^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood-Paley $g$-functions or $g_{\lambda}^\ast$-functions via first establishing an anisotropic Fefferman-Stein vector-valued inequality on the mixed-norm Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$. In addition, the authors also obtain the duality between $H_A^{\vec{p}}(\mathbb{R}^n)$ and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from $H_A^{\vec{p}}(\mathbb{R}^n)$ into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional $\delta$-type and non-convolutional $\beta$-order Calder\'{o}n-Zygmund operators from $H_A^{\vec{p}}(\mathbb{R}^n)$ to itself [or to $L^{\vec{p}}(\mathbb{R}^n)$]. As a corollary, the boundedness of anisotropic convolutional $\delta$-type Calder\'on-Zygmund operators on the mixed-norm Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$ with $\vec{p}\in(1,\infty)^n$ is also presented.

Intrinsic Square Function Characterizations of Variable Weak Hardy Spaces

Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, via using the atomic and Littlewood-Paley function characterizations of variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_{\lambda}^{\ast}$-function.

Sharp Weighted Estimates for Square Functions Associated to Operators on Spaces of Homogeneous Type

Let X be a metric space with a doubling measure and let L be a linear operator in $$L^2(X)$$ which generates a semigroup $$e^{-tL}$$ whose kernels $$p_t(x,y)$$, $$t > 0$$, satisfy the Gaussian upper bound. In this article, we prove sharp$$L^p_w$$ norm inequalities for a number of square functions associated to L including the vertical square functions, the Lusin area integral square functions and the Littlewood–Paley g-functions. We note that our conditions on the heat kernels $$p_t(x,y)$$ are mild in the sense that the associated kernels of the square functions do not possess enough regularity for those operators to belong to the standard class of Calderon–Zygmund operators.

Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type

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 $\varphi:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ satisfies that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights). Let $H_{\varphi,\,L}(\mathcal{X})$ be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors characterize the space $H_{\varphi,\,L}(\mathcal{X})$ by means of atoms, non-tangential and radial maximal functions associated with $L$. In particular, when $\mu(\mathcal{X})<\infty$, the local non-tangential and radial maximal function characterizations of $H_{\varphi,\,L}(\mathcal{X})$ are obtained. As applications, the authors obtain various maximal function and the atomic characterizations of the "geometric" Musielak-Orlicz-Hardy spaces $H_{\varphi,\,r}(\Omega)$ and $H_{\varphi,\,z}(\Omega)$ on the strongly Lipschitz domain $\Omega$ in $\mathbb{R}^n$ associated with second-order self-adjoint elliptic operators with the Dirichlet and the Neumann boundary conditions; even when $\varphi(x,t):=t$ for any $x\in\mathbb{R}^n$ and $t\in[0,\infty)$, the equivalent characterizations of $H_{\varphi,\,z}(\Omega)$ given in this article improve the known results via removing the assumption that $\Omega$ is unbounded.

Lusin area integrals related to Jacobi expansions

We investigate mixed Lusin area integrals associated with Jacobi trigonometric polynomial expansions. We prove that these operators can be viewed as vector-valued Calderón–Zygmund operators in the sense of the associated space of homogeneous type. Consequently, their various mapping properties, in particular on weighted L^p spaces, follow from the general theory.

Generalized Littlewood–Paley characterizations of fractional Sobolev spaces

In this paper, the authors characterize the Sobolev spaces [Formula: see text] with [Formula: see text] and [Formula: see text] via a generalized Lusin area function and its corresponding Littlewood–Paley [Formula: see text]-function. The range [Formula: see text] is also proved to be nearly sharp in the sense that these new characterizations are not true when [Formula: see text] and [Formula: see text]. Moreover, in the endpoint case [Formula: see text], the authors also obtain some weak type estimates. Since these generalized Littlewood–Paley functions are of wide generality, these results provide some new choices for introducing the notions of fractional Sobolev spaces on metric measure spaces.

Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Let $$\vec {a}:=(a_1,\ldots ,a_n)\in [1,\infty )^n$$ , $$\vec {p}:=(p_1,\ldots ,p_n)\in (0,\infty )^n$$ and $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ be the anisotropic mixed-norm Hardy space associated with $$\vec {a}$$ defined via the non-tangential grand maximal function. In this article, via first establishing a Calderon–Zygmund decomposition and a discrete Calderon reproducing formula, the authors then characterize $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ , respectively, by means of atoms, the Lusin area function, the Littlewood–Paley g-function or $$g_{\lambda }^*$$ -function. The obtained Littlewood–Paley g-function characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ coincidentally confirms a conjecture proposed by Hart et al. (Trans Am Math Soc, https://doi.org/10.1090/tran/7312 , 2017). Applying the aforementioned Calderon–Zygmund decomposition as well as the atomic characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ , the authors establish a finite atomic characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ , which further induces a criterion on the boundedness of sublinear operators from $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of anisotropic Calderon–Zygmund operators from $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ to itself [or to the mixed-norm Lebesgue space $$L^{\vec {p}}(\mathbb {R}^n)$$ ]. The obtained atomic characterizations of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ and boundedness of anisotropic Calderon–Zygmund operators on these Hardy-type spaces positively answer two questions mentioned by Cleanthous et al. (J Geom Anal 27:2758–2787, 2017). All these results are new even for the isotropic mixed-norm Hardy spaces on $$\mathbb {R}^n$$ .