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Michael Ruzhansky

Publications and source records attributed to Michael Ruzhansky.

At least 163 records · Page 9Linked to original sources
arXiv2022

Drift diffusion equations with fractional diffusion on compact Lie groups

In this work we investigate the well-posedness for difussion equations associated to subelliptic pseudo-differential operators on compact Lie groups. The diffusion by strongly elliptic operators is considered as a special case and in particular the fractional diffusion with respect to the Laplacian. The general case is studied within the Hörmander classes associated to a sub-Riemannian structure on the group (encoded by a Hörmander system of vector fields). Applications to diffusion equations for fractional sub-Laplacians, fractional powers of more general subelliptic operators, and the corresponding quasi-geostrophic model with drift $D$ are investigated. Examples on SU(2) for diffusion problems with fractional diffusion are analysed.

math.AP↗
arXiv2022

A note on best constants for Weighted Integral Hardy inequalities on homogeneous groups

The main aim of this note is to prove sharp weighted integral Hardy inequality and conjugate integral Hardy inequality on homogeneous Lie groups with any quasi-norm for the range $1<p\leq q<\infty.$ We also calculate the precise value of sharp constants in respective inequalities, improving the result of $[19]$ in the case of homogeneous groups.

math.AP↗
arXiv2021

On a class of anharmonic oscillators II. General case

In this work we study a class of anharmonic oscillators on $\mathbb{R}^n$ corresponding to Hamiltonians of the form $A(D)+V(x)$, where $A(ξ)$ and $V(x)$ are $C^{\infty}$ functions enjoying some regularity conditions. Our class includes fractional relativistic Schrödinger operators and anharmonic oscillators with fractional potentials. By associating a Hörmander metric we obtain spectral properties in terms of Schatten-von Neumann classes for their negative powers and derive from them estimates on the rate of growth for the eigenvalues of the operators $A(D)+V(x)$. This extends the analysis in the first part of our work, where the case of polynomial $A$ and $V$ has been analysed.

math.FA↗
arXiv2021

Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups

In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold $M$: \begin{equation*} \begin{cases} u_{t}-\mathfrak{L}_{M} u=f(u), \;x\in M, \;t>0, \\u(0,x)=u_{0}(x), \;x\in M, \end{cases} \end{equation*} for $u_{0}\geq 0$, where $\mathfrak{L}_{M}$ is a sub-Laplacian of $M$. In the case when $M$ is a connected unimodular Lie group $\mathbb G$, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with $u_{0}\not\equiv 0$, blow up in finite time if and only if $1 0$, we also show that the differential inequality $$ u_{t}-\mathfrak{L}_{M} u\geq f(u) $$ does not admit any nontrivial distributional (a function $u\in L^{p}_{loc}(Q)$ which satisfies the differential inequality in $\mathcal{D}^{\prime}(Q)$) solution $u\geq 0$ in $Q:=(0,\infty)\times\mathbb G$. Furthermore, in the case when $\mathbb G$ has exponential volume growth and $f:[0,\infty)\to[0,\infty)$ is a continuous increasing function such that $f(u)\leq K_{1}u^{p}$ for some $K_{1}>0$, we prove that the Cauchy problem has a global, classical solution for $1<p<\infty$ and some positive $u_{0}\in L^{q}(\mathbb G)$ with $1\leq q<\infty$. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds $M$.

math.AP↗
arXiv2021

Analytic functional calculus and Gårding inequality on graded Lie groups with applications to diffusion equations

In this paper we study the Cauchy problem for diffusion equations associated to a class of strongly hypoelliptic pseudo-differential operators on graded Lie groups. To do so, we develop a global complex functional calculus on graded Lie groups in order to analyse the corresponding energy estimates. One of the main aspects of this complex functional calculus is that for the $(ρ,δ)$-Euclidean Hörmander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the Gårding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953's inequality due to Gårding [26].

math.AP↗
arXiv2021

The heat equation with singular potentials. II: Hypoelliptic case

In this paper we consider the heat equation with a strongly singular potential and show that it has a very weak solution. Our analysis is devoted to general hypoelliptic operators and is developed in the setting of graded Lie groups. The current work continues and extends a previous work , where the classical heat equation on $\mathbb R^n$ was considered.

math.AP↗
arXiv2021

$L^p$-$L^q$ boundedness of $(k, a)$-Fourier multipliers with applications to Nonlinear equations

The $(k,a)$-generalised Fourier transform is the unitary operator defined using the $a$-deformed Dunkl harmonic oscillator.The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k, a)$-generalised Fourier multipliers. To show the boundedness we first establish Paley inequality and Hausdorff-Young-Paley inequality for $(k, a)$-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations.

math.FA↗
arXiv2021

Liouville theorems for Kirchhoff-type hypoelliptic Partial Differential Equations and systems. I. Heisenberg group

In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.

math.AP↗
arXiv2021

Subelliptic sharp Gårding inequality on compact Lie groups

In this work we establish a subelliptic sharp Gårding inequality on compact Lie groups for pseudo-differential operators with symbols belonging to global subelliptic Hörmander classes. In order for the inequality to hold we require the global matrix-valued symbol to satisfy the suitable classical nonnegativity condition in our setting. Our result extends to $\mathscr{S}^m_{ρ,δ}(G)$-classes, $0\leq δ<ρ$, the one in [26] about the validity of the sharp Gårding inequality for the class $\mathscr{S}^m_{1,0}(G)$. We remark that the result we prove here is already new and sharp in the case of the torus.

