arXiv subjects

David Norrbo

Publications and source records attributed to David Norrbo.

9 recordsLinked to original sources

arXiv2026

Sharp exponential integrability of conjugate functions

We prove that if a real-valued function $f\in L^1$ on the complex unit circle has a gap of width at least $\pi$ in its essential range, then $\exp(\widetilde f)$ is not integrable, where $\widetilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ satisfying $$ \int_0^\infty Y(x) e^{-x}\, dx = \infty. $$ We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, $\widetilde f$ fails to be $Y$-integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Finally, we obtain corresponding results for complex-valued functions.

math.CV↗

arXiv2026

The Boundedness Problem for Generalized Hilbert Operators on Hardy Spaces

Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H^p$ is characterized by the mean Lipschitz condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ when $1<p\leq2$, while the problem remains open for $2<p<\infty$. It has been recently proved that the condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ does not imply the boundedness of $\mathcal H_g$ on $H^p$, $2<p<\infty$ \cite{GuoTang2026}. We show that this condition is far from sufficient in the latter range: for every $2<p<\infty$, there exists a function $g\in\Lambda\left(p,\frac{1}{p}\right)$ such that $\mathcal H_g$ is not bounded even from $H^p$ into $H^1$. The main ingredient is an exact characterization of the boundedness of $\mathcal{H}_g:H^p\to H^2$ for all $1\leq p\leq\infty$. In particular, when $2<p<\infty$, this mapping is bounded if and only if $g'$ belongs to a certain mixed-norm space. For lacunary symbols, the same mixed-norm condition also characterizes the boundedness of $\mathcal H_g$ on $H^p$, and hence gives a complete solution of the open problem within this class of symbols. We also show that, for $1\leq q\leq\infty$, boundedness of $\mathcal H_g:H^1\to H^q$ is characterized by the condition $g'\in H^q$. We also characterize compactness of $\mathcal H_g$ in the aforementioned cases.

math.CV↗

arXiv2025

Weak null maximum and integration of families of multiplication operators

Let $X$ be a reflexive Hardy space or weighted Bergman space on the unit disk in the complex plane. For a bounded linear operator $S$ on $X$, let $\textrm{wem}(S):= \sup_{(f_n)} \limsup_n \|Sf_n\|$, that is, the supremum of cluster points of $n\mapsto \|S f_n\|$, where $(f_n)$ is any unit norm weakly null sequence. This quantity coincides with the essential norm on the reflexive weighted Bergman spaces. For a suitable family $\{ g_t : t\in]0,1[ \}$ of bounded analytic functions on the unit disk, we characterize when one can exchange $\textrm{wem}(\cdot)$ and integration over $t$ of the multiplication operators $M_{g_t}$, that is, when $\textrm{wem}( \int M_{g_t}\, dt ) = \int \textrm{wem}( M_{g_t} ) \, dt $; when the functions $g_t,t\in]0,1[$ can be continuously extended to the unit circle, we obtain a neat function-theoretic characterization.

math.FA↗

arXiv2025

Weak cluster points of maximizing sequences on Banach spaces satisfying $\boldsymbol{(M_p)}$

Given a bounded linear operator $T$ on a separable Banach space with property $(M_p)$, we prove that the smallest and the largest norm of weak cluster points of all maximizing sequences for $T$ can only take the values $0$ or $1$. The three classes of bounded linear operators emerging from the dichotomy of these extremal norm values coincides with the partition, created by considering the norm-attaining property and if the essential norm equals the norm.

math.FA↗

arXiv2025

Essential norm and integration of a family of weighted composition operators

We study the interchange of essential norm and integration of certain families of weighted composition operators acting on the standard weighted Bergman spaces $A^p_\alpha$, where $p>1$ and $\alpha\geq 0$. To be more precise, we give a sufficient condition for $ \|\int u_tC_{\phi_t}\, dt\|_e = \int \| u_tC_{\phi_t}\|_e \, dt $ to hold in terms of geometric properties of $u_t$ and $\phi_t$. We also provide some necessary conditions for the equality to hold and calculate the essential norm of some integral operators such as some Volterra operators.

math.FA↗

arXiv2025

Compactness and related properties of weighted composition operators on weighted BMOA spaces

It is shown that a large class of properties coincide for weighted composition operators on a large class of weighted VMOA spaces, including the ones with logarithmic weights and the ones with standard weights $(1-|z|)^{-c}, \ 0\leq c< 1/2$. Some of these properties are compactness, weak compactness, complete continuity and strict singularity. A function-theoretic characterization for these properties is also given. Similar results are also proved for many weighted composition operators on similarly weighted BMOA spaces. The main results extend the theorems given in [Proc. Amer. Math. Soc. 151 (2023), 1195--1207], and new test functions that are suitable for the weighted setting are developed.

math.FA↗

arXiv2024

Generalized Hilbert matrix operators acting on Bergman spaces

In this article we study the generalized Hilbert matrix operator $\Gamma_\mu$ acting on the Bergman spaces $A^p$ of the unit disc for $1\leq p<\infty$. In particular, we characterize the measures $\mu$ for which the operator $\Gamma_\mu$ is bounded and we provide estimates of its operator norm. Finally, we also describe when $\Gamma_\mu$ is compact by computing its essential norm.

math.CV↗

arXiv2022

Exact essential norm of generalized Hilbert matrix operators on classical analytic function spaces

We compute the exact value of the essential norm of a generalized Hilbert matrix operator acting on weighted Bergman spaces $A^p_v$ and weighted Banach spaces $H^\infty_v$ of analytic functions, where $v$ is a general radial weight. In particular, we obtain the exact value of the essential norm of the classical Hilbert matrix operator on standard weighted Bergman spaces $A^p_α$ for $p>2+α, \, α\ge 0,$ and on Korenblum spaces $H^\infty_α$ for $0 < α< 1.$ We also cover the Hardy space $H^p, \, 1 < p < \infty,$ case. In the weighted Bergman space case, the essential norm of the Hilbert matrix is equal to the conjectured value of its operator norm and similarly in the Hardy space case the essential norm and the operator norm coincide. We also compute the exact value of the norm of the Hilbert matrix on $H^\infty_{w_α}$ with weights $w_α(z)=(1-|z|)^α$ for all $0 < α< 1$. Also in this case, the values of the norm and essential norm coincide.

math.FA↗

arXiv2020

Unified approach to spectral properties of multipliers

Let $\mathbb B_n$ be the open unit ball in $\mathbb C^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on $\mathbb B_n$ satisfying some general conditions. These spaces include Bergman-Sobolev spaces $A^p_{α,β}$, Bloch-type spaces $\mathcal B_α$, weighted Hardy spaces $H^p_w$ with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces $H^2_β$. Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra.

math.FA↗