Littlewood-Paley Characterization Research Articles

Inclusion Relations Among Fractional Orlicz-Sobolev Spaces and a Littlewood-Paley Characterization

AbstractEmbeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. In particular, besides recovering standard embeddings for classical fractional Sobolev spaces, novel results are derived in borderline situations where the latter fail. For instance, limiting embeddings of Pohozhaev-Trudinger-Yudovich type into exponential spaces are offered. The equivalence of Gagliardo-Slobodeckij norms in fractional Orlicz-Sobolev spaces to norms defined via Littlewood-Paley decompositions, oscillations, or Besov type difference quotients is established as well. This equivalence, of independent interest, is a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.

Littlewood–Paley and wavelet characterization for mixed Morrey spaces

AbstractIn this paper, we consider the Littlewood–Paley characterization for mixed Morrey spaces and its predual spaces. The topology to converge the Littlewood–Paley decomposition for the element of mixed Morrey spaces is the weak‐* topology. If we consider the topology of mixed Morrey spaces, we must give other characterization by using the heat semigroup. As an application, we show the wavelet characterization for mixed Morrey spaces. In particular, this characterization can be shown without the Peetre maximal operator.

Weighted local Hardy spaces with variable exponents

AbstractThis paper defines local weighted Hardy spaces with variable exponents. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As applications, we obtain the boundedness of singular integral operators, the Littlewood–Paley characterization, and wavelet decomposition.

The T1 theorem for the generalized product Calderón-Zygmund operator on product endpoint function spaces over RD spaces

The Littlewood-Paley characterization of the product Lipschitz endpoint function spaces and the density argument in the weak sense is established on product RD spaces, where the measures satisfy the doubling condition and the reverse doubling condition, and the quasi-metric satisfies the Hölder regularities. Furthermore, we get the necessary and sufficient condition $T_1(1)=T_2(1)=0$ for the generalized product Calderón-Zygmund operator on product Lipschitz spaces.

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Littlewood—Paley Characterizations with Applications to Boundedness of Calderón—Zygmund Operators

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Littlewood—Paley Characterizations with Applications to Boundedness of Calderón—Zygmund Operators

Littlewood–Paley Characterization for Musielak–Orlicz–Hardy Spaces Associated with Self-Adjoint Operators

Let X , d , μ be a metric measure space endowed with a metric d and a non-negative Borel doubling measure μ . Let L be a non-negative self-adjoint operator on L 2 X . Assume that the (heat) kernel associated to the semigroup e − t L satisfies a Gaussian upper bound. In this paper, we prove that the Musielak–Orlicz–Hardy space H φ , L X associated with L in terms of the Lusin-area function and the Musielak–Orlicz–Hardy space H L , G , φ X associated with L in terms of the Littlewood–Paley function coincide and their norms are equivalent. To do this, we first establish the discrete characterization of these two spaces. It improves the known results in the literature.

Wavelet characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type and its applications

In this article, the authors establish the wavelet characterization of Besov and Triebel–Lizorkin spaces on a given space ( X , d , μ ) of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel–Lizorkin sequence spaces on X , and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel–Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood–Paley characterizations of Triebel–Lizorkin spaces on X . 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 property of X expressed via its equipped quasi-metric d , dyadic reference points, dyadic cubes, and wavelets.

Littlewood–Paley theory for matrix-weighted function spaces

We define the vector-valued, matrix-weighted function spaces \(\dot{F}^{\alpha q}_p(W)\) (homogeneous) and \(F^{\alpha q}_p(W)\) (inhomogeneous) on \({\mathbb {R}}^n\), for \(\alpha \in {\mathbb {R}}\), \(0<p<\infty \), \(0<q \le \infty \), with the matrix weight W belonging to the \(A_p\) class. For \(1<p<\infty \), we show that \(L^p(W) = \dot{F}^{0 2}_p(W)\), and, for \(k \in {\mathbb {N}}\), that \(F^{k 2}_p(W)\) coincides with the matrix-weighted Sobolev space \(L^p_k(W)\), thereby obtaining Littlewood-Paley characterizations of \(L^p(W)\) and \(L^p_k (W)\). We show that a vector-valued function belongs to \(\dot{F}^{\alpha q}_p(W)\) if and only if its wavelet or \(\varphi \)-transform coefficients belong to an associated sequence space \(\dot{f}^{\alpha q}_p(W)\). We also characterize these spaces in terms of reducing operators associated to W.

Littlewood-Paley Characterizations of Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces

Let X be a ball quasi-Banach function space on \({{\mathbb {R}}}^n\). In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X and is bounded on the associated space, the authors establish various Littlewood–Paley function characterizations of the Hardy space \(H_X({{\mathbb {R}}}^n)\) associated with X, under some weak assumptions on the Littlewood–Paley functions. To this end, the authors also establish a useful estimate on the change of angles in tent spaces associated with X. All these results have wide applications. Particularly, when \(X:=M_r^p({{\mathbb {R}}}^n)\) (the Morrey space), \(X:=L^{\vec {p}}({{\mathbb {R}}}^n)\) (the mixed-norm Lebesgue space), \(X:=L^{p(\cdot )}({{\mathbb {R}}}^n)\) (the variable Lebesgue space), \(X:=L_\omega ^p({{\mathbb {R}}}^n)\) (the weighted Lebesgue space) and \(X:=(E_\Phi ^r)_t({{\mathbb {R}}}^n)\) (the Orlicz-slice space), the Littlewood–Paley function characterizations of \(H_X({{\mathbb {R}}}^n)\) obtained in this article improve the existing results via weakening the assumptions on the Littlewood–Paley functions and widening the range of \(\lambda \) in the Littlewood–Paley \(g_\lambda ^*\)-function characterization of \(H_X({\mathbb {R}}^n)\).

