Paraproduct Operators Research Articles

Equivalent Norms of $$cmo^{p}(\mathbb {R}^{n})$$ and Applications

In this paper, we prove that $$cmo^{p}(\mathbb {R}^{n})$$ and $$\Lambda _{n(\frac{1}{p}-1)}$$ , the dual spaces of local Hardy space $$h^{p}(\mathbb {R}^{n})$$ , are coincide with equivalent norms for $$\frac{n}{n+1}<p\le 1$$ . Moreover, this space can be characterized by another simple norm. As an application, we prove the $$h^{p}(\mathbb {R}^{n})$$ boundedness of inhomogeneous para-product operators.

Littlewood–Paley decomposition in quantum calculus

ABSTRACT We introduce Littlewood–Paley decomposition related to q-Rubin's operator, this allows us to provide a dyadic characterization of Sobolev, Hölder and Lebesgue spaces associated with the q-Rubin's operators and to establish some embedding results for these spaces. We construct the paraproduct operators associated with the q-Rubin's operators and we establish its action on the Sobolev and Hölder spaces.

Characterization of a two-parameter matrix valued BMO by commutator with the Hilbert transform

In this paper, we prove that the space of two parameter matrix-valued BMO functions can be char\-ac\-terized by considering iterated commutators with the Hilbert transform. Specifically, we prove that $$ \| B \|_{BMO} \!\lesssim \! \| [[M_B, H_1],H_2] \|_{L^2(\mathbb{R} ^2;\mathbb{C} ^d) \rightarrow L^2(\mathbb{R} ^2;\mathbb{C} ^d)} \!\lesssim \! \| B \|_{BMO}. $$ The upper estimate relies on Petermichl's representation of the Hilbert transform as an average of dyadic shifts and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and Lacey's proof for the scalar case.

A Class of $\alpha$-Carleson Measures

In the present paper, we introduce a class of $\alpha$-Carleson measures $\mathcal{C}_{\alpha,v}(\mathbb{R}^{n+1}_+)$, which is called by the vanishing $\alpha$-Carleson measures. We prove that $\mathcal{C}_{1/p,v}(\mathbb{R}^{n+1}_+)$ is just a predual of the tent space $\widetilde{T}_{\infty}^p$ ($0 \lt p \lt 1$). Furthermore, we construct the $\alpha$-Carleson measures and the vanishing $\alpha$-Carleson measures by the Campanato functions and its a subclass, respectively. Moreover, a characterization of the vanishing $\alpha$-Carleson measure by the compactness of Poisson integral is given in this paper. Finally, as some applications, we give the $(L^{2/\alpha},L^2)$ boundedness and compactness for some paraproduct operators.

Multilinear dyadic operators and their commutators

We obtain a generalized paraproduct decomposition of the pointwise product of two or more functions that naturally gives rise to multilinear dyadic paraproducts and Haar multipliers. We then study the boundedness properties of these multilinear operators and their commutators with dyadic BMO functions. We also characterize the dyadic BMO functions via the boundedness of (a) certain paraproducts, and (b) the commutators of multilinear Haar multipliers and paraproduct operators.

Composition of dyadic paraproducts

We obtain necessary and sufficient conditions to characterize the boundedness of the composition of dyadic paraproduct operators.

Paraproducts via $H^{\infty}$-functional calculus

Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. In this paper, we introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated to $L$. We show various boundedness properties on $L^p(X)$ and the recently developed Hardy and BMO spaces $H^p_L(X)$ and $BMO_L(X)$. In generalization of standard paraproducts constructed via convolution operators, we show $L^2(X)$ off-diagonal estimates as a substitute for Calder\'on-Zygmund kernel estimates. As an application, we study differentiability properties of paraproducts in terms of fractional powers of the operator $L$. The results of this paper are fundamental for the proof of a T(1)-Theorem for operators beyond Calder\'on-Zygmund theory, which will be the subject of a forthcoming paper.

Generalized Besov Spaces and Their Applications

We define and study the Bessel potential and inhomogeneous Besov spaces associated with the Dunkl operators on $\mathbf{R}^d$. As applications on these spaces we construct the Sobolev type embedding theorem and the paraproduct operators associated with the Dunkl operators, as similar to those defined by Bony. We also establish Strichartz type estimates for the Dunkl-Schrodinger equation and finally we study the problem of well posedness of the generalized heat equation.

