Zygmund Operators Research Articles

Weighted Bounds for Multilinear Square Functions

Let $\vec {P}=(p_{1},\dotsc ,p_{m})$ with 1 < p 1, …, p m < ∞, 1/p 1+⋯+1/p m =1/p and $\vec {w}=(w_{1},\dotsc ,w_{m})\in A_{\vec {P}}$ . In this paper, we investigate the weighted bounds with dependence on aperture α for multilinear square functions $S_{\alpha ,\psi }(\vec {f})$ . We show that $$\|S_{\alpha,\psi}(\vec{f})\|_{L^{p}(\nu_{\vec{w}})} \leq C_{n,m,\psi,\vec{P}} \alpha^{mn}[\vec{w}]_{A_{\vec{P}}}^{\max(\frac{1}{2},\tfrac{p_{1}^{\prime}}{p},\dotsc,\tfrac{p_{m}^{\prime}}{p})} \prod\limits_{i=1}^{m} \|f_{i}\|_{L^{p_{i}}(w_{i})}. $$ This result extends the result in the linear case which was obtained by Lerner in 2014. Our proof is based on the local mean oscillation technique presented firstly to find the weighted bounds for Calderón–Zygmund operators. This method helps us avoiding intrinsic square functions in the proof of our main result.

Variable weak Hardy spaces and their applications

Let p(⋅):Rn→(0,∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on Rn, WHp(⋅)(Rn), via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of WHp(⋅)(Rn), respectively, by means of atoms, molecules, the Lusin area function, the Littlewood–Paley g-function or gλ⁎-function. As an application, the authors establish the boundedness of convolutional δ-type and non-convolutional γ-order Calderón–Zygmund operators from Hp(⋅)(Rn) to WHp(⋅)(Rn) including the critical case when p−=n/(n+δ) or when p−=n/(n+γ), where p−:=essinfx∈Rnp(x).

Characterizations of bounded mean oscillation through commutators of bilinear singular integral operators

We characterize bounded mean oscillation in terms of the boundedness of commutators of various bilinear singular integral operators with pointwise multiplication. In particular, we study commutators of a wide class of bilinear operators of convolution type, including bilinear Calderón–Zygmund operators and the bilinear fractional integral operators.

Orthogonal Polynomials on the Circle for the Weight w Satisfying Conditions $$w,w^{-1}\in \mathrm{BMO}$$ w , w - 1 ∈ BMO

For the weight w satisfying \(w,w^{-1}\in \mathrm{BMO}({\mathbb {T}})\), we prove the asymptotics of \(\{\Phi _n(e^{i\theta },w)\}\) in \(L^p[-\pi ,\pi ], 2\leqslant p<p_0\), where \(\{\Phi _n(z,w)\}\) are monic polynomials orthogonal with respect to w on the unit circle \({\mathbb {T}}\). Immediate applications include the estimates on the uniform norm and asymptotics of the polynomial entropies. The estimates on higher-order commutators between the Calderon–Zygmund operators and BMO functions play the key role in the proofs of main results.

Generalized commutators of multilinear Calderón–Zygmund type operators

Let $T$ be an $m$-linear Calderon–Zygmund operator with kernel $K$ and $T^*$ be the maximal operator of $T$. Let $S$ be a finite subset of $Z^+\times \{1,\dots,m\}$ and denote $d\vec{y}=dy_1\cdots dy_m$. Define the commutator $T_{\vec{b},S}$ of $T$, and $T^*_{\vec{b},S}$ of $T^*$ by $T_{\vec{b},S}(\vec{f})(x)= \int_{\mathbb{R}^{nm}}\prod_{(i,j)\in S}(b_i(x)-b_i(y_j)) K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_j(y_j)d\vec{y}$ and $T^*_{\vec{b},S}(\vec{f})(x)= \sup_{\delta>0}\big|\int_{{\sum_{j=1}^m|x-y_j|^2>\delta^2}} \prod_{(i,j)\in S}(b_i(x)-b_i(y_j))K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_{j}(y_j)d\vec{y}\big|$. These commutators are reflexible enough to generalize several kinds of commutators which already existed. We obtain the weighted strong and endpoint estimates for $T_{\vec{b},S}$ and $T^*_{\vec{b},S}$ with multiple weights. These results are based on an estimate of the Fefferman–Stein sharp maximal function of the commutators, which is proved in a pretty much more organized way than some known proofs. Similar results for the commutators of vector-valued multilinear Calderon–Zygmund operators are also given.

