arXiv subjects

Eskil Rydhe

Publications and source records attributed to Eskil Rydhe.

10 recordsLinked to original sources

arXiv2026

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↗

arXiv2024

A sharp higher order Sobolev embedding

We obtain sharp embeddings from the Sobolev space $W^{k,2}_0(-1,1)$ into the space $L^1(-1,1)$ and determine the extremal functions. This improves on a previous estimate of the sharp constants of these embeddings due to Kalyabin.

math.FA↗

arXiv2024

On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces

This paper studies the function spaces $\mathcal{D}(μ)$ by Richter and Aleman, and $\mathcal{D}_{\vecμ}$ by the second author. It is known that the forward shift $M_z$ is bounded and expansive on $\mathcal{D}(μ)$, and therefore $\mathcal{D}(μ)$ coincides with a de~Branges--Rovnyak space $\mathcal{H}[B]$. We show that such a $B$ is rational if and only if $μ$ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of $\mathcal{D}(μ)$ for finitely atomic $μ$. Similarly, we characterize the allowable tuples $\vecμ = (\frac{|dz|}{2π}, μ_1, \ldots, μ_{n-1})$ such that $M_z$ on $\mathcal{D}_{\vecμ}$ is expansive with finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples $\vecμ$.

math.FA↗

arXiv2020

A multi-parameter Hardy type inequality

This note contains two simple observations. First, by the weak factorization of product $H^1$ (Ferguson--Lacey, Lacey--Terwilleger), we obtain a multi-parameter analogue of Hardy's inequality. Second, as a dual statement, the Fourier transform of an essentially bounded function belongs to a certain product $\mathrm{BMO}$-Sobolev space.

math.FA↗

arXiv2020

On Laplace--Carleson embeddings, and $L^p$-mapping properties of the Fourier transform

We investigate so-called Laplace--Carleson embeddings for large exponents. In particular, we extend some results by Jacob, Partington, and Pott. We also discuss some related results for Sobolev- and Besov spaces, and mapping properties of the Fourier transform. These variants of the Hausdorff--Young theorem appear difficult to find in the literature. We conclude the paper with an example related to an open problem.

math.FA↗

arXiv2019

On $m$-isometric semigroups, and $2$-isometric cogenerators

It is known that a $C_0$-semigroup of Hilbert space operators is $m$-isometric if and only if its generator satisfies a certain condition, which we choose to call $m$-skew-symmetry. This paper contains two main results: We provide a Lumer--Phillips type characterization of generators of $m$-isometric semigroups. This is based on the simple observation that $m$-isometric semigroups are quasicontractive. We also characterize cogenerators of $2$-isometric semigroups. To this end, our main strategy is to construct a functional model for $2$-isometric semigroups with analytic cogenerators. The functional model yields numerous simple examples of $2$-isometric semigroups, but also allows for the construction of a closed, densely defined, $2$-skew-symmetric operator which is not a semigroup generator.

math.FA↗

arXiv2018

Cyclic $m$-isometries, and Dirichlet type spaces

We consider cyclic $m$-isometries on a complex separable Hilbert space. Such operators are characterized in terms of shifts on abstract spaces of weighted Dirichlet type. Our results resemble those of Agler and Stankus, but our model spaces are described in terms of Dirichlet integrals rather than analytic Dirichlet operators. The chosen point of view allows us to construct a variety of examples. An interesting feature among all of these is that the corresponding model spaces are contained in a certain subspace of the Hardy space $H^2$, depending only on the order of the corresponding operator. We also demonstrate how our framework allows for the construction of unbounded $m$-isometries.

math.FA↗

arXiv2017

Vectorial Hankel operators, Carleson embeddings, and notions of $BMOA$

We consider operators of the type $D^α:H^2(\mathcal{H})\to H^2(\mathcal{H})$, where $D^α$ denotes a fractional differentiation operator, and $Γ_ϕ$ is a Hankel operator. For $α>0$, we characterize boundedness in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries. The first is that our main result does not extend to $α=0$, i.e. Nehari-Page BMOA is not characterized by the natural anti-analytic Carleson embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of $Γ_ϕ$. The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.

math.FA↗

arXiv2016

On the characterization of Triebel--Lizorkin type spaces of analytic functions

We consider different characterizations of Triebel--Lizorkin type spaces of analytic functions on the unit disc. Even though our results appear in the folklore, detailed descriptions are hard to find, and in fact we are unable to discuss the full range of parameters. Without additional effort we work with vector-valued analytic functions, and also consider a generalized scale of function spaces, including for example so-called $Q$-spaces. The primary aim of this note is to generalize, and clarify, a remarkable result by Cohn and Verbitsky, on factorization of Triebel--Lizorkin spaces. Their result remains valid for functions taking values in an arbitrary Banach space, provided that the vector-valuedness "sits in the right factor". On the other hand, if we impose vector-valuedness on the "wrong" factor, then the factorization fails even for separable Hilbert spaces.

math.FA↗

arXiv2016

Two more counterexamples to the infinite dimensional Carleson embedding theorem

The existence of a counterexample to the infinite-dimensional Carleson embedding theorem has been established by Nazarov, Pisier, Treil, and Volberg. We provide an explicit construction of such an example. We also obtain a non-constructive example of particularly simple form; the density function of the measure (with respect to a certain weighted area measure) is the tensor-square of a Hilbert space-valued analytic function. This special structure of the measure has implications for Hankel-like operators appearing in control theory.

math.FA↗