In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{ρ,δ}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{ρ,δ}^m) \subset \calL(M^{p,q}(\R^n))$ to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space $M^{p,q}$ with $q \neq 2$. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on $M^{p,q}$ whose symbols are of the class $S_{1,δ}^0$ with $0<δ<1$.
We give the first natural examples of Calderón-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. This is achieved by showing that the purely imaginary Riesz and Bessel potentials on nested fractals with 3 or more boundary points are of this type. It follows that these operators are bounded on $L^{p}$, $1<p<\infty$ and satisfy weak 1-1 bounds. The analysis may be extended to infinite blow-ups of these fractals, and to product spaces based on the fractal or its blow-up.
In this paper, we obtain the necessary and sufficient conditions for the weak/strong boundedness of the Calderón-Zygmund operators in generalized weighted Orlicz-Morrey spaces. We also study the boundedness of the commutators of Calderón-Zygmund operators on these spaces. Moreover, the boundedness of Calderón-Zygmund operators in the vector-valued setting is given.
We propose a simple abstract version of Calderon--Zygmund theory, which is applicable to spaces with exponential volume growth, and then show that amenable Lie groups can be treated within this framework.
We state a new Calderon-Zygmund decomposition for Sobolev spaces on a doubling Riemannian manifold. Our hypotheses are weaker than those of the already known decomposition which used classical Poincare inequalities.
We give necessary and sufficient conditions for inhomogeneous Calderón-Zgymund operators to be bounded on the local hardy spaces $h^p(\mathbb{R}^n)$. We then give applications to local and truncated Riesz transforms, as well as pseudo-differential operators defined by amplitudes.
In this paper, we obtain the desired noncommutative maximal inequalities of the truncated Calderón-Zygmund operators of non-convolution type acting on operator-valued $L_p$-functions for all $1<p<\infty$, answering a question left open in the previous work \cite{HLX}.
In this paper, we prove strong type, weak type inequalities of Hardy-Littlewood maximal operator and fractional Hardy-Littlewood maximal operator on variable sequence spaces lp(Z). This is achieved using Calderon-Zygmund decomposition for sequences, properties of modular functional and Log Holder continuity.
In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted weak Morrey spaces. We prove that the operators are bounded on these spaces, under certain assumptions.
In this work we present a new approach to molecules on Goldberg's local Hardy spaces $h^p(\mathbb{R}^n)$, $0<p\leq1$, assuming an appropriate cancellation condition. As applications, we prove a version of Hardy's inequality and improved continuity results for inhomogeneous Calderón-Zygmund operators on these spaces.
Let $μ$ be an $n$-dimensional finite positive measure on $\mathbb{R}^m$. We obtain a $T1$ condition sufficient for the boundedness of Calderón-Zygmund operators on $\textrm{RBMO}(μ)$, the regular BMO space of Tolsa.
In this paper, we give the definition of local variable Morrey Lorentz spaces which are a new class of functions. Also, we prove the boundedness of the Hardy Littlewood maximal operator M and Calderon Zygmund operators T on these spaces. Finally, we apply these results to the Bochner Riesz operator, identity approximation and the Marcinkiewicz operator on these spaces.
In the paper we consider Calderón-Zygmund operators in nonhomogeneous spaces. We are going to prove the analogs of classical results for homogeneous spaces. Namely, we prove that a Calderón-Zygmund operator is of weak type if it is bounded in $L^2$. We also prove several versions of Cotlar's inequality for maximal singular operator. One version of Cotlar's inequality (a simpler one) is proved in Euclidean setting, another one in a more abstract setting when Besicovich covering lemma is not available. We obtain also the weak type of maximal singular operator from these inequalities.
We prove a new dyadic representation theorem with applications to the $T(1)$ and $A_2$ theorems. In particular, we obtain the non-homogeneous $T(1)$ theorem under weaker kernel regularity than the earlier approaches.
Let $T_m$ be the $m$-th Calderón-Zygmund type singular integral. In the paper, we consider the boundedness of $T_m$ on the generalized product local Morrey spaces $LM_{p_1,φ_1}^{\{x_0\}}\times LM_{p_2,φ_2}^{\{x_0\}}\times\dots\times LM_{p_m,φ_m}^{\{x_0\}}.$ And, the boundedness of the commutators of $T_m$ with local Campanato functions is obtained, also.