Calderón-Zygmund operators on Lipschitz spaces over RD spaces

. Suppose that (X, d, µ) is a metric measure space of homogeneous type in the sense of Coifman and Weiss with a reverse doubling condition. By establishing the Littlewood-Paley characterization of Lipschitz spaces, a density argument in the weak sense and almost orthogonality estimate we prove that a Caldeŕon-Zygmund operator T is bounded on Lipschitz space if and only if T 1 = 0.

The Boundedness of Calderón–Zygmund Operators on Lipschitz Spaces Over Spaces of Homogeneous Type

Let $$(\mathcal {X},\rho ,\mu )$$ be Ahlfors-1 regular metric spaces. By developing the Littlewood–Paley characterization of Lipschitz spaces over $$(\mathcal {X},\rho ,\mu )$$ and establishing a density argument in the weak sense, the authors give a necessary and sufficient condition for the boundedness of Calderon–Zygmund operators on the Lipschitz spaces.

Littlewood–Paley functions and Sobolev spaces

We establish Littlewood–Paley characterizations of the Sobolev spaces Wα,p in Euclidean spaces using several square functions defined via the spherical average, the ball average, the Bochner–Riesz means and some other well known operators. We provide a simple proof so that we are able to extend and improve many results published in recent papers.

Off-diagonal extrapolation on mixed variable Lebesgue spaces and its applications to strong fractional maximal operators

Abstract We establish off-diagonal extrapolation on mixed variable Lebesgue spaces. As its applications, we obtain the boundedness for strong fractional maximal operators. The vector-valued analogies are also considered. Additionally, the Littlewood–Paley characterization for mixed variable Lebesgue spaces is also established with the help of weighted norm inequalities and extrapolation.

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

Littlewood–Paley characterizations of fractional Sobolev spaces via averages on balls

By invoking some new ideas, we characterize Sobolev spaces Wα,p(ℝn) with the smoothness order α ∊ (0, 2] and p ∊ (max{1, 2n/(2α + n)},∞), via the Lusin area function and the Littlewood–Paley g*λ-function in terms of centred ball averages. We also show that the assumption p ∊ (max{1, 2n/(2α + n)},∞) is nearly sharp in the sense that these characterizations are no longer true when p ∊ (1, max{1, 2n/(2α + n)}). These characterizations provide a possible new way to introduce Sobolev spaces with smoothness order in (1, 2] on metric measure spaces.

LITTLEWOOD–PALEY CHARACTERIZATIONS OF ANISOTROPIC WEAK MUSIELAK–ORLICZ HARDY SPACES

Let $A$ be an expansive dilation on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D711}:\mathbb{R}^{n}\times [0,\infty )\rightarrow [0,\infty )$ an anisotropic growth function. In this article, the authors introduce the anisotropic weak Musielak–Orlicz Hardy space $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ via the nontangential grand maximal function and then obtain its Littlewood–Paley characterizations in terms of the anisotropic Lusin-area function, $g$-function or $g_{\unicode[STIX]{x1D706}}^{\ast }$-function, respectively. All these characterizations for anisotropic weak Hardy spaces $\mathit{WH}_{A}^{p}(\mathbb{R}^{n})$ (namely, $\unicode[STIX]{x1D711}(x,t):=t^{p}$ for all $t\in [0,\infty )$ and $x\in \mathbb{R}^{n}$ with $p\in (0,1]$) are new. Moreover, the range of $\unicode[STIX]{x1D706}$ in the anisotropic $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ coincides with the best known range of the $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of classical Hardy space $H^{p}(\mathbb{R}^{n})$ or its weighted variants, where $p\in (0,1]$.

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

AbstractIn this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .

Inhomogeneous Lipschitz spaces of variable order and their applications

In this article, the authors first give a Littlewood–Paley characterization for inhomogeneous Lipschitz spaces of variable order with the help of inhomogeneous Calderon identity and almost-orthogonality estimates. As applications, the boundedness of inhomogeneous Calderon–Zygmund singular integral operators of order (ϵ,σ) on these spaces has been presented. Finally, we note that a class of pseudodifferential operators Ta∈OpS1,10 are continuous on the inhomogeneous Lipschitz spaces of variable order as a corollary. We may observe that those operators are not, in general, continuous in L2.

Littlewood–Paley characterizations of Triebel–Lizorkin–Morrey spaces via ball averages

Let 1<p≤u<∞ and q∈(1,∞]. In this article, the authors establish equivalent characterizations of the Triebel–Lizorkin–Morrey space Eu,q,pα(Rn) with smoothness order α∈(0,2) in terms of the Lusin-area function and the gλ∗-function generated by the difference between f(x) and its ball average B2−jf(x):=1|B(x,2−j)|∫B(x,2−j)f(y)dy,∀j∈{1,2,…},∀x∈Rn, where, for any j∈{1,2,…} and x∈Rn, B(x,2−j):={y∈Rn:|y−x|<2−j}. As applications, the authors obtain several characterizations of Eu,∞,pα(Rn) via pointwise inequalities involving ball averages.