We prove an abstract interpolation theorem which interpolates the (r,2)-summing and (s,2)-mixing norm of a fixed operator in the image and the range space. Combined with interpolation formulas for spaces of operators we obtain as an application the original Bennett-Carl inequalities for identities acting between Minkowski spaces l_u as well as their analogues for Schatten classes S_u. Furthermore, our techniques motivate a study of Bennett-Carl inequalities within a more general setting of symmetric Banach sequence spaces and unitary ideals.
The Fréchet (resp.\ (LB)) sequence spaces $ces(p+) := \cap_{r > p} ces(r), 1 \leq p < \infty $ (resp.\ $ ces (p-) := \cup_{ 1 < r < p} ces (r), 1 < p \leq \infty),$ are known to be very different to the classical sequence spaces $ \ell_ {p+} $ (resp., $ \ell_{p_{-}}).$ Both of these classes of non-normable spaces $ ces (p+), ces (p-)$ are defined via the family of reflexive Banach sequence spaces $ ces (p), 1 < p < \infty .$ The dual Banach spaces $ d (q), 1 < q < \infty ,$ of the discrete Cesàro spaces $ ces (p), 1 < p < \infty,$ were studied by G.\ Bennett, A.\ Jagers and others. Our aim is …
We prove a substantial extension of a well-known result due to Bennett and Carl: The inclusion of a 2-concave symmetric Banach sequence space E into l_2 is (E,1)-summing, i.e. for every unconditionally summable sequence (x_n) in E the scalar sequence (||x_n||_2) is contained in E. Various applications are given, e.g. to the theory of eigenvalue distribution of compact operators and approximation theory.
Hytönen, McIntosh and Portal (J. Funct. Anal., 2008) proved two vector-valued generalizations of the classical Carleson embedding theorem, both of them requiring the boundedness of a new vector-valued maximal operator, and the other one also the type p property of the underlying Banach space as an assumption. We show that these conditions are also necessary for the respective embedding theorems, thereby obtaining new equivalences between analytic and geometric properties of Banach spaces.
We prove that for any two quasi-Banach spaces $X$ and $Y$ and any $α>0$ there exists a constant $γ_α>0$ such that $$ \sup_{1\le k\le n}k^αe_k(T)\le γ_α\sup_{1\le k\le n} k^αc_k(T) $$ holds for all linear and bounded operators $T:X\to Y$. Here $e_k(T)$ is the $k$-th entropy number of $T$ and $c_k(T)$ is the $k$-th Gelfand number of $T$. For Banach spaces $X$ and $Y$ this inequality is widely used and well-known as Carl's inequality. For general quasi-Banach spaces it is a new result.
We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition corresponds to the L^2 case in our theory. The result is applied to give a new proof for the L^p-boundedness of paraproducts with a BMO symbol. A novel feature of the argument is that all p are covered at once in a completely interpolation-free manner. This is achieved by using the L^1 Carleson norm, and indicates the usefulness of this notion. Our approach is…
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.
For two given symmetric sequence spaces $E$ and $F$ we study the $(E,F)$-multiplier space, that is the space all of matrices $M$ for which the Schur product $M\ast A$ maps $E$ into $F$ boundedly whenever $A$ does. We obtain several results asserting continuous embedding of $(E,F)$-multiplier space into the classical $(p,q)$-multiplier space (that is when $E=l_p$, $F=l_q$). Furthermore, we present many examples of symmetric sequence spaces $E$ and $F$ whose projective and injective tensor products are not isomorphic to any subspace of a Banach space with an unconditional basis, extending classi…
In this paper we give the isomorphic classification of atomic $H^1(X,d,μ)$, where $(X,d,μ)$ is a space of homogeneous type, hereby completing a line of investigation opened by the work of Bernard Maurey [Ma1], [Ma2], [Ma3] and continued by Lennard Carleson [C] and Przemyslaw Wojtaszczyk [Woj1], [Wpj2].
A description of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a reflexive complex Banach space is shown to remain true without the reflexivity requirement. A similar nature description of the entire vectors of exponential type, known for a normal operator in a complex Hilbert space, is generalized to the case of a scalar type spectral operator in a complex Banach space.
Given a domain $Ω$ in $\mathbb{C}^n$ and a collection of test functions $Ψ$ on $Ω$, we consider the complex-valued $Ψ$-Schur-Agler class associated to the pair $(Ω,\,Ψ)$. In this article, we characterize interpolating sequences for the associated Banach algebra of which the $Ψ$-Schur-Agler class is the closed unit ball. When $Ω$ is the unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$ and the class of test function includes only the identity function on $\mathbb{D}$, the aforementioned algebra is the algebra of bounded holomorphic functions on $\mathbb{D}$ and in this case, our characte…
We establish a Fredholm criterion for an arbitrary operator in the Banach algebra of singular integral operators with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over Carleson curves with logarithmic whirl points.
