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Sandra Pott

Publications and source records attributed to Sandra Pott.

At least 19 recordsLinked to original sources
arXiv2023

Multipliers for Hardy-Orlicz spaces and applications

Using real-variable methods, we characterise multipliers for general classes of Hardy--Orlicz spaces, unifying and extending several classical results due to Hardy and Littlewood; Duren and Shields; Paley; and others. Applications of our results include inequalities involving Fourier coefficients and Fourier transforms of elements of Hardy--Orlicz spaces and their duals, as well as embeddings into spaces of generalised smoothness, Sobolev type-embeddings and Paley-Wiener type theorems.

math.CA↗
arXiv2021

Laplace-Carleson embeddings and infinity-norm admissibility

A full characterization of the boundedness of Laplace--Carleson embeddings on $L^\infty$ is provided, in terms of the Carleson intensity of the respective measure and of a suitable weighted Berezin transform of the measure. Moreover, boundedness results, and in some cases full characterizations of boundedness, are proved for a large class of Orlicz spaces. These findings are crucial for characterizing admissibility of control operators for linear diagonal semigroup systems in a variety of contexts. A particular focus is laid on essentially bounded inputs.

math.FA↗
arXiv2021

Sparse Lerner operators in infinite dimensions

We use the principle of almost orthogonality to give a new and simple proof that a sparse Lerner operator is bounded on a matrix- or operator-weighted space $L_W^{2}(\mu)$, where $\mu$ is a doubling measure on $\R^d$ if and only if the weight $W$ satisfies the Muckenhoupt $A_2(\mu)$-condition, restricted to the sparse collection in question. Our method extends to the infinite-dimensional setting, thus allowing for applications to the multi-parameter setting. For the class of Muckenhoupt $A_2$-weights, we obtain bounds in terms of mixed $A_{2}(\mu)$-$A_{\infty}(\mu)$-conditions, which is independent of dimension and agrees with the best known bound in the finite-dimensional vectorial setting. As an application, we prove a matrix-weighted bound for the maximal Bergman projection, where we obtain a new sharper bound in terms of the B\'ekoll\'e-Bonami characteristic. Furthermore, we consider commutators of sparse Lerner operators on operator-valued weighted $L^{2}$-spaces and some applications to multi-parameters

math.FA↗
arXiv2020

Notes on $H^{\log} $: structural properties, dyadic variants, and bilinear $H^1$-$BMO$ mappings

This article is devoted to a study of the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space $H^1$ and a function in $BMO$ to distributions that belong to $H^{\log}$ based on dyadic paraproducts. We also point out analogues of classical results of Hardy-Littlewood, Zygmund, and Stein for $H^{\log}$ and related Musielak-Orlicz spaces.

math.CA↗
arXiv2020

Commutators in the two scalar and matrix weighted setting

In this paper we approach the two weighted boundedness of commutators via matrix weights. This approach provides both a sufficient and a necessary condition for the two weighted boundedness of commutators with an arbitrary linear operator in terms of one matrix weighted norm inequalities for this operator. Furthermore, using this approach, we surprisingly provide conditions that almost characterize the two matrix weighted boundedness of commutators with CZOs and completely arbitrary matrix weights, which is even new in the fully scalar one weighted setting. Finally, our method allows us to extend the two weighted Holmes/Lacey/Wick results to the fully matrix setting (two matrix weights and a matrix symbol), completing a line of research initiated by the first two authors.

math.CA↗
arXiv2019

Sharp $A_1$ weighted estimates for vector valued operators

Given $1\leq q<p<\infty$ quantitative weighted L^p estimates, in terms of Aq weights, for vector valued maximal functions, Calder\'on-Zygmund operators, commutators and maximal rough singular integrals are obtained. The results for singular operators will rely upon suitable convex body domination results, which in the case of commutators will be provided in this work, obtaining as a byproduct a new proof for the scalar case as well.

math.CA↗
arXiv2019

Weak endpoint bounds for matrix weights

We prove quantitative matrix weighted endpoint estimates for the matrix weighted Hardy-Littlewood maximal operator, Calder\'on-Zygmund operators, and commutators of CZOs with scalar BMO functions, when the matrix weight is in the class $A_1$ introduced by M.~Frazier and S.~Roudenko.

math.CA↗
arXiv2018

Laplace--Carleson embeddings on model spaces and boundedness of truncated Hankel and Toeplitz operators

A characterisation is given of bounded embeddings from weighted $L^2$ spaces on bounded intervals into $L^2$ spaces on the half-plane, induced by isomorphisms given by the Laplace transform onto weighted Hardy and Bergman spaces (Zen spaces). As an application necessary and sufficient conditions are given for the boundedness of truncated Hankel and Toeplitz integral operators, including the weighted case.

