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Michał Wojciechowski

Publications and source records attributed to Michał Wojciechowski.

At least 19 recordsLinked to original sources
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.

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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}.

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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.

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arXiv2026

Finite Riesz products and Ornstein non-inequalities on quantum tori

We demonstrate a construction of products on the quantum torus $\mathbb{T}_θ^2$ that generalises the usual construction of finite Riesz products on the commutative torus $\mathbb{T}^2$. We explain why the former constitutes a natural analogue of the latter in the non-commutative setting and, based on this construction, as well as on previous results by K. Kazaniecki and the second author, we prove a non-commutative version of an Ornstein non-inequality.

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arXiv2026

Nonexistence of Henkin type projections via a Wiener theorem for multipliers

Let $d\geq 2$, $l\geq 0$ and suppose $X$ is one of the function spaces $W^{l,1}(\mathbb{T}^{d})$, $W^{l,\infty }(\mathbb{T}^{d})$ or $C^{l}(\mathbb{T}^{d})$. We extend a result of Henkin (1967), showing that, for appropriate $N\times N$ matrix operators $A(D)$, the subspace of $X^{N}$ consisting of $A(D)-$free elements is noncomplemented. In order to prove this we establish a new property of the Fourier multipliers that are bounded on $X$: the kernel $k$ of any such multiplier obeys a weaker version of Wiener's theorem for the singularities of measures.

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arXiv2025

Factoring the Sobolev embedding operator

The paper studies the factorization and summing properties of the Sobolev embedding operator. We propose two different approaches. One shows that the Sobolev embedding operator $S:W^{1,1}(\mathbb{T}^2)\hookrightarrow L_2(\mathbb{T}^2)$ factorises through the identical embedding $\ell_Φ\hookrightarrow\ell_2$ for some Young function with Matuszewska-Orlicz index 1. Proof of this fact is based on two results of independent interest. First, a necessary and sufficient conditions on a Young function $Φ$ and weight $Ψ$ for boundedness of the embedding of the Sobolev space $W^{1,1}(\mathbb{T}^2)$ into Besov-Orlicz space $B^Ψ_{Φ,1}(\mathbb{T}^2)$. Second, a generalization of the Marcinkiewicz sampling theorem to the context of Orlicz spaces. Another approach is based on the extrapolation of $(p,1)$-summing norm.

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arXiv2025

Looking for a continuous version of Bennett--Carl theorem

We study absolute summability of inclusions of r.i. function spaces. It appears that such properties are closely related, or even determined by absolute summability of inclusions of subspaces spanned by the Rademacher system in respective r.i. spaces. Our main result states that for $1<p<2$ the inclusion $X_p\subset L^p$ is $(q,1)$-absolutely summing for each $p<q<2$, where $X_p$ is the unique r.i. Banach function space in which the Rademacher system spans copy of $l^p$. This result may be regarded as a continuous version of the well-known Carl--Bennett theorem. Two different approaches to the problem and extensive discussion on them are presented. We also conclude summability type of a kind of Sobolev embedding in the critical case.

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arXiv2024

Schur property for jump parts of gradient measures

We consider weakly null sequences in the Banach space of functions of bounded variation $\mathrm{BV}(\mathbb{R}^d)$. We prove that for any such sequence $\{f_n\}$ the jump parts of the gradients of functions $f_n$ tend to $0$ strongly as measures. It implies that Dunford--Pettis property for the space $\mathrm{SBV}$ is equivalent to the Dunford--Pettis property for the Sobolev space $W^{1,1}.$

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arXiv2024

On the relations between Auerbach or almost Auberbach Markushevich systems and Schauder bases

We establish that the summability of the series $\sum\varepsilon_n$ is the necessary and sufficient criterion ensuring that every $(1+\varepsilon_n)$ Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if $n\varepsilon_n\to \infty$, then in $\ell_2$ there exists a $(1+\varepsilon_n)$ Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis.

math.FA↗
arXiv2024

Trace operator on von Koch's snowflake

We study properties of the boundary trace operator on the Sobolev space $W^1_1(Ω)$. Using the density result by Koskela and Zhang, we define a surjective operator \mbox{$Tr: W^1_1(Ω_K)\rightarrow X(Ω_K)$}, where $Ω_K$ is von Koch's snowflake and $X(Ω_K)$ is a trace space with the quotient norm. Since $Ω_K$ is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý that there exists a right inverse to $Tr$, i.e. a linear operator $S: X(Ω_K) \rightarrow W^1_1(Ω_K)$ such that $Tr \circ S= Id_{X(Ω_K)}$. In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch's snowflake. Moreover we identify the isomorphism class of the trace space as $\ell_1$. As an additional consequence of our approach we obtain a simple proof of the Peetre's theorem about non-existence of the right inverse for domain $Ω$ with regular boundary, which explains Banach space geometry cause for this phenomenon.

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arXiv2023

Alberti's type rank one theorem for martingales

We prove that the polar decomposition of the singular part of a vector measure depends on its conditional expectations computed with respect to the $q$-regular filtration. This dependency is governed by a martingale analog of the so-called wave cone, which naturally corresponds to the result of De Philippis and Rindler about fine properties of PDE-constrained vector measures. As a corollary we obtain a martingale version of Alberti's rank-one theorem.

math.FA↗
arXiv2023

On Bernstein type quantitative estimates for Ornstein non-inequalities

For the sequence of multi-indexes $\{α_i\}_{i=1}^{m}$ and $β$ we study the inequality \[ \|D^β f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{α_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric polynomial of degree at most $N$ on $d$-dimensional torus. Assuming some natural geometric property of the set $\{α_j\}\cup\{β\}$ we show that \[ K_{N}\geq C \left(\ln N\right)^ϕ, \] where $ϕ<1$ depends only on the set $\{α_j\}\cup\{β\}$.

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arXiv2022

Dimensional estimates for measures on quaternionic spheres

In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansion. This estimate is an analog of a result previously obtained by the authors for the complex spheres.

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arXiv2021

On the Embedding of $BV$ Space into Besov-Orlicz Space

We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space.

math.FA↗
arXiv2021

Marcinkiewicz Sampling Theorem for Orlicz Spaces

In the article we generalize the Marcinkiewicz sampling theorem in the context of Orlicz spaces. We establish conditions under which sampling theorem holds in terms of restricted submultiplicativity and supermultiplicativity of an $N$-function $φ$, boundedness of the Hilbert transform and Matuszewska-Orlicz indices. In addition we give a new criterion for boundedness of Hilbert transform on Orlicz space.

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