In this work, we establish results on the continuity of strongly singular Calderón-Zygmund operators of type $σ$ on Hardy spaces $H^p(\mathbb{R}^n)$ for $0<p\leq 1$ assuming a weaker $L^{s}-$type Hörmander condition on the kernel. Operators of this type include appropriated classes of pseudodifferential operators $OpS^{m}_{σ,b}(\mathbb{R}^n)$ and operators associated to standard $δ$-kernels of type $σ$ introduced by Álvarez and Milman. As application, we show that strongly singular Calderón-Zygmund operators are bounded from $H^{p}_{w}(\mathbb{R}^n)$ to $L^{p}_{w}(\mathbb{R}^n)$, where $w$ bel…
The aim of this paper is to get the boundedness of the commutators of multi-sublinear operators generated by local campanato functions and multilinear Calderón-Zygmund operators on the product generalized local Morrey spaces.
In this paper we study some questions in connection with uniform rectifiability and the $L^2$ boundedness of Calderon-Zygmund operators. We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the Jones' $β$ numbers. We also use these new coefficients to prove that n-dimensional Calderon-Zygmund operators with odd kernel of type $C^2$ are bounded in $L^2(μ)$ if $μ$ is an n-dimensional uniformly rectifiable measure.
In the setting of $\R^d$ with an $n-$dimensional measure $μ,$ we give several characterizations of Lipschitz spaces in terms of mean oscillations involving $μ.$ We also show that Lipschitz spaces are preserved by those Calderon-Zygmund operators $T$ associated to the measure $μ$ for which T(1) is the Lipschitz class $0.$
In 2008, J. Parcet showed the $(1,1)$ weak-boundedness of Calderón-Zygmund operators acting on functions taking values in a von Neumann algebra. We propose a simplified version of his proof using the same tools : Cuculescu's projections and a pseudo-localisation theorem. This will unable us to recover the $L_p$-boundedness of Calderón-Zygmund operators with Hilbert valued kernels acting on operator valued functions for $1 < p < \infty$ and an $L_p$-pseudo-localisation result of P. Hytönen.
An RD-space $\mathcal{X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal{X}$. In this setting, the authors establish the boundedness of bilinear $θ$-type Calderón-Zygmund operator $T_θ$ and its commutator $[b_1,b_2,T_θ]$ generated by the function $b_1,b_2\in BMO(μ)$ and $T_θ$ on generalized weighted Morrey space $\mathcal{M}^{p,ϕ}(ω)$ and generalized weighted weak Morrey space $W\mathcal{M}^{p,ϕ}(ω)$ over RD-spaces.
For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
The Coifman-Fefferman inequality implies quite easily that a Calderon-Zygmund operator $T$ acts boundedly in a Banach lattice $X$ on $\mathbb R^n$ if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$. We discuss this phenomenon in some detail and establish a converse result under the assumption that $X$ is $p$-convex and $q$-concave with some $1 < p, q < \infty$ and satisfies the Fatou property: if a linear operator $T$ is bounded in $X$ and $T$ is nondegenerate in a certain sense (for example, if $T$ is a Riesz transform) then $M$ has to be bounded in both $X$ and $X'$…
We extend the recently much-studied Hardy factorization theorems to the weight case. The key point of this paper is to establish the factorization theorems without individual condition on the weight functions. As a direct application, we obtain the characterizations of $BMO(\mathbb{R}^n)$ space and Lipschitz spaces via the weighted boundedness of commutators of multilinear Calderón-Zygmund operators with the genuinely multilinear weights.
A proof is given for the "only if" part of the result stated in the previous paper of the series that a suitably nondegenerate Calderón-Zygmund operator $T$ is bounded in a Banach lattice $X$ on $\mathbb R^n$ if and only if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$, under the assumption that $X$ has the Fatou property and $X$ is $p$-convex and $q$-concave with some $1 < p, q < \infty$. We also get rid of a fixed point theorem in the proof of the main lemma and give an improved version of an earlier result concerning the divisibility of $\mathrm {BMO}$-regularity…
In this paper, we verify the $L^2$-boundedness for the jump functions and variations of Calderón-Zygmund singular integral operators with the underlying kernels satisfying \begin{align*}\int_{\varepsilon\leq |x-y|\leq N} K(x,y)dy=\int_{\varepsilon\leq |x-y|\leq N}K(x,y)dx=0\; \forall 0<\varepsilon\leq N<\infty,\end{align*} in addition to some proper size and smooth conditions. This result should be the first general criteria for the variational inequalities for kernels not necessarily of convolution type. The $L^2$-boundedness assumption that we verified here is also the starting point of the …
Let $(\cx,\,d,\,μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calderón-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its $L^p(μ)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(μ…
In this work, we give new sufficient conditions for a Littlewood-Paley-Stein square function and necessary and sufficient conditions for a Calderón-Zygmund operator 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 we 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 Calderón-Zygmund operators. These results are applied to prove that Bony paraproducts can be constructed such…
The aim of this paper is to get the boundedness of certain sublinear operators with rough kernel generated by Calderón-Zygmund operators on the generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutator operators formed by BMO functions and certain sublinear operators with rough kernel are also bounded on the generalized weighted Morrey spaces. Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.
