Littlewood Paley G-function 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).

The multilinear Littlewood-Paley square operators and their commutators on weighted Morrey spaces

In this paper, we prove the boundedness of the multilinear Littlewood-Paley square operators and their commutators on weighted Morrey spaces, then we give the boundedness and weak-type $$L\log L$$ estimates for the commutators of multilinear Littlewood-Paley g-functions and multilinear Marcinkiewicz integrals on weighted Morrey spaces in the form of corollaries.

Estimates for Littlewood–Paley Operators on Ball Campanato-Type Function Spaces

Let X be a ball quasi-Banach function space on $${{\mathbb {R}}}^n$$ and assume that the Hardy–Littlewood maximal operator satisfies the Fefferman–Stein vector-valued maximal inequality on X, and let $$q\in [1,\infty )$$ and $$d\in (0,\infty )$$ . In this article, the authors prove that, for any $$f\in {\mathcal {L}}_{X,q,0,d}({\mathbb {R}}^n)$$ (the ball Campanato-type function space associated with X), the Littlewood–Paley g-function g(f) is either infinite everywhere or finite almost everywhere and, in the latter case, g(f) is bounded on $${\mathcal {L}}_{X,q,0,d}({\mathbb {R}}^n)$$ . Similar results for both the Lusin-area function and the Littlewood–Paley $$g_\lambda ^*$$ -function are also obtained. All these results have a wide range of applications. In particular, even when X is the weighted Lebesgue space, or the mixed-norm Lebesgue space, or the variable Lebesgue space, or the Orlicz space, or the Orlicz-slice space, all these results are new. The proofs of all these results strongly depend on several delicate estimates of Littlewood–Paley operators on the mean oscillation of any locally integrable function f on $${\mathbb {R}}^n$$ . Moreover, the same ideas are also used to obtain the corresponding results for the special John–Nirenberg–Campanato space via congruent cubes.

Optimal orders of the best constants in the Littlewood-Paley inequalities

Let {Pt}t>0 be the classical Poisson semigroup on Rd and GP the associated Littlewood-Paley g-function operator:GP(f)=(∫0∞t|∂∂tPt(f)|2dt)12. The classical Littlewood-Paley g-function inequality asserts that for any 1<p<∞ there exist two positive constants Lt,pP and Lc,pP such that(Lt,pP)−1‖f‖p≤‖GP(f)‖p≤Lc,pP‖f‖p,f∈Lp(Rd). We determine the optimal orders of magnitude on p of these constants as p→1 and p→∞. We also consider similar problems for more general test functions in place of the Poisson kernel.The corresponding problem on the Littlewood-Paley dyadic square function inequality is investigated too. Let Δ be the partition of Rd into dyadic rectangles and SR the partial sum operator associated to R. The dyadic Littlewood-Paley square function of f isSΔ(f)=(∑R∈Δ|SR(f)|2)12. For 1<p<∞ there exist two positive constants Lc,p,dΔ and Lt,p,dΔ such that(Lt,p,dΔ)−1‖f‖p≤‖SΔ(f)‖p≤Lc,p,dΔ‖f‖p,f∈Lp(Rd). We show thatLt,p,dΔ≈d(Lt,p,1Δ)d and Lc,p,dΔ≈d(Lc,p,1Δ)d.All the previous results can be equally formulated for the d-torus Td. We prove a de Leeuw type transference principle in the vector-valued setting.

Boundedness of operators generated by fractional semigroups associated with Schrödinger operators on Campanato type spaces via T1 theorem

Let $${\mathcal {L}}=-\varDelta +V$$ be a Schrödinger operator, where the nonnegative potential V belongs to the reverse Hölder class $$B_{q}$$ . By the aid of the subordinative formula, we estimate the regularities of the fractional heat semigroup, $$\{e^{-t{\mathcal {L}}^{\alpha }}\}_{t>0},$$ associated with $${\mathcal {L}}$$ . As an application, we obtain the BMO $$^{\gamma }_{{\mathcal {L}}}$$ -boundedness of the maximal function, and the Littlewood–Paley g-functions associated with $${\mathcal {L}}$$ via T1 theorem, respectively.

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.

Generalized Homogeneous Littlewood–Paley g-Function on Some Function Spaces

Let $$({{\mathcal {X}}},d,\mu )$$ be a non-homogeneous metric measure space satisfying the so-called upper doubling and the geometrically doubling conditions in the sense of Hytonen. Under the assumption that the dominating function $$\lambda $$ satisfies weak reverse doubling conditions, in this paper, the authors prove that the generalized homogeneous Littlewood–Paley g-function $${\dot{g}}_{r} (r\in [2,\infty ))$$ is bounded from the Lipschitz spaces $${\mathrm{Lip}}_{\beta }(\mu )$$ into the Lipschitz spaces $${\mathrm{Lip}}_{\beta }(\mu )$$ for $$\beta \in (0,1)$$ , and the commutator $${\dot{g}}_{r,b}$$ generated by the $$b\in {\mathrm{Lip}}_{\beta }(\mu )$$ and the $${\dot{g}}_{r}$$ is bounded on the Lebesgue space $$L^{p}(\mu )$$ with $$p\in (1,+\infty )$$ . Furthermore, the boundedness of the $${\dot{g}}_{r}$$ and the commutator $${\dot{g}}_{r,b}$$ on generalized Morrey space $$L^{p,\phi }(\mu )$$ is also obtained, respectively.

