arXiv subjects

Konstantinos Bampouras

Publications and source records attributed to Konstantinos Bampouras.

9 recordsLinked to original sources

arXiv2026

Besov spaces and Schatten class Hankel operators for Hardy and Paley--Wiener spaces in higher dimensions

We consider Schatten class membership of Hankel operators on Paley--Wiener spaces of convex $Ω\subset \mathbb{R}^n$, both for bounded and unbounded domains. In particular, the classical product Hardy spaces fit within our theory. For admissible domains, we develop a framework and theory of Besov spaces of Paley--Wiener type, and prove that a Hankel operator belongs to the Schatten class $S^p$ if and only if its symbol belongs to a corresponding Besov space, for $1 \leq p \leq 2$. We extend this result to all $1 \leq p < \infty$ for the classical product Hardy spaces and to $1 \leq p < 2(n+1)/(n-1)$ for the Paley--Wiener space of a bounded smooth domain $Ω\subset \mathbb{R}^n$ of strictly positive curvature.

math.FA↗

arXiv2026

Helson Inequality, Hankel Operators, and Weak Factorization on Paley-Wiener spaces of Convex Domains

For any convex set $Ω\subset\mathbb{R}^n$ that does not contain affine lines, we prove the inequality $$\int_Ω\frac{|\hat{f}(x)|^2}{ω_Ω(x)}\,dx\leq C(n)\|f\|_{L^1}^2,\quad \supp\hat{f}\subsetΩ,$$ where $ω_Ω(x)=m(Ω\cap(2x-Ω))$. As a consequence, we derive a weak factorization for $$\PW^1(Ω)=\{f\in L^1(\mathbb{R}^n):\supp\hat{f}\subsetΩ\}.$$ Furthermore, we establish a complete characterization of Schatten class Hankel operators for polyhedra for all $1\leq p<\infty,$ extending the already known $1\leq p\leq 2$ range.

math.FA↗

arXiv2026

Boundary-Weighted Fourier Inequalities for Convex Domains

We consider the natural family of Fourier inequalities for the Paley--Wiener space $\mathrm{PW}^q(Ω)$, consisting of $L^q$-functions with Fourier support in a convex set $Ω\subset \mathbb{R}^n$, $n \geq 2$, free of affine lines. Namely, \[ \int_Ω\dfrac{|\hat{f}(x)|^p}{ω_Ω^d(x)}dx\leq C\|f\|_{L^q}^p,\quad f\in \mathrm{PW}^q(Ω). \] Here $\hat{f}$ is the Fourier transform of $f$, $1 \leq p, q < \infty$, $d \in \mathbb{R}$, and $ω_Ω$ is the frequency multiplier weight associated with the Paley--Wiener space of $Ω$, \[ ω_Ω(x)=m(Ω\cap (2x-Ω)), \qquad x \in Ω. \] For an arbitrary polyhedron $P$, we completely characterize the triples $(p,q,d)$ which yield valid Fourier inequalities. For a ball $B$, we characterize the valid triples when $p \geq 2$. When $p < 2$, the situation is different for the ball, and natural critical inequalities fail. However, we show that the spherical restriction conjecture implies a family of subcritical Fourier inequalities for the ball, which in turn imply the Kakeya conjecture (in its Minkowski-form). Finally, we link our family of Fourier inequalities to the theory of truncated Hankel operators acting on the Paley--Wiener space of $Ω$.

math.CA↗

arXiv2026

The norm of the backward shift on $H^4$ is $\sqrt[4]φ$

We prove that the backward shift operator on $H^4$ has norm equal to $\sqrt[4]φ$, with $φ= \frac{1 + \sqrt{5}}{2}$. Furthermore, we characterize all extremal functions; they are precisely the functions of the form \[ f(z) = μ\left( I(z) - \sqrt{\frac{1}{2φ}}\right), \] where $μ\in \mathbb{C}$ and $I$ is an inner function with $I(0) = \sqrt{\fracφ{2}}$.

