Michael Ruzhansky
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View article: Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups
Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups Open
In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic e…
View article: Functional calculus for Safarov pseudo-differential operators
Functional calculus for Safarov pseudo-differential operators Open
Given a smooth, closed Riemannian manifold $(M,g)$ equipped with a linear connection $\nabla$ (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes $Ψ_{ρ,…
View article: Subelliptic Pseudo-differential Operators and Fourier Integral Operators on Compact Lie Groups
Subelliptic Pseudo-differential Operators and Fourier Integral Operators on Compact Lie Groups Open
In this memoir we extend the theory of global pseudo-differential operators to the setting of arbitrary sub-Riemannian structures on a compact Lie group. More precisely, given a compact Lie group $G$, and the sub-Laplacian $\mathcal{L}$ as…
View article: A Closed-form Approximation for Impulse Response of Fractionally Damped Oscillators
A Closed-form Approximation for Impulse Response of Fractionally Damped Oscillators Open
We consider a fractionally damped oscillator, where the damping term is expressed by the Caputo fractional derivative of order $β\in (0,1).$ The impulse response of this oscillator can be expressed in terms of the bivariate Mittag-Leffler …
View article: Schatten-von Neumann classes of tensors of invariant operators
Schatten-von Neumann classes of tensors of invariant operators Open
In this work we study Schatten-von Neumann classes of tensor products of invariant operators on Hilbert spaces. In the first part we first deduce some spectral properties for tensors of anharmonic oscillators thanks to the knowledge on cor…
View article: Anharmonic semigroups and applications to global well-posedness of nonlinear heat equations
Anharmonic semigroups and applications to global well-posedness of nonlinear heat equations Open
In this work we consider the semigroup e−tAk,ℓγ for γ > 0 associated to an anharmonic oscillator of the form Ak,ℓ=(−Δ)ℓ+|x|2k where k, ℓ are integers ≥1. By introducing a suitable Hörmander metric on the phase-space we analyze the semig…
View article: Schatten-von Neumann classes of tensors of invariant operators
Schatten-von Neumann classes of tensors of invariant operators Open
In this work we study Schatten-von Neumann classes of tensor products of invariant operators on Hilbert spaces. In the first part we first deduce some spectral properties for tensors of anharmonic oscillators thanks to the knowledge on cor…
View article: Very weak solutions of the heat equation with anisotropically singular time-dependent diffusivity
Very weak solutions of the heat equation with anisotropically singular time-dependent diffusivity Open
We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very …
View article: $$L^p$$-$$L^q$$ boundedness of continuous linear operators on smooth manifolds
$$L^p$$-$$L^q$$ boundedness of continuous linear operators on smooth manifolds Open
In this paper, we study the boundedness of global continuous linear operators on smooth manifolds. Using the notion of a global symbol, we extend a classical condition of Hörmander type to guarantee the $$L^p$$ - $$L^q$$ -boundedne…
View article: Expansion of traces and Dixmier traceability for global pseudo-differential operators on manifolds with boundary
Expansion of traces and Dixmier traceability for global pseudo-differential operators on manifolds with boundary Open
Given a smooth manifold M (with or without boundary), in this paper we study the regularisation of traces for the global pseudo-differential calculus in the context of non-harmonic analysis. Indeed, using the global pseudo-differential cal…
View article: PointExplainer: Towards Transparent Parkinson's Disease Diagnosis
PointExplainer: Towards Transparent Parkinson's Disease Diagnosis Open
Deep neural networks have shown potential in analyzing digitized hand-drawn signals for early diagnosis of Parkinson's disease. However, the lack of clear interpretability in existing diagnostic methods presents a challenge to clinical tru…
View article: Lp$L^p$‐bounds in Safarov pseudo‐differential calculus on manifolds with bounded geometry
Lp$L^p$‐bounds in Safarov pseudo‐differential calculus on manifolds with bounded geometry Open
Given a smooth complete Riemannian manifold with bounded geometry and a linear connection on it (not necessarily a metric one), we prove the ‐boundedness of operators belonging to the global pseudo‐differential classes constructed by Safar…
View article: A note on $$L^p$$-$$L^q$$ boundedness of Fourier multipliers on noncommutative spaces
A note on $$L^p$$-$$L^q$$ boundedness of Fourier multipliers on noncommutative spaces Open
In this work, we study Fourier multipliers on noncommutative spaces. In particular, we show a simple proof of $$L^p$$ - $$L^q$$ estimate of Fourier multipliers on general noncommutative spaces associated with semifinite von Neumann…