SOME OPERATORS ACTING ON WEIGHTED SEQUENCE BESOV SPACES AND APPLICATIONS

In this article, we study the boundedness of matrix operators acting on weighted sequence Besov spaces $\dot{b}_{p,w}^{\alpha,q}$. First we obtain the necessary and sufficient condition for the boundedness of diagonal matrices acting on weighted sequence Besov space $\dot{b}_{p,w}^{\alpha,q}$, and investigate the duals of $\dot{b}_{p,w}^{\alpha,q}$, where the weight is non-negative and locally integrable. In particular, when $0 \lt p \lt 1$, we find a type of new sequence sapces which characterize the dual space of $\dot{b}^{\alpha,q}_{p,w}$. We also use the duals of $\dot{b}_{p,w}^{\alpha,q}$ to characterize an algebra of matrix operators acting on weighted sequence Besov spaces $\dot{b}_{p,w}^{\alpha,q}$ and find the necessary and sufficient conditions to such a characterization. Note that we do not require that the given weight satisfies the doubling condition in this situation. Using these results, we give some applications to characterize the boundedness of Fourier-Haar multipliers and paraproduct operators. In this situation, we need to require that the weight $w$ is an $A_p$ weight.

SOME OPERATORS ACTING ON WEIGHTED SEQUENCE BESOV SPACES AND APPLICATIONS

In this article, we study the boundedness of matrix operators acting on weighted sequence Besov spaces $\dot{b}_{p,w}^{\alpha,q}$. First we obtain the necessary and sufficient condition for the boundedness of diagonal matrices acting on weighted sequence Besov space $\dot{b}_{p,w}^{\alpha,q}$, and investigate the duals of $\dot{b}_{p,w}^{\alpha,q}$, where the weight is non-negative and locally integrable. In particular, when $0<p< 1$, we find a type of new sequence sapces which characterize the dual space of $\dot{b}^{\alpha,q}_{p,w}$. We also use the duals of $\dot{b}_{p,w}^{\alpha,q}$ to characterize an algebra of matrix operators acting on weighted sequence Besov spaces $\dot{b}_{p,w}^{\alpha,q}$ and find the necessary and sufficient conditions to such a characterization. Note that we do not require that the given weight satisfies the doubling condition in this situation. Using these results, we give some applications to characterize the boundedness of Fourier-Haar multipliers and paraproduct operators. In this situation, we need to require that the weight $w$ is an $A_p$ weight.

Generalized Carleson Measure Spaces and Their Applications

We introduce the generalized Carleson measure spaces CM that extend BMO. Using Frazier and Jawerth′s φ‐transform and sequence spaces, we show that, for α ∈ ℝ and 0 &lt; p ≤ 1, the duals of homogeneous Triebel‐Lizorkin spaces for 1 &lt; q &lt; ∞ and 0 &lt; q ≤ 1 are CM and CM (for any r ∈ ℝ), respectively. As applications, we give the necessary and sufficient conditions for the boundedness of wavelet multipliers and paraproduct operators acting on homogeneous Triebel‐Lizorkin spaces.

Duals of homogeneous weighted sequence Besov spaces and applications

In this article, we study the duals of homogeneous weighted sequence Besov spaces , where the weight w is non-negative and locally integrable. In particular, when 0 < p < 1, we find a type of new sequence spaces which characterize the duals of . Also, we find the necessary and sufficient conditions for the boundedness of diagonal matrices acting on homogeneous weighted sequence Besov spaces. Using these results, we give some applications to characterize the boundedness of Fourier–Haar multipliers and paraproduct operators. In this situation, we need to require that the weight w is an Ap weight.

Boundedness of paraproduct operators on RD-spaces

Let (X, d,µ) be an RD-space with “dimension” n, namely, a space of homogeneous type in the sense of Coifman and Weiss satisfying a certain reverse doubling condition. Using the Calderon reproducing formula, the authors hereby establish boundedness for paraproduct operators from the product of Hardy spaces H p (X) × H q (X) to the Hardy space H r (X), where p, q, r ∈ (n/(n + 1),∞) satisfy 1/p + 1/q = 1/r. Certain endpoint estimates are also obtained. In view of the lack of the Fourier transform in this setting, the proofs are based on the derivation of appropriate kernel estimates.