math.AP↗
arXiv2021

Titchmarsh theorems, Hausdorff-Young-Paley inequality and $L^p$-$L^q$ boundedness of Fourier multipliers on harmonic $NA$ groups

In this paper we extend classical Titchmarsh theorems on the Fourier transform of H$\ddot{\text{o}}$lder-Lipschitz functions to the setting of harmonic $NA$ groups, which relate smoothness properties of functions to the growth and integrability of their Fourier transform. We prove a Fourier multiplier theorem for $L^2$-H$\ddot{\text{o}}$lder-Lipschitz spaces on Harmonic $NA$ groups. We also derive conditions and a characterisation of Dini-Lipschitz classes on Harmonic $NA$ groups in terms of the behaviour of their Fourier transform. Then, we shift our attention to the spherical analysis on Harmonic $NA$ group. Since the spherical analysis on these groups fits well in the setting of Jacobi analysis we prefer to work in the Jacobi setting. We prove $L^p$-$L^q$ boundedness of Fourier multipliers by extending a classical theorem of H$\ddot{\text{o}}$rmander to the Jacobi analysis setting. On the way to accomplish this classical result we prove Paley-type inequality and Hausdorff-Young-Paley inequality. We also establish $L^p$-$L^q$ boundedness of spectral multipliers of the Jacobi Laplacian.

math.FA↗
arXiv2021

$L^p$-$L^q$ Multipliers on commutative hypergroups

The main purpose of this paper is to prove Hörmander's $L^p$-$L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing Paley inequality and Hausdorff-Young-Paley inequality for commutative hypergroups. We show the $L^p$-$L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli-Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$-$L^q$ norms of the heat kernel for generalised radial Laplacian. Finally, we present applications of the obtained results to study the well-posedness of nonlinear partial differential equations.

math.FA↗
arXiv2021

Logarithmic Hardy-Rellich inequalities on Lie groups

In this paper we obtain logarithmic Hardy and Rellich inequalities on general Lie groups. In the case of graded groups, we also show their refinements using the homogeneous Sobolev norms. In fact, we derive a family of weighted logarithmic Hardy-Rellich inequalities, for which logarithmic Hardy and Rellich inequalities are special cases. As a consequence of these inequalities, we also derive a Gross type logarithmic Hardy inequality on general stratified groups. An interesting feature of such estimate is that we consider the measure which is Gaussian only on the first stratum of the group. Such choice of the measure is natural in view of the known Gross type logarithmic Sobolev inequalities on stratified groups. The obtained results are new already in the setting of the Euclidean space $\mathbb R^n.$ Finally, we also present a simple argument for getting a logarithmic Poincaré inequality, as well as the logarithmic Hardy inequality for the fractional $p$-sub-Laplacian on homogeneous groups.

math.AP↗
arXiv2021

Well-posedness of Tricomi-Gellerstedt-Keldysh-type fractional elliptic problems

In this paper Tricomi-Gellerstedt-Keldysh-type fractional elliptic equations are studied. The results on the well-posedness of fractional elliptic boundary value problems are obtained for general positive operators with discrete spectrum and for Fourier multipliers with positive symbols. As examples, we discuss results in half-cylinder, star-shaped graph, half-space and other domains.

math.AP↗
arXiv2021

Sharpness of Seeger-Sogge-Stein orders for the weak (1,1) boundedness of Fourier integral operators

Let $X$ and $Y$ be two smooth manifolds of the same dimension. It was proved by Seeger, Sogge and Stein in \cite{SSS} that the Fourier integral operators with real non-degenerate phase functions in the class $I^μ_1(X,Y;Λ),$ $μ\leq -(n-1)/2,$ are bounded from $H^1$ to $L^1.$ The sharpness of the order $-(n-1)/2,$ for any elliptic operator was also proved in \cite{SSS} and extended to other types of canonical relations in \cite{Ruzhansky1999}. That the operators in the class $I^μ_1(X,Y;Λ),$ $μ\leq -(n-1)/2,$ satisfy the weak (1,1) inequality was proved by Tao \cite{Tao:weak11}. In this note, we prove that the weak (1,1) inequality for the order $ -(n-1)/2$ is sharp for any elliptic Fourier integral operator, as well as its versions for canonical relations satisfying additional rank conditions.

math.AP↗
arXiv2021

Dixmier traces, Wodzicki residues, and determinants on compact Lie groups: the paradigm of the global quantisation

\begin{abstract} By following the paradigm of the global quantisation, instead of the analysis under changes of coordinates, in this work we establish a global analysis for the explicit computation of the Dixmier trace and the Wodzicki residue of (elliptic and subelliptic) pseudo-differential operators on compact Lie groups. The regularised determinant for the Dixmier trace is also computed. We obtain these formulae in terms of the global symbol of the corresponding operators. In particular, our approach links the Dixmier trace and Wodzicki residue to the representation theory of the group. Although we start by analysing the case of compact Lie groups, we also compute the Dixmier trace and its regularised determinant on arbitrary closed manifolds $M$, for the class of invariant pseudo-differential operators in terms of their matrix-valued symbols. This analysis includes e.g. the family of positive and elliptic pseudo-differential operators on $M$.

math.DG↗
arXiv2021

Fractional Klein-Gordon equation with singular mass. II: Hypoelliptic case

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part of this work which was devoted to the classical Euclidean Klein-Gordon equation.

math.AP↗
arXiv2021

Schatten-von Neumann classes of integral operators

In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in ${\mathbb R}^n$ of finite measure, and conditions for operators on ${\mathbb R}^n$ given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups.

math.FA↗