arXiv subjects
Publications and source records attributed to Michał Wojciechowski.
In this paper we study the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. This is achieved by adapting the method of Brummelhuis to the non-analytic case. As an application we obtain a general form of uncertainty principle for measures on the complex sphere which subsumes certain classical results about pluriharmonic measures.
We provide an estimate from below for the lower Hausdorff dimension of measures on the unit circle based on the arithmetic properties of their spectra. We obtain our bounds via application of a general result for abstract $q$-regular martingales to the Gundy--Varopoulos backwards martingale. To show the sharpness of our method, we improve the best known numerical lower bound for the Hausdorff dimension of certain Riesz products.
In this paper we quantify the notion of antisymmetry of the Fourier transform of certain vector valued measures. The introduced scale is related to the condition appearing in Uchiyama's theorem and is used to give a lower bound for the rectifiable dimension of those measures. Moreover, we obtain an estimate of the lower Hausdorff dimension assuming certain more restrictive version of the $2$-wave cone condition. Results of our considerations can be viewed as an uncertainty-type principle in the following way: it is impossible to simultaneously localize a (bundle) measure and a direction of its Fourier transform on small sets. The investigated class is modeled on the example of gradients of BV functions. The article contains also a theorem concerning regularity: we prove that elements of considered class vanish on 1-purely unrectifiable sets. Our results can be applied to studying the properties of PDE-constrainted measures.
The well known result of Bourgain and Kwapień states that the projection $P_{\leq m}$ onto the subspace of the Hilbert space $L^2\left(Ω^\infty\right)$ spanned by functions dependent on at most $m$ variables is bounded in $L^p$ with norm $\leq c_p^m$ for $1<p<\infty$. We will be concerned with two kinds of endpoint estimates. We prove that $P_{\leq m}$ is bounded on the space $H^1\left(\mathbb{D}^\infty\right)$ of functions in $L^1\left(\mathbb{T}^\infty\right)$ analytic in each variable. We also prove that $P_{\leq 2}$ is bounded on the martingale Hardy space associated with a natural double-indexed filtration and, more generally, we exhibit a multiple indexed martingale Hardy space which contains $H^1\left(\mathbb{D}^\infty\right)$ as a subspace and $P_{\leq m}$ is bounded on it.
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.
We consider Auerbach bases in Banach spaces of dimension n>2. We show that there exists at least (n-1)n/2+1 such bases. This estimate follows from the calculation of the Lusternik-Schnirelmann category of the flag variety. A better estimate is obtained for generic Banach spaces by the Morse theory.
Knots in proteins have been proposed to resist proteasomal degradation. Ample evidence associates proteasomal degradation with neurodegeneration. One interesting possibility is that indeed knotted conformers stall this machinery leading to toxicity. However, although the proteasome is known to unfold mechanically its substrates, at present there are no experimental methods to emulate this particular traction geometry. Here, we consider several dynamical models of the proteasome in which the complex is represented by an effective potential with an added pulling force. This force is meant to induce translocation of a protein or a polypeptide into the catalytic chamber. The force is either constant or applied periodically. The translocated proteins are modelled in a coarse-grained fashion. We do comparative analysis of several knotted globular proteins and the transiently knotted polyglutamine tracts of length 60 alone and fused in exon 1 of the huntingtin protein. Huntingtin is associated with Huntington disease, a well-known genetically-determined neurodegenerative disease. We show that the presence of a knot hinders and sometimes even jams translocation. We demonstrate that the probability to do so depends on the protein, the model of the proteasome, the magnitude of the pulling force, and the choice of the pulled terminus. In any case, the net effect would be a hindrance in the proteasomal degradation process in the cell. This would then yield toxicity \textit{via} two different mechanisms: one through toxic monomers compromising degradation and another by the formation of toxic oligomers. Our work paves the way to the mechanistic investigation of the mechanical unfolding of knotted structures by the proteasome and its relation to toxicity and disease.
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.
In this paper we construct an injection from the linear space of trigonometric polynomials defined on $\mathbb{T}^d$ with bounded degrees with respect to each variable to a suitable linear subspace $L^1_E\subset L^1(\mathbb{T})$. We give such a quantitative condition on $L^1_E$ that this injection is a isomorphism of a Banach spaces equipped with $L^1$ norm and the norm of the isomorphism is independent on the dimension $d$.
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).
We construct examples of uncountable compact subsets of complex numbers with the property that any Borel measure on the circle group taking values of its Fourier coefficients from this set has natural spectrum. For measures with Fourier coefficients tending to 0 we construct tho open set with this property.
We provide a new technique to prove Ornstein's non-inequalities for derivatives with some geometrical dependence on their indexes.
In this paper we proof that every Fourier multiplier on homogeneous sobolev space $\dot{W}_1^1(\mathbb{R}^d)$ is a continuous function. Our theorem is generalization of A. Bonami and S. Poornima result for Fourier multipliers, which are homogeneous functions of degree zero.