Sparse Domination Research Articles

Lebesgue Space Estimates for Spherical Maximal Functions on Heisenberg Groups

Abstract We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse domination for global maximal operators. We also consider lacunary variants and extensions to Métivier groups.

Sparse domination results for compactness on weighted spaces

By means of appropriate sparse bounds, we deduce compactness on weighted $$L^p(w)$$ spaces, $$1<p<\infty$$ , for all Calderón–Zygmund operators having compact extensions on $$L^2({\mathbb {R}}^n)$$ . Similar methods lead to new results on boundedness and compactness of Haar multipliers on weighted spaces. In particular, we prove weighted bounds for weights in a class strictly larger than the typical $$A_p$$ class.

Singular stochastic integral operators

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on $\mathbb{R}^d$ and smooth and angular domains.

Bilinear Spherical Maximal Functions of Product Type

In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calderón–Zygmund theory. This operator is different from the bilinear spherical maximal function considered by Geba et al. in (Math Res Lett 20(4):675–694, 2013). We deal with lacunary and full versions of this operator, and prove weighted estimates with respect to genuine bilinear weights beyond the Banach range. Our results are implied by sharp sparse domination for both the operators, following ideas by Lacey (J Anal Math 139(2):613–635, 2019). In the case of the lacunary maximal operator we also use interpolation of analytic families of operators to address the weighted boundedness for the whole range of tuples.

Musielak-Orlicz-Bumps and Bloom Type Estimates for Commutators of Calderón-Zygmund and Fractional Integral Operators on Generalized Zygmund Spaces Via Sparse Operators

We study continuity properties for commutators of Calderon-Zygmund and fractional integral operators between generalized Zygmund spaces of L log L type, in the variable exponent setting with different weights. In order to reach this goal we use two different approaches: the first one is related to generalized bump conditions on a pair of weights, allowing us to handle with a wide class of symbol involved with the commutator. The other approaches give Bloom type estimates restricting the class of symbols. The techniques involved in both type of results are related with the theory of sparse domination.

Sparse domination via the helicoidal method

Using exclusively the localized estimates upon which the helicoidal method was built by the authors, we show how sparse estimates can also be obtained. This approach yields a sparse domination for scalar and multiple vector-valued extensions of operators alike. We illustrate these ideas for an n -linear Fourier multiplier whose symbol is singular along a k -dimensional subspace of \Gamma=\{\xi_1+\cdots+\xi_{n+1}=0\} , where k &lt; (n+1)/{2} , and for the variational Carleson operator.

Weighted estimates for maximal bilinear rough singular integrals via sparse dominations

Let $$x=(x_1,x_2)$$ with $$x_1,x_2 \in \mathbb {R}^n$$ and let $$K(x)={\Omega \big ({x}/{|x|}\big )}{\big |x\big |^{-2n}}$$ , where $$\Omega \in L^{\infty }(\mathbb {S}^{2n-1})$$ and satisfies $$\int _{\mathbb {S}^{2n-1}}\Omega =0$$ . We show that the maximal truncated bilinear singular integrals with rough kernel $$K(x_1,x_2)$$ satisfy a sparse bound by (p, p, p)-averages for all $$p>1$$ . As consequences, we obtain some quantitative weighted estimates for these rough singular integrals. A pointwise sparse domination for commutators of bilinear rough singular integrals were also established, which can be used to establish some weighted inqualities.

Sparse domination of weighted composition operators on weighted Bergman spaces

The purpose of this paper is to study sparse domination estimates of composition operators in the setting of complex function theory. The method originates from proofs of the A2 theorem for Calderón-Zygmund operators in harmonic analysis. Using this tool from harmonic analysis, some new characterizations are given for the boundedness and compactness of weighted composition operators acting between weighted Bergman spaces in the upper half plane. Moreover, we establish a new weighted type estimate for the holomorphic Bergman-class functions, for a new class of weights, which is adapted to Sawyer–testing conditions. We also extend our results to the unit ball B in Cn.

Variation of Calderón–Zygmund operators with matrix weight

Let [Formula: see text], [Formula: see text] and [Formula: see text] be a matrix [Formula: see text] weight. In this paper, we introduce a version of variation [Formula: see text] for matrix Calderón–Zygmund operators with modulus of continuity satisfying the Dini condition. We then obtain the [Formula: see text]-boundedness of [Formula: see text] with norm [Formula: see text] by first proving a sparse domination of the variation of the scalar Calderón–Zygmund operator, and then providing a convex body sparse domination of the variation of the matrix Calderón–Zygmund operator. The key step here is a weak type estimate of a local grand maximal truncated operator with respect to the scalar Calderón–Zygmund operator.

On Pointwise ell ^r-Sparse Domination in a Space of Homogeneous Type

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual ell ^1-sum in the sparse operator is replaced by an ell ^r-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the A_2-theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.

Sharp weighted estimates for multi-linear Calderón–Zygmund operators on non-homogeneous spaces

In this article, we address pointwise sparse domination for multilinear Calderon—Zygmund operators on upper doubling, geometrically doubling metric measure spaces. As a consequence, we have obtained sharp quantitative weighted estimates for multilinear Calderon–Zygmund operators.