Triebel-Lizorkin Spaces of Para-Accretive Type and a Tb Theorem

In this article, we use a discrete Calderon-type reproducing formula and Plancherel-Polya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type \(\dot{F}^{\alpha,q}_{b,p}\) , which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a necessary and sufficient condition for the \(\dot{F}^{0,q}_{1,p}-\dot{F}^{0,q}_{b,p}\) boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with MbTMb∈WBP is bounded from \(\dot{F}^{0,q}_{1,p}\) to \(\dot{F}^{0,q}_{b,p}\) if and only if \(Tb\in \dot{F}^{0,q}_{b,\infty}\) and T*b=0 for \(\frac{n}{n+\varepsilon}<p\le1\) and \(\frac{n}{n+\varepsilon}<q\le 2\) , where e is the regularity exponent of the kernel of T.

Paraproducts in one and several parameters

For multiparameter bilinear paraproduct operators B we prove the estimate B : L pL q 7 L r ; 1 < p; qay: Here, 1=p þ 1=q ¼ 1=r and special attention is paid to the case of 0 < r < 1. (Note that the families of multiparameter paraproducts are much richer than in the one parameter case.) These estimates are the essential step in the version of the multiparameter Coifman-Meyer theorem proved by C. Muscalu, J. Pipher, T. Tao, and C. Thiele (10, 11). We oer a dierent proof of these inequalities.

Paraproduit sur le groupe de Heisenberg et applications

We adapt the homogeneous Littlewood–Paley decomposition on the Heisenberg group constructed by H. Bahouri, P. Gérard and C.-J Xu in [4] to the inhomogeneous case, which enables us to build paraproduct operators, similar to those defined by J.-M. Bony in [5]; although there is no simple formula for the Fourier transform of the product of two functions, some spectral localization properties of the classical case are preserved on the Heisenberg group after the product has been taken. Using the dyadic decomposition and the paraproduct algorithm, we prove the Gagliardo–Nirenberg inequality on the Heisenberg group; the smoothness of solutions of subelliptic, semi-linear systems is also studied, as well as semi-linear wave equations. Résumé En adaptant au cas inhomogène la décomposition de Littlewood–Paley homogène sur le groupe de Heisenberg introduite par H. Bahouri, P. Gérard et C.-.J Xu dans [4] on construit des opérateurs de paraproduit analogues à ceux définis par J.-.M Bony dans [5]; malgré le fait que l'on ne dispose pas de formule simple pour la transformée de Fourier d'un produit des propriétés de localisation spectrale du cas classique sont préservées sur le groupe de Heisenberg après passage au produit. A partir du découpage dyadique et du paraproduit on démontre l'inégalité de Gagliardo–Nirenberg sur le groupe de Heisenberg et l'on étudie la régularité des solutions de systèmes sous-elliptiques semi-linéaires ainsi que des équations d'ondes semi-linéaires.

Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

We analyze the stability of Muckenhoupt’s R H p d \mathbf {RH}_{\mathbf {p}}^{\mathbf {d}} and A P d \mathbf {A}_{\mathbf {P}}^{\mathbf {d}} classes of weights under a nonlinear operation, the λ \lambda -operation. We prove that the dyadic doubling reverse Hölder classes R H p d \mathbf {RH}_{\mathbf {p}}^{\mathbf {d}} are not preserved under the λ \lambda -operation, but the dyadic doubling A p A_p classes A P d \mathbf {A}_{\mathbf {P}}^{\mathbf {d}} are preserved for 0 ≤ λ ≤ 1 0\leq \lambda \leq 1 . We give an application to the structure of resolvent sets of dyadic paraproduct operators.

The Generalization of Paraproducts and the Full T1 Theorem for Sobolev and Triebel–Lizorkin Spaces

We study the boundedness of generalized Calderón–Zygmund operators acting on Sobolev and, more generally, Triebel–Lizorkin spaces of arbitrary order of smoothness. We are able to relax the assumptionsT(xγ)=0 and/orT*(xγ)=0, which have been required in earlier results by other authors. To do this, we consider generalized paraproduct operators. We obtain the sharpness of our assumptions for some special cases. Using the same technique, we also obtain some sharper results for “norming” and “molecular” families.

The boundedness of certain operators on Holder and Sobolev spaces

The main topic in this note is to discuss the boundedness of pseudo-differential operators and para-product operators on the Holder and Sobolev spaces, respectively. It is the preparation of the theory of Gibbs-Butzer defferential operators and differential equations.

Para-product operators and para-linearization on locally compact Vilenkin groups

The concept of para-product operators over locally compact Vilenkin groups is established and the applications to the para-linearization in nonlinear problems are studied. This kind of operators plays a special role in dealing with those functions which do not have the classical derivatives.