View article: Hardy and Rellich Inequalities with Bessel Pairs
Hardy and Rellich Inequalities with Bessel Pairs Open
In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $\Omega \subset \mathbb{R}^n$ in order to have the following Hardy inequality: \begin{equation*} \int_{\Omega} W(x)…
View article: Schatten classes on noncommutative tori: Kernel conditions
Schatten classes on noncommutative tori: Kernel conditions Open
In this note, we give criteria on noncommutative integral kernels ensuring that integral operators on quantum torus belong to Schatten classes. With the engagement of a noncommutative Schwartz’ kernel theorem on the quantum torus, a specif…
View article: Hormander-Mikhlin type theorem on non-commutative spaces
Hormander-Mikhlin type theorem on non-commutative spaces Open
In this paper, we introduce a Fourier-type formalism on non-commutative spaces. As a result, we obtain two versions of Hormander-Mikhlin Lp-multiplier theorem: on locally compact Kac groups and on semi-finite von Neumann algebras, respecti…
View article: Hardy-Hilbert type inequalities on homogeneous groups-An introduction and generalization to the kernel case
Hardy-Hilbert type inequalities on homogeneous groups-An introduction and generalization to the kernel case Open
There is a lot of information available concerning Hardy-Hilbert type inequalities in one or more dimensions. In this paper we introduce the development of such inequalities on homogeneous groups. Moreover, we point out a unification of se…
View article: Zero modes and Dirac-(logarithmic) Sobolev-type inequalities
Zero modes and Dirac-(logarithmic) Sobolev-type inequalities Open
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 …
View article: Singular Klein-Gordon equation on a bounded domain
Singular Klein-Gordon equation on a bounded domain Open
In this paper, we consider the wave equation for the Laplace operator with potential, initial data, and nonhomogeneous Dirichlet boundary condition. We establish a weak solution by using traces and extension domains. We also establish the …
View article: Besov and Triebel-Lizorkin spaces on homogeneous groups
Besov and Triebel-Lizorkin spaces on homogeneous groups Open
This paper develops a theory of Besov spaces $\dot{\mathbf{B}}^σ_{p,q} (N)$ and Triebel-Lizorkin spaces $\dot{\mathbf{F}}^σ_{p,q} (N)$ on an arbitrary homogeneous group $N$ for the full range of parameters $p, q \in (0, \infty]$ and $σ\in …
View article: Levin-Cochran-Lee inequalities and best constants on homogeneous groups
Levin-Cochran-Lee inequalities and best constants on homogeneous groups Open
In this paper, we apply a direct method instead of a limit approach, for proving the Levin-Cochran-Lee inequalities. First, we state and prove Levin-Cochran-Lee type inequalities on a homogeneous group $\mathbb{G}$ with parameters $0
View article: Hardy-Hilbert type inequalities on homogeneous groups-An introduction and generalization to the kernel case
Hardy-Hilbert type inequalities on homogeneous groups-An introduction and generalization to the kernel case Open
There is a lot of information available concerning Hardy-Hilbert type inequalities in one or more dimensions. In this paper we introduce the development of such inequalities on homogeneous groups. Moreover, we point out a unification of se…
View article: The Prabhakar fractional q-integral and q-differential operators, and their properties
The Prabhakar fractional q-integral and q-differential operators, and their properties Open
In this paper, we have introduced the Prabhakar fractional q-integral and q-differential operators. We first study the semigroup property of the Prabhakar fractional q-integral operator, which allowed us to introduce the corresponding q-di…
View article: Parabolic equations with concave non-linearity
Parabolic equations with concave non-linearity Open
In this paper we prove the existence and uniqueness of positive mild solutions for the semilinear parabolic equations of the form $u_t+\mathcal{L}u=f+h\cdot G(u)$, where $h$ is a positive function and $G$ a positive concave function (for e…
View article: Titchmarsh theorems on Damek–Ricci spaces via moduli of continuity of higher order
Titchmarsh theorems on Damek–Ricci spaces via moduli of continuity of higher order Open
A classical theorem of Titchmarsh relates the [Formula: see text]-Lipschitz functions and decay of the Fourier transform of the functions. In this note, we prove the Titchmarsh theorem for Damek–Ricci space (also known as harmonic [Formula…
View article: Global well‐posedness of space‐time fractional diffusion equation with Rockland operator on graded lie group
Global well‐posedness of space‐time fractional diffusion equation with Rockland operator on graded lie group Open
In this article, we examine the general space‐time fractional diffusion equation for left‐invariant hypoelliptic homogeneous operators on graded Lie groups. Our study covers important examples such as the time fractional diffusion equation…