arXiv subjects

Marianna Chatzakou

Publications and source records attributed to Marianna Chatzakou.

At least 19 recordsLinked to original sources

arXiv2026

A short note on the dimension of the CC-geodesic Kakeya sets in the first Heisenberg group

We study CC-geodesic Kakeya sets in the first Heisenberg group, namely Borel sets $E$ such that, for every unit-speed CC-geodesic segment of length $1$ issuing from the identity, some left translate of the segment is contained in $E$. The natural analogue of the Kakeya conjecture would predict full Heisenberg Hausdorff dimension 4 for such sets. We show that this prediction fails: the sharp lower bound for their Heisenberg Hausdorff dimension is 3, and it remains sharp even among compact CC-geodesic Kakeya sets. By adjoining a Lebesgue-null set of full Heisenberg Hausdorff dimension, we obtain a CC-geodesic Kakeya set of full Heisenberg Hausdorff dimension 4 and zero Lebesgue measure. Finally, when one prescribes only geodesic segments with one fixed nonzero curvature parameter $κ$, rather than segments of all curvatures, the condition is weaker. For every $κ\in(0,2π]$, we construct a compact curvature-$κ$ Kakeya set of zero Lebesgue measure whose Euclidean and Heisenberg Hausdorff dimensions are both equal to $1$.

math.CA↗

arXiv2026

Generic simplicity for self-adjoint operators under bounded potential perturbations

We are interested in the generic simplicity of the spectrum of self-adjoint operators under bounded potential perturbations. More precisely, given a semibounded self-adjoint operator with compact resolvent and a suitable space of real-valued bounded perturbations, we study whether all eigenvalues of the perturbed operator are simple for a generic choice of the potential. In the first part of this paper we prove an abstract criterion which ensures that the set of perturbations giving only simple eigenvalues is residual. In the second part, we apply this criterion to several geometric and analytic settings, including sub-Laplacians and maximally hypoelliptic operators on compact manifolds, Laplacians on bounded domains with different boundary conditions, and Schrödinger-type operators on non-compact spaces.

math.SP↗

arXiv2026

Geometric Hardy inequalities on the Heisenberg groups via convexity

We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group ${\mathbb{H}}^n$, $n\geq 1$. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is then implemented for the distance defined by the gauge quasi-norm related to the fundamental solution of the horizontal Laplacian when the domain is a half-space or a convex polytope. Finally it is implemented for the Carnot-Carathéodory distance on half-spaces and arbitrary bounded convex domains of ${\mathbb{H}}^n$. In all cases the constant $((p-1)/p)^p$ is obtained. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak $H$-concavity of the Euclidean distance to the boundary, thus obtaining a proof of the $L^p$-Hardy inequality on convex domains.

math.AP↗

arXiv2026

Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators

In this work we give a sufficient condition under which the global Poincaré inequality on Carnot groups holds true for a large family of probability measures absolutely continuous with respect to the Lebesgue measure. The density of such probability measure is given in terms of homogeneous quasi-norm on the group. We provide examples to which our condition applies including the most known families of Carnot groups. This, in particular, allows to extend the results in the previous work [CFZ21]. A consequence of our result is that the associated Schrödinger operators have a spectral gap.

math.FA↗

arXiv2025

Fujita exponent for heat equation with Hörmander vector fields

In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on $\Rn$ satisfying Hörmander's rank condition with a non-linearity of the form $f(u)$, where $f$ is a suitable function and $u$ is the solution. In particular, when $f(u)=u^p$, we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type $φ(t)f(u)$.

math.AP↗

arXiv2025

Zero modes and Dirac-(logarithmic) Sobolev-type inequalities

We study the decay rate of the zero modes of the Dirac operator with a matrix-valued potential that is considered here without any regularity assumptions, compared to the existing literature. For the Dirac operator and for Clifford-valued functions we prove the $L^p$-$L^2$ Dirac Sobolev inequality with explicit constant, as well as the $L^p$-$L^q$ Dirac-Sobolev inequalities. We prove its logarithmic counterpart for $q=2$, extending it to its Gaussian version of Gross, as well as show Nash and Poincaré inequalities in this setting, with explicit values for constants.

math.AP↗

arXiv2024

Nearness and solvability of non-invariant equations on stratified groups

We prove the well-posedness of the differential equation $Au=f$ in the setting of a stratified group $\mathbb{G}$ when the considered second-order differential operator $A$ can be non-invariant and non-linear. Our approach follows the Campanato theory of nearness of operators, allowing one to treat equations with only bounded coefficients, without any regularity assumptions. Our analysis becomes explicit in the particular case of the Heisenberg group $\mathbb{H}^n$ of any dimension and on the Euclidean case $\mathbb{R}^n$, where in the latter case our results also extend the known results by treating the unbounded domain setting.

math.AP↗

arXiv2024

On global solutions of heat equations with time-dependent nonlinearities on unimodular Lie groups

