Zygmund Operators Research Articles

Weighted norm inequalities for multilinear strongly singular Calderón–Zygmund operators on RD‐spaces

AbstractLet be an RD‐space, namely, a space of homogeneous type in the sense of Coifman and Weiss with the Borel measure μ satisfying the reverse doubling condition on . Based on this space, the authors define a multilinear strongly singular Calderón–Zygmund operator whose kernel does not need any size condition and has more singularities near the diagonal than that of a standard multilinear Calderón–Zygmund operator. For such an operator, we establish its boundedness on product of weighted Lebesgue spaces by means of the pointwise estimate for the sharp maximal function. In addition, the endpoint estimates of the type are also obtained. Moreover, we prove weighted boundedness results for multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions. These results contribute to the extension of multilinear strongly singular Calderón–Zygmund operator theory in the Euclidean case to the context of space of homogeneous type.

Strongly singular Calderón–Zygmund operators on Hardy spaces associated with ball quasi-Banach function spaces

Strongly singular Calderón–Zygmund operators on Hardy spaces associated with ball quasi-Banach function spaces

$$\Theta $$-type Calderón–Zygmund operator and its commutator on (grand) generalized weighted variable exponent Morrey space over RD-spaces

$$\Theta $$-type Calderón–Zygmund operator and its commutator on (grand) generalized weighted variable exponent Morrey space over RD-spaces

Weighted Norm Inequalities for Calderón–Zygmund Operators of $$\phi$$-Type and Their Commutators

Weighted Norm Inequalities for Calderón–Zygmund Operators of $$\phi$$-Type and Their Commutators

The commutators of multilinear Calderón–Zygmund operators on weighted Hardy spaces

In this paper, we study the behaviours of the commutators $[\vec b,\,T]$ generated by multilinear Calderón–Zygmund operators $T$ with $\vec b=(b_1,\,\ldots,\,b_m)\in L_{\rm loc}(\mathbb {R}^n)$ on weighted Hardy spaces. We show that for some $p_i\in (0,\,1]$ with $1/p=1/p_1+\cdots +1/p_m$ , $\omega \in A_\infty$ and $b_i\in \mathcal {BMO}_{\omega,p_i}$ ( $1\le i\le m$ ), which are a class of non-trivial subspaces of ${\rm BMO}$ , the commutators $[\vec b,\,T]$ are bounded from $H^{p_1}(\omega )\times \cdots \times H^{p_m}(\omega )$ to $L^p(\omega )$ . Meanwhile, we also establish the corresponding results for a class of maximal truncated multilinear commutators $T_{\vec b}^*$ .

Causal sparse domination of Beurling maximal regularity operators

We prove boundedness of Calderön–Zygmund operators acting in Banach function spaces on domains, defined by the L1 Carleson functional and Lq (1 < q < ∞) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.

Multilinear commutators of multilinear strongly singular integral operators with generalized kernels

Abstract In [S. Yang and Y. Lin, Multilinear strongly singular integral operators with generalized kernels and applications, AIMS Math. 6 2021, 12, 13533–13551], the authors of the present paper further weaken the smoothness condition of kernel functions with multilinear strongly singular Calderón–Zygmund operators of [Y. Lin, Multilinear theory of strongly singular Calderón–Zygmund operators and applications, Nonlinear Anal. 192 2020, Article ID 111699]. They defined a new class of multilinear strongly singular integral operators, and studied its weighted L p {L^{p}} boundedness, variable exponent L p ⁢ ( ⋅ ) {L^{p(\cdot)}} boundedness and endpoint estimates. In this paper, we naturally consider the boundedness of multilinear commutators and multilinear iterated commutators, which are generated by multilinear strongly singular integral operators with generalized kernels and Lipschitz functions. Our results include the corresponding results of multilinear strongly singular Calderón–Zygmund operators and classical multilinear Calderón–Zygmund operators, respectively.

Calderón–Zygmund operators, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents

We obtain the boundedness of Calderón–Zygmund operators on the entire Hardy–Morrey spaces with variable exponents. We obtain this result by refining the extrapolation theory. This refined extrapolation theory also gives the boundedness of Fourier multipliers, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents.

