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Ole Fredrik Brevig

Publications and source records attributed to Ole Fredrik Brevig.

At least 19 recordsLinked to original sources
arXiv2026

Contractivity of Rudin's diagonal restriction operator for $H^p$ of the polydisc

We establish that Rudin's diagonal restriction operator is contractive from the Hardy space $H^p$ of the bidisc to the Bergman space $A^p$ of the disc for $2\leq p<\infty$, and characterize all cases of equality. We extend the result to polydiscs and obtain applications to polynomial point evaluation among others. The proof relies on an explicit formula, deduced from certain Green's formulas, for the defect between the $p$th powers of the two $L^p$ norms.

math.FA↗
arXiv2026

Mean values, order, and convergence in the Hardy space of Dirichlet series

Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if $f$ in $\mathscr{H}^2$ has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values \[\|f\|_{\mathscr{H}^2}^2 = \lim_{σ\to 0^+} \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T \lvert f(σ+it) \rvert^2\,dt.\] We investigate to what extent the assumption that the analytic continuation be bounded can be relaxed, and obtain optimal results in two different directions. The first direction concerns the order of the analytic continuation, and yields as a corollary a version of the classical Landau--Schnee convergence theorem for $\mathscr{H}^2$. The second direction concerns the abscissa of convergence, and resolves a problem posed by McCarthy.

math.CA↗
arXiv2026

Bohr sets for multiplier algebras of complete Pick spaces

We reexamine F. Wiener's classical proof that the Bohr radius of $H^\infty$ is $1/3$. We replace $H^\infty$ by the multiplier algebras of complete Pick spaces and describe the Bohr set for a wide class of such algebras. In particular, we show that the Bohr set of the multiplier algebra of the Drury--Arveson space $H^2_d$ is the closed $\ell^2$-ball of radius $1/3$. Other cases covered by our description of Bohr sets include Dirichlet-type spaces of analytic functions in the unit disc and a class of Hilbert spaces of ordinary Dirichlet series, studied earlier by McCarthy and Shalit.

math.FA↗
arXiv2026

A bootstrap proof of Bohr's theorem

Let $f$ be an analytic function in the unit disc whose modulus is bounded by $1$. We present a new proof of Bohr's theorem, which asserts that its majorant function $Mf(r)$ is bounded by $1$ for $0\leq r \leq 1/3$ and that $1/3$ is optimal. The argument also yields a sharp upper bound for $Mf(r)$ in terms of $\lvert f(0) \rvert$, complementing a result due to Bombieri.

math.CV↗
arXiv2026

A simple proof that the Riesz projection is bounded on $L^p(\mathbb{T})$ for $1<p<\infty$

Let $\mathbf{P}$ denote the Riesz projection on the unit circle $\mathbb{T}$ and suppose that $1<p<\infty$. We present a simple proof of the bound $\|\mathbf{P}f\|_p \leq \max(p,q) \|f\|_p$, where $f$ is in $L^p(\mathbb{T})$ and $p^{-1}+q^{-1}=1$. Our proof is a variation of a classical argument due to M. Riesz demonstrating that the Hilbert transform is bounded on $L^p(\mathbb{T})$.

math.CA↗
arXiv2026

Harald Bohr's splitting theorem

We present a new elementary proof of a theorem due to Harald Bohr, which states that an unbounded, analytic, and almost periodic function in a half-plane can be written as the sum of two analytic functions: the first is unbounded and periodic, while the second is bounded and almost periodic. The proof is based on a well-known arithmetical property of translation numbers of almost periodic functions.

math.CA↗
arXiv2025

Contractive Hardy--Littlewood inequalities in the Dirichlet range

The class $A_α^p$ consists of those analytic functions $f$ in the unit disc such that \[\|f\|_{α,p}^p := |f(0)|^p+\int_0^1 \left(\frac{d}{dr} M_p^p(r,f)\right) (1-r^2)^{α-1} \,dr < \infty,\] where $M_p^p(r,f)$ is the radial integral mean of $|f|^p$ and $0<α, p <\infty$. For $α>1$, $A_α^p$ is the standard weighted Bergman space, and $A_1^p=H^p$. We consider $A_α^p$ for $0<α<1$ and show that (weighted) isometric conformal invariance extends to this range, and we also clarify the relation between $A_α^p$ and the classical Besov spaces. Our main result is the contractive inequality $\|f\|_{β,q} \leq \|f\|_{α,p}$, valid when $0<α<β<\infty$ and $α/p=β/q$. We also identify the functions for which equality is attained. We thus extend recent results of the second-named author ($1\leq α<β$) and Llinares ($β=1$ and $p=2$). The extension of results from the classical range $1\leq α< \infty$ to the Dirichlet range $0<α<1$ uses arguments relying on analytic continuation.

math.CV↗
arXiv2025

Carlson's theorem and vertical limit functions

We extend a classical theorem of Carlson on moments of Dirichlet series from $p=2$ to $1 \leq p < \infty$. When combined with the ergodic theorem for the Kronecker flow, a coherent approach to almost sure properties of vertical limit functions in $H^p$ spaces of Dirichlet series is obtained. This allows us to establish an almost sure analytic continuation of vertical limit functions to the right half-plane that can be used to compute the $H^p$ norm and to prove a version of Fatou's theorem.

math.CA↗
arXiv2025

Two footnotes to the F. & M. Riesz theorem

We present a new proof of the F. & M. Riesz theorem on analytic measures of the unit circle $\mathbb{T}$ that is based the following elementary inequality: If $f$ is analytic in the unit disc $\mathbb{D}$ and $0 \leq r \leq \varrho < 1$, then \[\|f_r-f_\varrho\|_1 \leq 2 \sqrt{\|f_\varrho\|_1^2-\|f_r\|_1^2},\] where $f_r(e^{iθ})=f(r e^{iθ})$ and where $\|\cdot\|_1$ denotes the norm of $L^1(\mathbb{T})$. The proof extends to the infinite-dimensional torus $\mathbb{T}^\infty$, where it clarifies the relationship between Hilbert's criterion for $H^1(\mathbb{T}^\infty)$ and the F. & M. Riesz theorem.