math.CV↗

arXiv2026

Recovering Product BMO from Schatten Hankel operators

We prove that if a small Hankel operator on the product Hardy space belongs to some Schatten class $S^p$, $p < \infty$, then it has a symbol in product BMO. In other words, the conclusion of Nehari's theorem holds under the hypothesis that the operator belongs to a Schatten class.

math.FA↗

arXiv2026

Nehari's Theorem and Hardy's inequality for Paley--Wiener spaces

Recently it was proven that for a convex subset of $\mathbb{R}^{n}$ that has infinitely many extreme vectors, the Nehari theorem fails, that is, there exists a bounded Hankel operator $\Ha_ϕ$ on the Paley--Wiener space $\PW(Ω)$ that does not admit a bounded symbol. In this paper we examine whether Nehari's theorem can hold under the stronger assumption that the Hankel operator $\Ha_ϕ$ is in the Schatten class $S^{p}(\PW(Ω))$. We prove that this fails for $p>4$ for any convex subset of $\mathbb{R}^{n}$, $n\geq2$, of boundary with a $C^{2}$ neighborhood of nonzero curvature. Furthermore we prove that for a polytope $P$ in $\mathbb{R}^{n}$, the inequality $$\int_{2P}\dfrac{|\widehat{f}(x)|}{m(P\cap (x-P))}dx\leq C(P)\|f\|_{L^{1}},$$ holds for all $f\in \PW^{1}(2P)$, and consequently any Hilbert--Schmidt Hankel operator on a Paley--Wiener space of a polytope is generated by a bounded function.

math.FA↗

arXiv2025

Inner Functions, Möbius Distortion and Angular Derivatives

We prove that an inner function has finite $\mathcal{L} (p)$-entropy if and only if its accumulated Möbius distortion is in $L^p$, $0<p<\infty$. We also study the support of the positive singular measures such that their corresponding singular inner functions have finite $\mathcal{L} (p)$-entropy.

math.CV↗

arXiv2024

Norm attaining vectors and Hilbert points

Let $H$ be a Hilbert space that can be embedded as a dense subspace of a Banach space $X$ such that the norm of the embedding is equal to $1$. We consider the following statements for a nonzero vector $φ$ in $H$: (A) $\|φ\|_X = \|φ\|_H$. (H) $\|φ+f\|_X \geq \|φ\|_X$ for every $f$ in $H$ such that $\langle f, φ\rangle =0$. We use duality arguments to establish that (A) $\implies$ (H), before turning our attention to the special case when the Hilbert space in question is the Hardy space $H^2(\mathbb{T}^d)$ and the Banach space is either the Hardy space $H^1(\mathbb{T}^d)$ or the weak product space $H^2(\mathbb{T}^d) \odot H^2(\mathbb{T}^d)$. If $d=1$, then the two Banach spaces are equal and it is known that (H) $\implies$ (A). If $d\geq2$, then the Banach spaces do not coincide and a case study of the polynomials $φ_α(z) = z_1^2 + αz_1 z_2 + z_2^2$ for $α\geq0$ illustrates that the statements (A) and (H) for the two Banach spaces describe four distinct sets of functions.

math.FA↗

arXiv2023

On the failure of the Nehari Theorem for Paley-Wiener spaces

Let $Ω$ be a nonempty, open and convex subset of $\mathbb{R}^{n}$. The Paley-Wiener space with respect to $Ω$ is defined to be the closed subspace of $L^{2}(\mathbb{R}^{n})$ of functions with Fourier transform supported in $2Ω$. For a tempered distribution $ϕ$ we define a Hankel operator to be the densely defined operator: $$\widehat{H_ϕf}(x)=\int_Ω\widehat{f}(y)\widehatϕ(x+y)dy,\text{ for $x\inΩ$}.$$ We say that the Nehari theorem is true for $Ω$, if every bounded Hankel operator is generated by a bounded function. In this paper we prove that the Nehari theorem fails for any convex set in $\mathbb{R}^{n}$ that has infinitely many extreme points. In particular, it fails for all convex bounded sets which are not polytopes. Furthermore, in the setting of $R^{2}$, it fails for all non-polyhedral sets, bounded or unbounded.

math.FA↗