math.FA↗
arXiv2018

A matrix weighted bilinear Carleson Lemma and Maximal Function

We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and Treil that features a matrix Carleson measure and a matrix weight. It is well known that a Carleson embedding theorem implies a Doob's maximal inequality and this holds true in the matrix weighted setting with an appropriately defined maximal operator. It is also known that a dimensional growth must occur in the Carleson embedding theorem with matrix Carleson measure, even with trivial weight. We give a definition of a maximal type function whose norm in the matrix weighted setting does not grow with dimension.

math.CA↗
arXiv2017

Endpoint compactness of singular integrals and perturbations of the Cauchy Integral

We prove sufficient and necessary conditions for compactness of Calder\'on-Zygmund operators on the endpoint from $L^{\infty }(\mathbb R)$ into ${\rm CMO}(\mathbb R)$. We use this result to prove compactness on $L^{p}(\mathbb R)$ with $1<p<\infty $ of certain perturbations of the Cauchy integral on curves with normal derivatives satisfying a ${\rm CMO}$-condition.

math.CA↗
arXiv2017

Sharp bounds and T1 theorem for Calder\'on-Zygmund operators with matrix kernel on matrix weighted spaces

For a matrix A_2 weight W on R^p, we introduce a new notion of W-Calder\'on-Zygmund matrix kernels, following earlier work in by Isralowitz. We state and prove a T1 theorem for such operators and give a representation theorem in terms of dyadic W-Haar shifts and paraproducts, in the spirit of Hyt\"onen's Representation Theorem. Finally, by means of a Bellman function argument, we give sharp bounds for such operators in terms of bounds for weighted matrix martingale transforms and paraproducts.

math.CA↗
arXiv2015

$\beta$-admissibility of observation and control operators for hypercontractive semigroups

We prove a Weiss conjecture on $\beta$-admissibility of control and observation operators for discrete and continuous $\gamma$-hypercontractive semigroups of operators, by representing them in terms of shifts on weighted Bergman spaces and using a reproducing kernel thesis for Hankel operators. Particular attention is paid to the case $\gamma=2$, which corresponds to the unweighted Bergman shift.

math.AP↗
arXiv2015

Bounds for Calder\'on-Zygmund operators with matrix $A_2$ weights

It is well-known that dyadic martingale transforms are a good model for Calder\'on-Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that, if $W$ is an $A_2$ matrix weight, then the weighted $L^2$-norm of a Calder\'on-Zygmund operator with cancellation has the same dependence on the $A_2$ characteristic of $W$ as the weighted $L^2$-norm of the martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calder\'on-Zygmund operators on the $A_2$ characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calder\'on-Zygmund operators with even kernel. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hyt\"onen to extend the result to general Calder\'on-Zygmund operators.

math.CA↗
arXiv2015

Matrix weighted norm inequalities for commutators and paraproducts with matrix symbols

Let $B$ be a locally integrable matrix function, $W$ a matrix A${}_p$ weight with $1 < p < \infty$, and $T$ be any of the Riesz transforms. We will characterize the boundedness of the commutator $[T, B]$ on $L^p(W)$ in terms of the membership of $B$ in a natural matrix weighted BMO space. To do this, we will characterize the boundedness of dyadic paraproducts on $L^p(W)$ via a new matrix weighted Carleson embedding theorem. Finally, we will use some of the ideas from these proofs to (among other things) obtain quantitative weighted norm inequalities for these operators and also use them to prove sharp $L^2$ bounds for the Christ/Goldberg matrix weighted maximal function associated with matrix A${}_2$ weights.

math.CA↗
arXiv2014

Thin sequences and the Gram matrix

We provide a new proof of Volberg's Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-$2$ class perturbation of a unitary operator. We extend this characterization from $2$ to $p$, where $2 \le p \le \infty$.

math.CV↗
arXiv2014

A matrix weighted $T1$ theorem for matrix kernelled Calderon Zygmund operators - I

In this series of two papers, we will prove a natural matrix weighted $T1$ theorem for matrix kernelled CZOs. In the current paper, we will prove matrix weighted norm inequalities for matrix symbolled paraproducts via a general matrix weighted Carleson embedding theorem. Along the way, we will also provide a stopping time proof of the identification of $L^p(W)$ as a weighted Triebel-Lizorkin space when $W$ is a matrix A${}_p$ weight.

math.CA↗
arXiv2013

Linear bounds for Calder\'{o}n-Zygmund operators with even kernel on UMD spaces

It is well-known that several classical results about Calder\'{o}n-Zygmund singular integral operators can be extended to \(X\)-valued functions if and only if the Banach space \(X\) has the UMD property. The dependence of the norm of an \(X\)-valued Calder\'{o}n-Zygmund operator on the UMD constant of the space \(X\) is conjectured to be linear. We prove that this is indeed the case for sufficiently smooth Calder\'{o}n-Zygmund operators with cancellation, associated to an even kernel. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hyt\"{o}nen to extend the result to general Calder\'{o}n-Zygmund operators.

math.CA↗