Let $p(\cdot):\ \mathbb{R}^n\to(0,\infty]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. Let $H_A^{p(\cdot)}(\mathbb{R}^n)$ be the variable anisotropic Hardy space associated with $A$ defined via the non-tangential grand maximal function. In this article, via the known atomic characterization of $H_A^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its molecular characterization with the known best possible decay of molecules. As an application, the author obtains a criterion on the boundedness of …
In this paper we establish a T1 criterion for the boundedness of Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space naturally associated to the Hermite operator H. We apply this criterion in a systematic way to prove the boundedness on BMO_H(R^n) of certain harmonic analysis operators related to H (Riesz transforms, maximal operators, Littlewood-Paley g-functions and variation operators).
Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of "Markov metric" governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderó…
One defines a non-homogeneous space $(X, μ)$ as a metric space equipped with a non-doubling measure $μ$ so that the volume of the ball with center $x$, radius $r$ has an upper bound of the form $r^n$ for some $n> 0$. The aim of this paper is to study the boundedness of a Calderón-Zygmund operator $T$ as well as the boundedness of certain related singular integrals associated with $T$ on various function spaces on $(X, μ)$ such as the Hardy spaces, the $L^p$ spaces and the regularized BMO spaces. This article thus extends the work of X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n…
Given a general dyadic grid ${\mathscr{D}}$ and a sparse family of cubes ${\mathcal S}=\{Q_j^k\}\in {\mathscr{D}}$, define a dyadic positive operator ${\mathcal A}_{\mathscr{D},{\mathcal S}}$ by $${\mathcal A}_{\mathscr{D},{\mathcal S}}f(x)=\sum_{j,k}f_{Q_j^k}χ_{Q_j^k}(x).$$ Given a Banach function space $X({\mathbb R}^n)$ and the maximal Calderón-Zygmund operator $T_{\natural}$, we show that $$\|T_{\natural}f\|_X\le c(n,T)\sup_{\mathscr{D},{\mathcal S}}\|{\mathcal A}_{\mathscr{D},{\mathcal S}}f\|_{X}.$$ This result is applied to weighted inequalities. In particular, it implies: (i) the "two-w…
We prove a multilinear local $T(b)$ theorem that differs from previously considered multilinear local $T(b)$ theorems in using exclusively general testing functions $b$ as opposed to a mix of general testing functions and indicator functions. The main new feature is a set of relations between the various testing functions $b$ that to our knowledge has not been observed in the literature and is necessitated by our approach. For simplicity we restrict attention to the perfect dyadic model.
Calderón-Zygmund operators with noncommuting kernels may fail to be Lp-bounded for $p \neq 2$, even for kernels with good size and smoothness properties. Matrix-valued paraproducts, Fourier multipliers on group vNa's or noncommutative martingale transforms are frameworks where we find such difficulties. We obtain weak type estimates for perfect dyadic CZO's and cancellative Haar shifts associated to noncommuting kernels in terms of a row/column decomposition of the function. Arbitrary CZO's satisfy $H_1 \to L_1$ type estimates. In conjunction with $L_\infty \to BMO$, we get certain row/column …
Given a Radon measure $μ$ on $R^d$, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties that hold when $μ$ is doubling remain valid for the space BMO introduced in this paper, without assuming $μ$ doubling. For instance, Calderon-Zygmund operators which are bounded in $L^2$ are bounded from $L^\infty$ into the new BMO space. Moreover, a John-Nirenberg inequality is satisfied, and the predual of BMO is an atomic space $H^1$. Using a sharp maximal function it is proved that operators bounded from $L^\infty$ into BMO and from …
