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Przemysław Ohrysko

Publications and source records attributed to Przemysław Ohrysko.

17 recordsLinked to original sources
arXiv2026

A proof of Kahane's conjecture in the Wiener algebra

We prove Kahane's conjecture: if a continuous unimodular function $F$ on the circle satisfies $\|F^n\|_{A(\mathbb{T})}=o(\log|n|)$ as $|n|\to\infty$, $n\in\mathbb{Z}$, then $F(x)=e^{i(mx+b)}$ for some $m\in\mathbb{Z}$ and $b\in\mathbb{R}$. The main step is a logarithmic lower bound for averages: for every nonconstant real-valued $φ\in A(\mathbb{T})$, $\frac{1}{T}\int_a^{a+T}\|e^{itφ}\|_{A(\mathbb{T})}dt \ge c_φ\log T \quad(a\in\mathbb{R},\ T\ge T_φ)$, where $c_φ>0$ and $T_φ>1$ are independent of $a$. The proof uses exact sampling on finite subgroups of the circle, uniformly bounded localized modulations, and the weighted non-harmonic Littlewood inequality of Jaming--Kellay--Saba. A suitable cutoff and a small linear perturbation of the phase ensure that a fixed positive proportion of the coefficients at the beginning of the frequency ordering have modulus one. Passing to integer powers and removing the degree of the circle map yield Kahane's conjecture without any additional smoothness assumptions. We also give equivalent phase, convolution, and operator formulations, and derive consequences for homomorphisms of weighted Wiener algebras and for higher-dimensional tori.

math.CA↗
arXiv2026

Strongly small-2 sets that are not Riesz sets on compact abelian groups

We prove that every infinite compact abelian group $G$ admits a countably infinite proper set $E\subset\widehat G$ that is strongly small-2 but not Riesz. More precisely, $|α|*|β|$ is absolutely continuous for all $α,β\in M_E(G)$, whereas $M_E(G)$ contains a singular complex measure of total mass one with exact Fourier support $E$ whose convolution square belongs to $L^2(G)$. When the image of the doubling map on $\widehat G$ is infinite, $E$ may moreover be chosen so that $\#(E\cap(γ-E))<\infty$ for every $γ$ and $E\cap(-E)=\{0\}$. When that image is finite, we reduce the problem to a nonrectangular construction on the Walsh group, for which $E\cap(E+γ)$ is finite at every nonzero shift.

math.FA↗
arXiv2026

Norm-Controlled Inversion in Measure Algebras

We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every $δ>1/2$ there is a constant $C_M(δ)$, independent of the group $G$, such that $\|μ\|_{M(G)}\le1$ and $\inf_{γ\in\widehat G}|\widehatμ(γ)|\geδ$ imply that $μ$ is invertible in $M(G)$ and $\|μ^{-1}\|_{M(G)}\le C_M(δ)$. Together with Nikolski's negative results, this shows that $1/2$ is the sharp universal threshold. The main step is a uniform inversion theorem for Fourier algebras $A(K)$ of compact Abelian groups; its proof is based on a profile decomposition and a resolvent argument. We also obtain norm-controlled inversion for Toeplitz operators with non-vanishing Wiener symbols of winding number zero and give direct endpoint constructions showing failure of norm control at $δ=1/2$ in $A(K)$ for every infinite compact Abelian group $K$, and in the unitization of $L^1(G)$ for every nondiscrete locally compact Abelian group $G$.

math.FA↗
arXiv2026

The $H^1(\T^d)$-multiplier norm of Rajchman measures, Zafran-type phenomena, and Walsh--Stieltjes series

We study convolution by complex Rajchman measures as Fourier multipliers on the mean-zero part of the classical isotropic Hardy space $H^1(\T^d)$. Our main technical result gives a lower bound for the multiplier norm in terms of the amount of mass carried by sets of small lower Minkowski dimension. The estimate holds for every $d\ge1$; the argument is first established for real measures and then extended to complex measures by a finite phase decomposition. Since the testing functions have mean zero, a corresponding lower bound also holds on the full isotropic Hardy space. The estimate yields multidimensional Zafran-type criteria for non-natural spectra and, in particular, shows that positive Rajchman probability measures carried by sets of lower Minkowski dimension zero induce convolution operators with non-natural spectrum. In dimension one, for real measures, we also obtain the analogous estimate on the analytic space $H^1_0(\T)$ and frequency-localized failures of the natural-spectrum property. Finally, the binary form of the underlying combinatorial lemma gives Haar block estimates, which transfer to uncertainty principles and carrier-dimension results for Walsh--Stieltjes coefficients.