Atomic Characterization of Musielak–Orlicz–Lorentz Hardy Spaces and Its Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators

Atomic Characterization of Musielak–Orlicz–Lorentz Hardy Spaces and Its Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators

Fractional Bloom boundedness and compactness of commutators

Abstract Let T be a non-degenerate Calderón–Zygmund operator and let b : ℝ d → ℂ {b:\mathbb{R}^{d}\to\mathbb{C}} be locally integrable. Let 1 &lt; p ≤ q &lt; ∞ {1&lt;p\leq q&lt;\infty} and let μ p ∈ A p {\mu^{p}\in A_{p}} and λ q ∈ A q {\lambda^{q}\in A_{q}} , where A p {A_{p}} denotes the usual class of Muckenhoupt weights. We show that ∥ [ b , T ] ∥ L μ p → L λ q ∼ ∥ b ∥ BMO ν α , [ b , T ] ∈ 𝒦 ⁢ ( L μ p , L λ q ) iff b ∈ VMO ν α , \lVert[b,T]\rVert_{L^{p}_{\mu}\to L^{q}_{\lambda}}\sim\lVert b\rVert_{% \operatorname{BMO}_{\nu}^{\alpha}},\quad[b,T]\in\mathcal{K}(L^{p}_{\mu},L^{q}_% {\lambda})\quad\text{iff}\quad b\in\operatorname{VMO}_{\nu}^{\alpha}, where L μ p = L p ⁢ ( μ p ) {L^{p}_{\mu}=L^{p}(\mu^{p})} and α / d = 1 / p - 1 / q {\alpha/d=1/p-1/q} , the symbol 𝒦 {\mathcal{K}} stands for the class of compact operators between the given spaces, and the fractional weighted BMO ν α {\operatorname{BMO}_{\nu}^{\alpha}} and VMO ν α {\operatorname{VMO}_{\nu}^{\alpha}} spaces are defined through the following fractional oscillation and Bloom weight: 𝒪 ν α ⁢ ( b ; Q ) = ν ⁢ ( Q ) - α / d ⁢ ( 1 ν ⁢ ( Q ) ⁢ ∫ Q | b - 〈 b 〉 Q | ) , ν = ( μ λ ) β , β = ( 1 + α / d ) - 1 . \mathcal{O}_{\nu}^{\alpha}(b;Q)=\nu(Q)^{-\alpha/d}\biggl{(}\frac{1}{\nu(Q)}% \int_{Q}\lvert b-\langle b\rangle_{Q}\rvert\biggr{)},\quad\nu=\biggl{(}\frac{% \mu}{\lambda}\biggr{)}^{\beta},\quad\beta=(1+\alpha/d)^{-1}. The key novelty is dealing with the off-diagonal range p &lt; q {p&lt;q} , whereas the case p = q {p=q} was previously studied by Lacey and Li. However, another novelty in both cases is that our approach allows complex-valued functions b, while other arguments based on the median of b on a set are inherently real-valued.

Maximal Operator, Cotlar’s Inequality and Pointwise Convergence for Singular Integral Operators in Dunkl Setting

We establish the maximal operator, Cotlar’s inequality and pointwise convergence in the Dunkl setting for the (nonconvolution type) Dunkl–Calderón–Zygmund operators introduced recently in Tan et al. (https://arxiv.org/abs/2204.01886). The fundamental geometry of the Dunkl setting involves two nonequivalent metrics: the Euclidean metric and the Dunkl metric deduced by finite reflection groups, and hence the classical methods do not apply directly. The key idea is to introduce truncated singular integrals and the maximal singular integrals by the Dunkl metric and the Euclidean metric. We show that these two kind of truncated singular integrals are dominated by the Hardy–Littlewood maximal function, which yields the Cotlar’s inequalities and hence the boundedness of maximal Dunkl–Calderón–Zygmund operators. Further, as applications, two equivalent pointwise convergences for Dunkl–Calderón–Zygmund operators are obtained.

Existence of unconditional wavelet bases for Lp-norm over a local fields of positive characteristic

In this paper, we defined Calder[Formula: see text]n–Zygmund operator [Formula: see text] as [Formula: see text] on local fields ([Formula: see text]) under certain conditions on [Formula: see text] and then find the boundedness of operator [Formula: see text]. Using result on boundedness of [Formula: see text] and orthonormal wavelet basis of [Formula: see text], we provide unconditional wavelet bases for [Formula: see text]-norm over a local fields of positive characteristic by the wavelet coefficients.

Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights

Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights

The Boundedness of Commutators of Sublinear Operators on Herz Triebel–Lizorkin Spaces with Variable Exponent

In this paper, the authors first discuss the characterization of Herz Triebel–Lizorkin spaces with variable exponent via two families of operators. By this characterization, the authors prove that the Lipschitz commutators of sublinear operators is bounded from Herz spaces with variable exponent to Herz Triebel–Lizorkin spaces with variable exponent. As applications, the corresponding boundedness estimates for the commutators of maximal operator, Riesz potential operator and Calderón–Zygmund operator are established.

Multi-Parameter Hardy Spaces Theory and Endpoint Estimates for Multi-Parameter Singular Integrals

The main purpose of this paper is to establish the theory of the multi-parameter Hardy spaces H p H^p ( 0 &gt; p ≤ 1 0&gt;p\leq 1 ) associated to a class of multi-parameter singular integrals extensively studied in the recent book of B. Street (2014), where the L p L^p ( 1 &gt; p &gt; ∞ ) (1&gt;p&gt;\infty ) estimates are proved for this class of singular integrals. This class of multi-parameter singular integrals are intrinsic to the underlying multi-parameter Carnot-Carathéodory geometry, where the quantitative Frobenius theorem was established by B. Street (2011), and are closely related to both the one-parameter and multi-parameter settings of singular Radon transforms considered by Stein and Street (2011, 2012a, 2012b, 2013). More precisely, Street (2014) studied the L p L^p ( 1 &gt; p &gt; ∞ ) (1&gt;p&gt;\infty ) boundedness, using elementary operators, of a type of generalized multi-parameter Calderón Zygmund operators on smooth and compact manifolds, which include a certain type of singular Radon transforms. In this work, we are interested in the endpoint estimates for the singular integral operators in both one and multi-parameter settings considered by Street (2014). Actually, using the discrete Littlewood-Paley-Stein analysis, we will introduce the Hardy space H p H^p ( 0 &gt; p ≤ 1 0&gt;p\leq 1 ) associated with the multi-parameter structures arising from the multi-parameter Carnot-Carathéodory metrics using the appropriate discrete Littlewood-Paley-Stein square functions, and then establish the Hardy space boundedness of singular integrals in both the single and multi-parameter settings. Our approach is much inspired by the work of Street (2014) where he introduced the notions of elementary operators so that the type of singular integrals under consideration can be decomposed into elementary operators.

Commutators of multilinear Calderón–Zygmund operators with kernels of Dini’s type on generalized weighted Morrey spaces and applications

Commutators of multilinear Calderón–Zygmund operators with kernels of Dini’s type on generalized weighted Morrey spaces and applications

Musielak–Orlicz Hardy space estimates for commutators of Calderón–Zygmund operators

AbstractLet and T be a δ‐Calderón–Zygmund operator. When and , it is well‐known (see the work by Harboure, Segovia, and Torrea [Illinois J. Math. 41 (1997), no. 4, 676–700]) that the commutator is not bounded from the Hardy space into the Lebesgue space if b is not a constant function. Let φ be a Musielak–Orlicz function satisfying that, for any , belongs to the Muckenhoupt weight class with the critical weight exponent and is an Orlicz function with the critical lower type . In this paper, we find a proper subspace of such that, if then is bounded from the Musielak–Orlicz Hardy space into the Musielak–Orlicz space . Conversely, if and the commutators of the classical Riesz transforms are bounded from into , then . Our results generalize some recent results by Huy and Ky [Vietnam J. Math. (2020). https://doi.org/10.1007/s10013‐020‐00406‐2] and Liang, Ky, and Yang [Proc. Amer. Math. Soc. 144 (2016), no. 12, 5171–5181].

Weighted Variable Hardy Spaces Associated with Para-Accretive Functions and Boundedness of Calderón–Zygmund Operators

Weighted Variable Hardy Spaces Associated with Para-Accretive Functions and Boundedness of Calderón–Zygmund Operators

Weighted Lipschitz estimates for commutators of multilinear Calderón–Zygmund operators with Dini type kernels

In this paper, we discuss the properties for commutators and iterated commutators generated by the multilinear ω-CZO and weighted Lipschitz functions on Lebesgue space.

Calderón–Zygmund operators on RBMO

Let μ \mu be an n n -dimensional finite positive measure on R m \mathbb {R}^m . We obtain a T 1 T1 condition sufficient for the boundedness of Calderón–Zygmund operators on R B M O ( μ ) \mathrm {RBMO}(\mu ) , the regular BMO space of Tolsa [Math. Ann. 319 (2001), pp. 89–149].