Multilinear integral operators in weighted grand Lebesgue spaces

Abstract The boundedness of multi(sub)linear Hardy&amp;ndash;Littlewood maximal, Calder&amp;oacute;n&amp;ndash;Zygmund and fractional integral operators defined on metric measure spaces is established in weighted grand Lebesgue spaces. In particular, we derive the one-weight inequality for maximal and singular integrals under the Muckenhoupt type conditions, weighted Sobolev type theorem and trace type inequality for fractional integrals.

A two weight theorem for $\alpha$-fractional singular integrals with an energy side condition

Let \sigma and \omega be locally finite positive Borel measures on \mathbb{R}^{n} with no common point masses, and let T^{\alpha} be a standard \alpha -fractional Calderón–Zygmund operator on \mathbb{R}^{n} with 0 \leq \alpha &lt; n . Furthermore, assume as side conditions the \mathcal{A}_{2}^{\alpha} conditions and certain \alpha -energy conditions . Then we show that T^{\alpha} is bounded from L^{2}(\sigma ) to L^{2}( \omega ) if the cube testing conditions hold for T^{\alpha} and its dual, and if the weak boundedness property holds for T^{\alpha} . Conversely, if T^{\alpha} is bounded from L^{2}( \sigma ) to L^{2}( \omega ) , then the testing conditions hold, and the weak boundedness condition holds. If the vector of \alpha -fractional Riesz transforms \mathbf{R}_{\sigma }^{\alpha} (or more generally a strongly elliptic vector of transforms) is bounded from L^{2}( \sigma) to L^{2}( \omega ) , then the \mathcal{A}_{2}^{\alpha} conditions hold. We do not know if our energy conditions are necessary when n \geq 2 . The innovations in this higher dimensional setting are the control of functional energy by energy modulo \mathcal{A}_{2}^{\alpha} , the necessity of the \mathcal{A}_{2}^{\alpha} conditions for elliptic vectors, the extension of certain one-dimensional arguments to higher dimensions in light of the differing Poisson integrals used in \mathcal A_2 and energy conditions, and the treatment of certain complications arising from the Lacey–Wick monotonicity lemma. The main obstacle in higher dimensions is thus identified as the pair of energy conditions.

A Calderón–Zygmund operator of higher order Schrödinger type

We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We show that an operator of higher order Schrödinger type is a Calderón–Zygmund operator. We also show that there exist potentials which satisfy our assumptions but are not nonnegative polynomials.

Boundedness of multi-parameter Fourier multiplier operators on Triebel–Lizorkin and Besov–Lipschitz spaces

The main purpose of this paper is three-fold. First, we prove that under the limited smoothness conditions, multi-parameter Fourier multiplier operators are bounded on multi-parameter Triebel–Lizorkin and Besov–Lipschitz spaces by the Littlewood–Paley decomposition and the strong maximal operator. Second, we offer a different and more direct method to deal with the boundedness instead of transforming Fourier multiplier operators into multi-parameter Calderón–Zygmund operators. Third, we also prove the boundedness of multi-parameter Fourier multiplier operators on weighted multi-parameter Triebel–Lizorkin and Besov–Lipschitz spaces when the Fourier multiplier is only assumed with limited smoothness.

Higher order Journé commutators and characterizations of multi-parameter BMO

We characterize Lp boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms. We obtain a two-sided norm estimate that shows that such operators are bounded on Lp if and only if the symbol belongs to the appropriate multi-parameter BMO class. We extend our results to a much more intricate situation; commutators of multiplication by a symbol function and paraproduct-free Journé operators. We show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi-parameter BMO class. In this sense the tensor products of Riesz transforms are a representative testing class for Journé operators.Previous results in this direction do not apply to tensor products and only to Journé operators which can be reduced to Calderón–Zygmund operators. Upper norm estimates of Journé commutators are new even in the case of no iterations. Lower norm estimates for iterated commutators only existed when no tensor products were present. In the case of one dimension, lower estimates were known for products of two Hilbert transforms, and without iterations. New methods using Journé operators are developed to obtain these lower norm estimates in the multi-parameter real variable setting.

Weak Musielak–Orlicz Hardy spaces and applications

Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.

Localized Frames and Compactness

We introduce the concept of weak localization for continuous frames and use this concept to define a class of weakly localized operators. This class contains many important classes of operators, including: short time Fourier transform multipliers, Calderon–Toeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, some pseudodifferential operators, some Calderon–Zygmund operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transforms.

Weighted Calderón–Zygmund Decomposition with Some Applications to Interpolation

Let X be an A1-regular lattice of measurable functions and let Q be a projection that is also a Calderon–Zygmund operator. In this case, it is possible to define a space XQ consisting of functions f ∈ X for which Qf = f in a certain sense. By using the Bourgain approach to interpolation, we show that the couple (L 1 , XQ) is K-closed in (L1, X). This result is sharp in the sense that, in general, A1-regularity cannot be replaced by weaker conditions such as Ap-regularity for p > 1.