math.FA↗
arXiv2026

Wiener-Pitt sets for compact Abelian groups

Suppose that $G$ is a compact Hausdorff Abelian group. We say $μ\in M(G)$ is strongly continuous if $|μ|(x+H)=0$ for any $x \in G$ and any $H \leq G$ that is closed and of infinite index. We prove that for any sufficiently rapidly decreasing sequence $(a_{n})_{n=1}^{\infty}\in c_{0}(\mathbb{N})$, for every strongly continuous $μ\in M(G)$ with $\|μ\| \leq 1$ and $\widehatμ(\widehat{G})\subset \{a_n: n \in \mathbb{N}\}\cup\{0\}$, the measure $μ\astμ$ is absolutely continuous with respect to Haar measure on $G$. This implies that $μ$ does not exhibit the so-called Wiener-Pitt phenomenon. The paper is a continuation of investigations started in \cite{ow}.

math.FA↗
arXiv2026

Nevanlinna--Pick norms on scattered and Cantor spectra

We introduce Nevanlinna--Pick norms associated with finite families of characters on a commutative semisimple Banach algebra $A$ and study the class $NP_\infty$ of algebras for which all these norms coincide with the $\ell^\infty$ norm. Our positive result is topological: if $A\in NP_\infty$ and $K\subset Δ(A)$ is compact scattered Hausdorff, then the restriction algebra $NP(A,K)$ is isometrically isomorphic to $C(K)$. In particular, if $Δ(A)$ is compact scattered, then \[ A\in NP_\infty \iff A\cong C(Δ(A)) \quad\text{isometrically}. \] By contrast, we show that this rigidity fails in the presence of Cantor structure: for every compact Hausdorff space $S$ containing a Cantor subset, there exists a commutative semisimple unital Banach algebra $A\in NP_\infty$ with $Δ(A)\cong S$ such that $A\neq C(S)$. As a consequence, for compact metrizable spectra, scatteredness is exactly the topological condition forcing rigidity.

math.FA↗
arXiv2019

The spectral radius formula for Fourier-Stieltjes algebras

In this short note we first extend the validity of the spectral radius formula obtained in \cite{ag} to Fourier--Stieltjes algebras. The second part is devoted to showing that for the measure algebra on any locally compact non-discrete Abelian group there are no non-trivial constraints between three quantities: the norm, the spectral radius and the supremum of the Fourier--Stieltjes transform even if we restrict our attention to measures with all convolution powers singular with respect to Haar measure.

math.FA↗
arXiv2019

Spectrally reasonable measures II

A measure on a locally compact Abelian group is said to have a natural spectrum if its spectrum is equal to the closure of the range of the Fourier-Stieltjes transform. In this paper we continue the study of spectrally reasonable measures (measures perturbing any measure with a natural spectrum to a measure with a natural spectrum) initiated in \cite{ow}. Particularly, we provide a full characterization of such measures for certain class of locally compact Abelian groups which includes the circle and the real line. We also elaborate on the spectral properties of measures with non-natural but real spectra constructed by F. Parreau.

math.FA↗
arXiv2018

Spectral theory of Fourier-Stieltjes algebras

In this paper we start studying spectral properties of the Fourier-Stieltjes algebras, largely following Zafran's work on the algebra of measures on a locally compact group. We show that for a large class of discrete groups the Wiener-Pitt phenomenon occurs, i.e. the spectrum of an element of the Fourier-Stieltjes algebra is not captured by its range. We also investigate the notions of absolute continuity and mutual singularity in this setting; non-commutativity forces upon us two distinct versions of support of an element, indicating a crucial difference between this setup and the realm of Abelian groups. In spite of these difficulties, we also show that one can introduce and use generalised characters to prove a criterion on belonging of a multiplicative-linear functional to the Shilov boundary of the Fourier-Stieltjes algebra.

math.FA↗
arXiv2016

Spectrally reasonable measures

In this paper we investigate the problems related to measures with a natural spectrum (equal to the closure of the set of the values of the Fourier-Stieltjes transform). Since it is known that the set of all such measures does not have a Banach algebra structure we consider the set of all suitable perturbations called spectrally reasonable measures. In particular, we exhibit a broad class of spectrally reasonable measures which contains absolutely continuous ones. On the other hand, we show that except trivial cases all discrete (purely atomic) measures do not posses this property.

math.FA↗
arXiv2014

On the Gelfand space of the measure algebra on the circle group

This paper is devoted to studying certain topological properties of the maximal ideal space of the measure algebra on the circle group. In particular, we focus on Cech cohomologies of this space. Moreover, we show that the Gelfand space of $M(\mathbb{T})$ is not separable. On the other hand, we give a direct procedure to recover many copies of $β\mathbb{Z}$ in $\mathfrak{M}(M(\mathbb{T}))$, but we also show that this is result is not accessible in the most natural way (namely, by the canonical mapping induced by homomorphism assigning to measure its Fourier - Stieltjes transform).

math.FA↗