In this work, we study the global well-posedeness of the heat equation with variable time-dependent nonlinearity of the form $φ(t)f(u)$ on unimodular Lie groups when the differential operator arises as the sum of squares of Hörmander vector fields. For general unimodular Lie groups, we derive the necessary conditions for the nonexistence of global positive solutions. This gives different conditions in the cases of compact, polynomial, and exponential volume growth groups. In the case of the Heisenberg groups $\mathbb{H}^{n}$, we also derive sufficient conditions, which coincide with the necessary ones in the case of $\mathbb{H}^{1}$ (and this is also true for $\mathbb{R}^{n}$). In particular, in the case of the Heisenberg group $\mathbb{H}^{1}$ we obtain the necessary and sufficient conditions under which the aforesaid initial value problem with variable nonlinearity has a global positive solution.

math.AP↗

arXiv2024

Logarithmic Sobolev-type inequalities on Lie groups

In this paper we show a number of logarithmic inequalities on several classes of Lie groups: log-Sobolev inequalities on general Lie groups, log-Sobolev (weighted and unweighted), log-Gagliardo-Nirenberg and log-Caffarelli-Kohn-Nirenberg inequalities on graded Lie groups. Furthermore, on stratified groups, we show that one of the obtained inequalities is equivalent to a Gross-type log-Sobolev inequality with the horizontal gradient. As a result, we obtain the Gross log-Sobolev inequality on general stratified groups but, {\bf very interestingly}, with the Gaussian measure on the first stratum of the group. Moreover, our methods also yield weighted versions of the Gross log-Sobolev inequality. In particular, we also obtain new weighted Gross-type log-Sobolev inequalities on $\mathbb R^n$ for arbitrary choices of homogeneous quasi-norms. As another consequence we derive the Nash inequalities on graded groups and an example application to the decay rate for the heat equations for sub-Laplacians on stratified groups. We also obtain weighted versions of log-Sobolev and Nash inequalities for general Lie groups.

math.AP↗

arXiv2024

Sharp upper bound for anisotropic Rényi entropy and Heisenberg uncertainty principle

In this paper, we prove the anisotropic Shannon inequality for the Renyi entropy with the best constant on Folland-Stein homogeneous Lie groups. As a consequence, we also prove the optimal Shannon inequality in the same setting. Using a logarithmic Sobolev inequality in the setting of stratified groups, we prove a Heisenberg-type uncertainty principle in the latter setting.

math.FA↗

arXiv2024

Geometric logarithmic-Hardy and Hardy-Poincaré inequalities on stratified groups

We develop a unified strategy to obtain the geometric logarithmic Hardy inequality on any open set M of a stratified group, provided the validity of the Hardy inequality in this setting, where the so-called "weight" is regarded to be any measurable non-negative function on M . Provided the legitimacy of the latter for some open set and for some weight, we also show an inequality that is an extension of the "generalised Poincaré inequality" introduced by Beckner with the addition of a weight, and this is referred to as the "geometric Hardy-Poincaré inequality". The aforesaid inequalities become explicit in the case where M is the half-space of the group and the weight is the distance function from the boundary, and in the case where M is just the whole group (or any open set in the group), in which case the weight is the "horizontal norm" on the first stratum of the group. For the second case, the semi-Gaussian analogue of the derived inequalities is proved, when the Gaussian measure is regarded with respect to the first stratum of the group. Applying our results to the case where the group is just the (abelian) Euclidean space we generalise the classical probabilistic Poincaré inequality by adding weights.

math.FA↗

arXiv2024

Anharmonic semigroups and applications to global well-posedness of nonlinear heat equations

In this work we consider the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ for $γ>0$ associated to an anharmonic oscillator of the form $ \mathcal{A}_{k,\,\ell}=(-Δ)^{\ell}+|x|^{2k}$ where $k,\ell$ are integers $\geq 1$. By introducing a suitable Hörmander metric on the phase-space we analyse the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ within the framework of Hörmander $S(M,g)$ classes and obtain mapping properties in the scale of modulation spaces $M^{p,q},\, 0<p,q\leq \infty,$ with respect to an anharmonic modulation weight. As an application, we apply the obtained bounds to establish the well-posedness for the nonlinear heat equation associated with $\mathcal{A}_{k,\,\ell}^γ$. It is worth noting that the results presented in this paper are novel, even in the case where $γ=1.$

math.AP↗

arXiv2023

LSTM-CNN: An efficient diagnostic network for Parkinson's disease utilizing dynamic handwriting analysis

Background and objectives: Dynamic handwriting analysis, due to its non-invasive and readily accessible nature, has recently emerged as a vital adjunctive method for the early diagnosis of Parkinson's disease. In this study, we design a compact and efficient network architecture to analyse the distinctive handwriting patterns of patients' dynamic handwriting signals, thereby providing an objective identification for the Parkinson's disease diagnosis. Methods: The proposed network is based on a hybrid deep learning approach that fully leverages the advantages of both long short-term memory (LSTM) and convolutional neural networks (CNNs). Specifically, the LSTM block is adopted to extract the time-varying features, while the CNN-based block is implemented using one-dimensional convolution for low computational cost. Moreover, the hybrid model architecture is continuously refined under ablation studies for superior performance. Finally, we evaluate the proposed method with its generalization under a five-fold cross-validation, which validates its efficiency and robustness. Results: The proposed network demonstrates its versatility by achieving impressive classification accuracies on both our new DraWritePD dataset ($96.2\%$) and the well-established PaHaW dataset ($90.7\%$). Moreover, the network architecture also stands out for its excellent lightweight design, occupying a mere $0.084$M of parameters, with a total of only $0.59$M floating-point operations. It also exhibits near real-time CPU inference performance, with inference times ranging from $0.106$ to $0.220$s. Conclusions: We present a series of experiments with extensive analysis, which systematically demonstrate the effectiveness and efficiency of the proposed hybrid neural network in extracting distinctive handwriting patterns for precise diagnosis of Parkinson's disease.