Banach-Valued Modulation Invariant Carleson Embeddings and Outer-L^p Spaces: The Walsh Case

We prove modulation invariant embedding bounds from Bochner spaces L^p(mathbb {W};X) on the Walsh group to outer-L^p spaces on the Walsh extended phase plane. The Banach space X is assumed to be UMD and sufficiently close to a Hilbert space in an interpolative sense. Our embedding bounds imply L^p bounds and sparse domination for the Banach-valued tritile operator, a discrete model of the Banach-valued bilinear Hilbert transform.

$$\mathbf {L^p\rightarrow L^q}$$ bounds for spherical maximal operators

Let \(f\in L^p({\mathbb {R}}^d)\), \(d\ge 3\), and let \(A_t f(x)\) be the average of f over the sphere with radius t centered at x. For a subset E of [1, 2] we prove close to sharp \(L^p\rightarrow L^q\) estimates for the maximal function \(\sup _{t\in E} |A_t f|\). A new feature is the dependence of the results on both the upper Minkowski dimension of E and the Assouad dimension of E. The result can be applied to prove sparse domination bounds for a related global spherical maximal function.

Weighted norm inequalities in a bounded domain by the sparse domination method

We prove a local two-weight Poincaré inequality for cubes using the sparse domination method that has been influential in harmonic analysis. The proof involves a localized version of the Fefferman–Stein inequality for the sharp maximal function. By establishing a local-to-global result in a bounded domain satisfying a Boman chain condition, we show a two-weight p-Poincaré inequality in such domains. As an application we show that certain nonnegative supersolutions of the p-Laplace equation and distance weights are p-admissible in a bounded domain, in the sense that they support versions of the p-Poincaré inequality.

Vector-valued operators, optimal weighted estimates and the Cp condition

In this paper some new results concerning the Cp classes introduced by Muckenhoupt (1981) and later extended by Sawyer (1983), are provided. In particular, we extend the result to the full expected range p < 0, to the weak norm, to other operators and to their vector-valued extensions. Some of those results rely upon sparse domination, which in the vector-valued case are provided as well. We will also provide sharp weighted estimates for vector-valued extensions relying on those sparse domination results.

A sparse domination for the Marcinkiewicz integral with rough kernel and applications

Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mu_{\Omega}$ be the higher-dimensional Marcinkiewicz integral defined by $$\mu_\Omega(f)(x)= \Big(\int_0^\infty\Big|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-1}}f(y)dy\Big|^2\frac{dt}{t^3}\Big)^{1/2}. $$ In this paper, the authors establish a bilinear sparse domination for $\mu_{\Omega}$ under the assumption $\Omega\in L^{\infty}(S^{n-1})$. As applications, some quantitative weighted bounds for $\mu_{\Omega}$ are obtained.

Sharp Matrix Weighted Strong Type Inequalities for the Dyadic Square Function

In this paper we refine the recent sparse domination of the integrated p = 2 matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic square functions. We then use this to prove sharp two matrix weighted strong type inequalities for matrix weighted dyadic square functions when 1 < p ≤ 2.

On Bloom type estimates for iterated commutators of fractional integrals

In this paper we provide quantitative Bloom type estimates for iterated commutators of fractional integrals improving and extending results from a work of Holmes, Rahm and Spencer. We give new proofs for those inequalities relying upon a new sparse domination that we provide as well in this paper and also in techniques developed in a recent paper due to Lerner, Ombrosi and the third author. We extend as well the necessity established in the work of Holmes, Rahm and Spencer to iterated commutators providing a new proof. As a consequence of the preceding results we recover the one weight estimates in works of Cruz-Uribe and Moen and B\'enyi, Martell, Moen, Stachura, Torres and establish the sharpness in the iterated case. Our result provides as well a new characterization of the BMO space.

Analysis of hyper-singular, fractional, and order-zero singular integral operators

In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional derivatives, pseudodifferential operators, Calder\'on-Zygmund operators, and many others. The main results of this article are built around the notion of an operator calculus that connects operators with different kernel singularities via vanishing moment conditions and composition with fractional derivative operators. We also provide several boundedness results on weighted and unweighted distribution spaces, including homogeneous Sobolev, Besov, and Triebel-Lizorkin spaces, that are necessary and sufficient for the operator's vanishing moment properties, as well as certain behaviors for the operator under composition with fractional derivative and integral operators. As applications, we prove $T1$ type theorems for singular integral operators with different singularities, boundedness results for pseudodifferential operators belonging to the forbidden class $S_{1,1}^0$, fractional order and hyper-singular paraproduct boundedness, a smooth-oscillating decomposition for singular integrals, sparse domination estimates that quantify regularity and oscillation, and several operator calculus results. It is of particular interest that many of these results do not require $L^2$-boundedness of the operator, and furthermore, we apply our results to some operators that are known not to be $L^2$-bounded.

A sparse approach to mixed weak type inequalities

In this paper we provide some quantitative mixed weak-type estimates assuming conditions that imply that $$uv\in A_{\infty }$$ for Calderon–Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in Domingo-Salazar et al. (Bull Lond Math Soc 48(1):63–73, 2016) and extended in Lerner et al. (Adv Math 319:153–181, 2017) and Li et al. (J Geom Anal, 2018).