math.CV↗
arXiv2025

Almost periodicity and boundary values of Dirichlet series

We employ almost periodicity to establish analogues of the Hardy--Stein identity and the Littlewood--Paley formula for Hardy spaces of Dirichlet series. A construction of Saksman and Seip shows that the limits in this Littlewood--Paley formula cannot be interchanged. We apply this construction to show that the limits in the definition of the mean counting function for Dirichlet series cannot be interchanged. These are essentially statements about the two different kinds of boundary values that we associate with Dirichlet series that converge to a bounded analytic function in a half-plane. The treatment of the mean counting function also involves an investigation of the zero sets and Blaschke products of such Dirichlet series.

math.CA↗
arXiv2024

Critical exponents of the Riesz projection

Let $\mathfrak{p}_d(q)$ denote the critical exponent of the Riesz projection from $L^q(\mathbb{T}^d)$ to the Hardy space $H^p(\mathbb{T}^d)$, where $\mathbb{T}$ is the unit circle. We present the state-of-the-art on the conjecture that $\mathfrak{p}_1(q) = 4(1-1/q)$ for $1 \leq q \leq \infty$ and prove that it holds in the endpoint case $q = 1$. We then extend the conjecture to \[\mathfrak{p}_d(q) = 2+\cfrac{2}{d+\cfrac{2}{q-2}}\] for $d\geq1$ and $\frac{2d}{d+1} \leq q \leq \infty$ and establish that if the conjecture holds for $d=1$, then it also holds for $d=2$. When $d=2$, we verify that the conjecture holds in the endpoint case $q = 4/3$.

math.FA↗
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

Point evaluation in Paley--Wiener spaces

We study the norm of point evaluation at the origin in the Paley--Wiener space $PW^p$ for $0 < p < \infty$, i. e., we search for the smallest positive constant $C$, called $\mathscr{C}_p$, such that the inequality $|f(0)|^p \leq C \|f\|_p^p$ holds for every $f$ in $PW^p$. We present evidence and prove several results supporting the following monotonicity conjecture: The function $p\mapsto \mathscr{C}_p/p$ is strictly decreasing on the half-line $(0,\infty)$. Our main result implies that $\mathscr{C}_p p/2$ for $1 \leq p < 2$. We also estimate the asymptotic behavior of $\mathscr{C}_p$ as $p \to \infty$ and as $p \to 0^+$. Our approach is based on expressing $\mathscr{C}_p$ as the solution of an extremal problem. Extremal functions exist for all $0<p<\infty$; they are real entire functions with only real zeros, and the extremal functions are known to be unique for $1\leq p < \infty$. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau--Pollak--Slepian operator of time--frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range $1<p<\infty$, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to $\mathscr{C}_p$ and the extremal functions.

math.CA↗
arXiv2023

An extension of Bohr's theorem

The following extension of Bohr's theorem is established: If a somewhere convergent Dirichlet series $f$ has an analytic continuation to the half-plane $\mathbb{C}_θ= \{s = σ+it\,:\, σ>θ\}$ that maps $\mathbb{C}_θ$ to $\mathbb{C} \setminus \{α,β\}$ for complex numbers $α\neq β$, then $f$ converges uniformly in $\mathbb{C}_{θ+\varepsilon}$ for any $\varepsilon>0$. The extension is optimal in the sense that the assertion no longer holds should $\mathbb{C}\setminus \{α,β\}$ be replaced with $\mathbb{C}\setminus \{α\}$.

math.CV↗
arXiv2023

The best constant in a Hilbert-type inequality

We establish that \[\sum_{m=1}^\infty \sum_{n=1}^\infty a_m \overline{a_n} \frac{mn}{(\max(m,n))^3} \leq \frac{4}{3}\sum_{m=1}^\infty |a_m|^2\] holds for every square-summable sequence of complex numbers $a = (a_1,a_2,\ldots)$ and that the constant $4/3$ cannot be replaced by any smaller number. Our proof is rooted in a seminal 1911 paper concerning bilinear forms due to Schur, and we include for expositional reasons an elaboration on his approach.

math.CA↗
arXiv2023

Maximal norm Hankel operators

A Hankel operator $\mathbf{H}_φ$ on the Hardy space $H^2$ of the unit circle with analytic symbol $φ$ has minimal norm if $\|\mathbf{H}_φ\|=\|φ\|_2$ and maximal norm if $\|\mathbf{H}_φ\| = \|φ\|_\infty$. The Hankel operator $\mathbf{H}_φ$ has both minimal and maximal norm if and only if $|φ|$ is constant almost everywhere on the unit circle or, equivalently, if and only if $φ$ is a constant multiple of an inner function. We show that if $\mathbf{H}_φ$ is norm-attaining and has maximal norm, then $\mathbf{H}_φ$ has minimal norm. If $|φ|$ is continuous but not constant, then $\mathbf{H}_φ$ has maximal norm if and only if the set at which $|φ|=\|φ\|_{\infty}$ has nonempty intersection with the spectrum of the inner factor of $φ$. We obtain further results illustrating that the case of maximal norm is in general related to "irregular" behavior of $\log |φ|$ or the argument of $φ$ near a "maximum point" of $|φ|$. The role of certain positive functions coined apical Helson--Szegő weights is discussed in the former context.

math.FA↗