Let $L$ be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces $L^{p}(R^{n};X)$ of $X$-valued functions on $R^n$. We characterize Kato's square root estimates $\|\sqrt{L}u\|_{p} \eqsim \|\nabla u\|_{p}$ and the $H^{\infty}$-functional calculus of $L$ in terms of R-boundedness properties of the resolvent of $L$, when $X$ is a Banach function lattice with the UMD property, or a noncommutative $L^{p}$ space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. I…
We extend Rubio de Francia's extrapolation theorem for functions valued in UMD Banach function spaces, leading to short proofs of some new and known results. In particular we prove Littlewood-Paley-Rubio de Francia-type estimates and boundedness of variational Carleson operators for Banach function spaces with UMD concavifications.
A Banach space valued graphon is a function $W:(Ω, \mathcal{A},π)^2\to\mathcal{Z}$ from a probability space to a Banach space with a separable predual, measurable in a suitable sense, and lying in appropriate $L^p$-spaces. As such we may consider $W(x,y)$ as a two-variable random element of the Banach space. A two-dimensional analogue of moments can be defined with the help of graphs and weak-* evaluations, and a natural question that then arises is whether these generalized moments determine the function $W$ uniquely -- up to measure preserving transformations. The main motivation comes from …
For $p\in(0,1),$ let $Q_p$ spaces be the space of all analytic functions on the unit disk $\mathbb{D}$ such that $|f'(z) | ^2 (1-| z| ^2)^p dA(z)$ is a $p$ - Carleson measure. In this paper, we prove that the Wolff's Ideal Theorem on $H^\infty{(\mathbb{D})}$ can be extended to the Banach algebra $H^{\infty}(\mathbb{D})\cap Q_{p}$, and also to the multiplier algebra on $Q_p$ spaces.
We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the lattice operation of taking the positive part; in case of $\mathbb R^d$-valued neural networks this corresponds to the ReLU activation function. We prove inverse and direct approximation theorems with Monte-Carlo rates for a certain class of functions, extending existing results for the finite-dimensional case. In the second part of the paper, we study, from …
The purpose of this article is to develop a technique to estimate certain bounds for entropy numbers of diagonal operator on spaces of p-summable sequences for finite p greater than 1. The approximation method we develop in this direction works for a very general class of operators between Banach spaces, in particular reflexive spaces. As a consequence of this technique we also obtain that the entropy number of a bounded linear operator T between two separable Hilbert spaces is equal to the entropy number of the adjoint of T. This gives a complete answer to the question posed by B. Carl [4] in…
We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, defined by the $L_1$ Carleson functional and $L_q$ ($1<q<\infty$) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.
For a fixed Banach operator ideal $\mathcal A$, we use the notion of $\mathcal A$-compact sets of Carl and Stephani to study $\mathcal A$-compact polynomials and $\mathcal A$-compact holomorphic mappings. Namely, those mappings $g\colon X\rightarrow Y$ such that every $x \in X$ has a neighborhood $V_x$ such that $g(V_x)$ is relatively $\mathcal A$-compact. We show that the behavior of $\mathcal A$-compact polynomials is determined by its behavior in any neighborhood of any point. We transfer some known properties of $\mathcal A$-compact operators to $\mathcal A$-compact polynomials. In order t…
We prove a Fredholm criterion for operators in the Banach algebra of singular integral operators with matrix piecewise continuous coefficients acting on a variable Lebesgue space with a radial oscillating weight over a logarithmic Carleson curve. The local spectra of these operators are massive and have a shape of spiralic horns depending on the value of the variable exponent, the spirality indices of the curve, and the Matuszewska-Orlicz indices of the weight at each point. These results extend (partially) the results of A. Böttcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spac…
The purpose of the paper is to study the operators on the weighted Bergman spaces on the unit disk ${\mathbb{D}}$, denoted by $A^{p}_{λ,w}({\mathbb{D}})$, that are associated with a class of generalized analytic functions, named the $λ$-analytic functions, and with a class of radial weight functions $w$. For $λ\ge0$, a $C^2$ function $f$ on ${\mathbb D}$ is said to be $λ$-analytic if $D_{\bar{z}}f=0$, where $D_{\bar{z}}$ is the (complex) Dunkl operator given by $D_{\bar{z}}f=\partial_{\bar{z}}f-λ(f(z)-f(\bar{z}))/(z-\bar{z})$. It is shown that, for $2λ/(2λ+1)\le p\le1$, the boundedness of an o…