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.

Weighted estimates for bilinear square functions with non-smooth kernels and commutators

Under weaker conditions on the kernel functions, we discuss the boundedness of bilinear square functions associated with non-smooth kernels on the product of weighted Lebesgue spaces. Moreover, we investigate the weighted boundedness of the commutators of bilinear square functions (with symbols which are BMO functions and their weighted version, respectively) on the product of Lebesgue spaces. As an application, we deduce the corresponding boundedness of bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley g-functions.

Hardy–Littlewood–Sobolev type inequalities associated with the Weinstein operator

We introduce the Riesz potential associated with the Weinstein operator of order s, as where is its heat semigroup. Next, we define the fractional Littlewood–Paley g-function denoted of order s, as where is the Poisson integral of f. We establish, for the two operators, the following Hardy–Littlewood–Sobolev type inequalities. For , p>1 and , there exists a positive constant such that for every in the appropriate Lebesgue space

Equivalent characterizations of Hardy spaces with variable exponent via wavelets

Via the boundedness of intrinsic g-functions from the Hardy spaces with variable exponent, ℋp(·)(ℝn), into Lebesgue spaces with variable exponent, Lp(·)(ℝn), and establishing some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of ℋp(·)(ℝn) in terms of wavelets, which extend the wavelet characterizations of the classical Hardy spaces. The main ingredients are that, we overcome the difficulties of the quasi-norms of ℋp(·)(ℝn) by elaborately using an observation and the Fefferman-Stein vector-valued maximal inequality on Lp(·)(ℝn), and also overcome the difficulty of the failure of q = 2 in the atomic decomposition of ℋp(·)(ℝn) by a known idea.

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 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 $$\mathbb {R}^n$$ .

Non-homogeneous Tb Theorem for Bi-parameter g-function

The main result of this paper is a bi-parameter Tb theorem for Littlewood–Paley g-function, where b is a tensor product of two pseudo-accretive function. Instead of the doubling measure, we work with a product measure μ = μn × μm, where the measures μn and μm are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter b-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again.

Littlewood–Paley and Finite Atomic Characterizations of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Let $$p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]$$ be a variable exponent function satisfying the globally log-Holder continuous condition, $$q\in (0,\infty ]$$ and A be a general expansive matrix on $${\mathbb {R}}^n$$ . Let $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ by means of the Littlewood–Paley g-function or the Littlewood–Paley $$g_\lambda ^*$$ -function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ . Moreover, the finite atomic characterization of $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ to $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.

Littlewood-Paley g-function and Radon Transform on the Heisenberg Group

In this paper, we consider Radon transform on the Heisenberg group $\textbf{H}^{n}$, and obtain new inversion formulas via dual Radon transforms and Poisson integrals. We prove that the Radon transform is a unitary operator from Sobelov space $W$ into $L^{2}(\textbf{H}^{n})$. Moreover, we use the Radon transform to define the Littlewood-Paley $g$-function on a hyperplane and obtain the Littlewood-Paley theory.

Littlewood-Paley characterizations of anisotropic Hardy-Lorentz spaces

Let p∈ (0,1], q ∈(0,∞] and A be a general expansive matrix on ℝn. Let HAp,q(ℝn) be the anisotropic Hardy-Lorentz spaces associated with A defined via the non-tangential grand maximal function. In this article, the authors characterize HAp,q(ℝn) in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley gλ*-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space Lp,qℝn. All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on ℝn. Moreover, the range of λ in the gλ*-function characterization of HAp,q(ℝn) coincides with the best known one in the classical Hardy space Hp(ℝn) or in the anisotropic Hardy space HAp(ℝn).

Tent spaces and Littlewood-Paley g-functions associated with Bergman spaces in the unit ball of ℂn

ABSTRACTIn this paper, a family of holomorphic spaces of tent type in the unit ball of is introduced, which is closely related to maximal and area integral functions in terms of the Bergman metric. It is shown that these spaces coincide with Bergman spaces. Furthermore, Littlewood-Paley type g-functions for the Bergman spaces are introduced in terms of the radial derivative, the complex gradient and the invariant gradient. The corresponding characterizations for Bergman spaces are obtained as well. As an application, we obtain new maximal and area integral characterizations for Besov spaces.

Littlewood–Paley g-function associated with the Weinstein operator

ABSTRACTUsing harmonic analysis associated with the Weinstein operator on we study the associated Poisson semigroup. Next, in this context, we study the Littlewood–Paley function denoted . Especially, we establish that for , there exist two positive constants and such that for every f in the appropriate Lebesgue space ,

Endpoint estimates of generalized homogeneous Littlewood–Paley g-functions over non-homogeneous metric measure spaces

Let (X, d, μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Under the weak reverse doubling condition, the authors prove that the generalized homogeneous Littlewood–Paley g-function ġr (r ∈ [2,∞)) is bounded from Hardy space H1(μ) into L1(μ). Moreover, the authors show that, if f ∈ RBMO(μ), then [ġr(f)]r is either infinite everywhere or finite almost everywhere, and in the latter case, [ġr(f)]r belongs to RBLO(μ) with the norm no more than ‖f‖RBMO(μ) multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of ġr from RBMO(μ) into RBLO(μ). The vector valued Calderon–Zygmund theory over (X, d, μ) is also established with details in this paper.