Hardy Space Estimates for Littlewood–Paley–Stein Square Functions and Calderón–Zygmund Operators

In this work, we give new sufficient conditions for Littlewood–Paley–Stein square function operators and necessary and sufficient conditions for Calderon–Zygmund operators to be bounded on Hardy spaces $$H^p$$ with indices smaller than 1. New Carleson measure type conditions are defined for Littlewood–Paley–Stein operators, and the authors show that they are sufficient for the associated square function to be bounded from $$H^p$$ into $$L^p$$ . New polynomial growth $$BMO$$ conditions are also introduced for Calderon–Zygmund operators. These results are applied to prove that Bony paraproducts can be constructed such that they are bounded on Hardy spaces with exponents ranging all the way down to zero.

Riesz transform related to a Schrödinger type operator on the Heisenberg group

Abstract In this paper, we consider the Schrödinger type operator H 2 = ( - Δ ℍ n ) 2 + V 2 ${H_2=(-\Delta _{\mathbb {H}^n})^2+V^2}$ on the Heisenberg group ℍ n , where Δ ℍ n $\Delta _{\mathbb {H}^n}$ is the sub-Laplacian and the non-negative potential V belongs to the reverse Hölder class B q 1 for q 1 &gt; Q 2 ${q_1&amp;gt;\frac{Q}{2}}$ and Q &gt; 5, where Q = 2 n + 2 ${Q=2n+2}$ is the homogeneous dimension of ℍ n . The Lp and weak type (1,1) estimates of the Riesz transform ∇ ℍ n 2 H 2 - 1 / 2 $\nabla _{\mathbb {H}^n}^2 H^{-1/2}_2$ related to the Schrödinger type operator H 2 are obtained, where ∇ ℍ n ${\nabla _{\mathbb {H}^n}}$ is the gradient operator on ℍ n . Moreover, we show that ∇ ℍ n 2 H 2 - 1 / 2 $\nabla _{\mathbb {H}^n}^2 H^{-1/2}_2$ is a Calderón–Zygmund operator if V ∈ B Q / 2 ${V\in B_{Q/2}}$ and there exists a positive constant C such that V ( g ) ≤ C m ( g , V ) 2 ${V(g)\le Cm(g,V)^2}$ .

A new [formula omitted]–[formula omitted] estimate for Calderón–Zygmund operators in spaces of homogeneous type

In this note, we study the Ap–A∞ estimate for Calderón–Zygmund operators in terms of the weak A∞ characteristics in spaces of homogeneous type. The weak A∞ class was introduced recently by Anderson, Hytönen and Tapiola. Our estimate is new even in Euclidean space.

A T(P) theorem for Sobolev spaces on domains

Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given 0<s⩽1, 1<p<∞ with sp>2 and a Lipschitz domain Ω⊂C, the Beurling transform Bf=−p.v.1πz2⁎f is bounded in the Sobolev space Ws,p(Ω) if and only if BχΩ∈Ws,p(Ω).In this paper we obtain a generalized version of the former result valid for any s∈N and for a larger family of Calderón–Zygmund operators in any ambient space Rd as long as p>d. In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally we find a sufficient condition in terms of Carleson measures for p⩽d. In the particular case s=1, this condition is in fact necessary, which yields a complete characterization.

Leibniz’s Rule, Sampling and Wavelets on Mixed Lebesgue Spaces

Several new results about Leibniz’s rule, sampling, and wavelets in the context of mixed Lebesgue spaces are presented. The results obtained rely in part on vector-valued estimates for Calderon–Zygmund operators and the use of appropriate maximal functions and Littlewood–Paley estimates.

A characterization of compactness for singular integrals

We prove a T(1) theorem to completely characterize compact Calderón–Zygmund operators. The result provides sufficient and necessary conditions for the compactness of singular integral operators acting on Lp(R) with 1<p<∞.

On the Local Tb Theorem: A Direct Proof under the Duality Assumption

AbstractWe give a new direct proof of the local Tb theorem in the Euclidean setting and under the assumption of dual exponents. This theorem provides a flexible framework for proving the boundedness of a Calderón–Zygmund operator, supposing the existence of systems of local accretive functions. We assume that the integrability exponents on these systems of functions are of the form 1/p + 1/q ⩽ 1, the ‘dual case’ 1/p + 1/q = 1 being the most difficult one. Our proof is direct: it avoids a reduction to the perfect dyadic case unlike some previous approaches. The principal point of interest is in the use of random grids and the corresponding construction of the corona. We also use certain twisted martingale transform inequalities.