cs.AI↗

arXiv2023

Logarithmic Sobolev, Hardy and Poincaré inequalities on the Heisenberg group

In this paper we first prove a number of important inequalities with explicit constants in the setting of the Heisenberg group. This includes the fractional and integer Sobolev, Gagliardo-Nirenberg, (weighted) Hardy-Sobolev, Nash inequalities, and their logarithmic versions. In the case of the first order Sobolev inequality, our constant recovers the sharp constant of Jerison and Lee. Remarkably, we also establish the analogue of the Gross inequality with a semi-probability measure on the Heisenberg group that allows -- as it happens in the Euclidean setting -- an extension to infinite dimensions, and particularly can be regarded as an inequality on the infinite dimensional $\mathbb{H}^{\infty}$. Finally, we prove the so-called generalised Poincaré inequality on the Heisenberg group both with respect to the aforementioned semi-probability measure and the Haar measure, also with explicit constants.

math.AP↗

arXiv2023

Comparison of One- Two- and Three- Dimensional CNN models for Drawing-Test-Based Diagnostics of the Parkinson's Disease

Subject: In this article, convolutional networks of one, two, and three dimensions are compared with respect to their ability to distinguish between the drawing tests produced by Parkinson's disease patients and healthy control subjects. Motivation: The application of deep learning techniques for the analysis of drawing tests to support the diagnosis of Parkinson's disease has become a growing trend in the area of Artificial Intelligence. Method: The dynamic features of the handwriting signal are embedded in the static test data to generate one-dimensional time series, two-dimensional RGB images and three-dimensional voxelized point clouds, and then one-, two-, and three-dimensional CNN can be used to automatically extract features for effective diagnosis. Novelty: While there are many results that describe the application of two-dimensional convolutional models to the problem, to the best knowledge of the authors, there are no results based on the application of three-dimensional models and very few using one-dimensional models. Main result: The accuracy of the one-, two- and three-dimensional CNN models was 62.50%, 77.78% and 83.34% in the DraWritePD dataset (acquired by the authors) and 73.33%, 80.00% and 86.67% in the PaHaW dataset (well known from the literature), respectively. For these two data sets, the proposed three-dimensional convolutional classification method exhibits the best diagnostic performance.

math.AP↗

arXiv2023

$q$-Poincaré inequalities on Carnot Groups with filiform type Lie algebra

In this paper, we prove (global) $q$-Poincaré inequalities for probability measures on nilpotent Lie groups with filiform Lie algebra of any length. The probability measures under consideration have a density with respect to the Haar measure given as a function of a suitable homogeneous norm.

math.FA↗

arXiv2023

Semi-classical pseudo-differential operators on $\hbar\mathbb{Z}^n$ and applications

In this paper we consider the semiclassical version of pseudo-differential operators on the lattice space $\hbar \mathbb{Z}^n$. The current work is an extension of a previous work and agrees with it in the limit of the parameter $\hbar \rightarrow 1$. The various representations of the operators will be studied as well as the composition, transpose, adjoint and the link between ellipticity and parametrix of operators. We also give the conditions for the $\ell^p(\hbar \mathbb{Z}^n)$, weighted $\ell^2(\hbar \mathbb{Z}^n)$ boundedness and $\ell^p(\hbar \mathbb{Z}^n)$ compactness of operators. We investigate the relation between the classical and semi-classical quantization and employ its applications to Schatten-Von Neumann classes on $\ell^2( \hbar \mathbb{Z}^n)$. We establish Gårding and sharp Gårding inequalities, with an application to the well-posedness of parabolic equations on the lattice $\hbar \mathbb{Z}^n$. Finally we verify that in the limiting case where $\hbar \rightarrow 0$ the semi-classical calculus of pseudo-differential operators recovers the classical Euclidean calculus, but with a twist.

math.AP↗

arXiv2023

Discrete Heat Equation with irregular thermal conductivity and tempered distributional data

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb{Z}^n$. We establish the well-posedeness of such Cauchy equations in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell^{2}_{s}(\hbar \mathbb{Z}^n)$, $s \in \mathbb{R}$, which is enough to prove the well-posedness in the space of tempered distributions $\mathcal{S}'(\hbar \mathbb{Z}^n)$. Notably, when $s=0$, we show that for $\hbar \rightarrow 0$, the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb{R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb{Z}^n$.

math.AP↗