arXiv subjects

Michael Ruzhansky

Publications and source records attributed to Michael Ruzhansky.

At least 73 records · Page 4Linked to original sources

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

Well-posedness for a class of pseudo-differential hyperbolic equations on the torus

In this paper we establish the well-posedness of the Cauchy problem for a class of pseudo-differential hyperbolic equations on the torus. The class considered here includes a space-like fractional order Laplacians. By applying the toroidal pseudo-differential calculus we establish regularity estimates, existence and uniqueness in the scale of the standard Sobolev spaces on the torus

math.AP↗

arXiv2024

$L^p-L^q$ estimates for non-local heat and wave type equations on locally compact groups

We prove the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat and wave type equations on a locally compact separable unimodular group $G$ by using an integro-differential operator in time and any positive left invariant operator (maybe unbounded) on $G$. We complement our studies by giving asymptotic time estimates for the solutions, which in some cases are sharp.

math.AP↗

arXiv2024

Bilinear Hardy inequalities on metric measure spaces

In this paper, we discuss the Hardy inequality with bilinear operators on general metric measure spaces. We give the characterization of weights for the bilinear Hardy inequality to hold on general metric measure spaces having polar decompositions. We also provide several examples of the results, finding conditions on the weights for integral Hardy inequalities on homogeneous Lie groups, as well as on hyperbolic spaces and more generally on Cartan-Hadamard manifolds.

math.FA↗

arXiv2024

Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces

Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $ν\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-γ}(\mathbb G), γ>0$. In the framework of Sobolev spaces of negative order, we prove that $p_{\text{Crit}}(Q, γ, ν) :=1+\frac{2ν}{Q+2γ}$ is the new critical exponent for $γ\in (0, \frac{Q}{2})$. More precisely, we show the global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{Crit}}(Q, γ, ν) $ in the energy evolution space $ \mathcal{C}\left([0, T], H^{s}(\mathbb{G})\right), s\in (0, 1]$. Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for $1<p<p_{\text{Crit}}(Q, γ, ν)$. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical cases. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups.

math.AP↗

arXiv2024

Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^n$ with power type nonlinearity $|u|^p$ and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space $\dot H^{-γ}_{\mathcal{L}}(\mathbb{H}^n), γ>0$, on $\mathbb{H}^n$. In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent $p_{\text{crit}}(Q, γ)=1+\frac{4}{Q+2γ},$ for some $γ\in (0, \frac{Q}{2})$, where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. More precisely, we establish a global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{crit}}(Q, γ)$ in the energy evolution space; a finite time blow-up of weak solutions for $1<p<p_{\text{crit}}(Q, γ)$ under certain conditions on the initial data by using the test function method. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.

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

Functional inequalities on symmetric spaces of noncompact type and applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein-Weiss inequality, also known as a weighted Hardy-Littlewood-Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein-Weiss inequality, we deduce Hardy-Sobolev, Hardy-Littlewood-Sobolev, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace-Beltrami operator on symmetric spaces.

math.AP↗

arXiv2024

A Direct Method of Moving Planes for Logarithmic Schrödinger Operator

In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schr$\ddot{\text{o}}$dinger operator $(\mathcal{I}-Δ)^{\log}$ corresponding to the logarithmic symbol $\log(1 + |ξ|^2)$, which is a singular integral operator given by $$(\mathcal{I}-Δ)^{\log}u(x) =c_{N}P.V.\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N}}κ(|x-y|)dy,$$ where $c_{N}=π^{-\frac{N}{2}}$, $κ(r)=2^{1-\frac{N}{2}}r^{\frac{N}{2}}\mathcal{K}_{\frac{N}{2}}(r)$ and $\mathcal{K}_ν$ is the modified Bessel function of second kind with index $ν$. The proof hinges on a direct method of moving planes for the logarithmic Schr$\ddot{\text{o}}$dinger operator.

math.AP↗

arXiv2024

Vector Valued Gårding Inequality for pseudo-differential operators on compact homogeneous manifolds

We prove sufficient conditions in order to obtain a sharp Gårding inequality for pseudo-differential operators acting on vector-valued functions on compact Lie groups. As a consequence, we obtain a sharp Gårding inequality for compact homogeneous vector bundles and compact homogeneous manifolds. The sharp Gårding inequality is the strongest lower bound estimate known to hold for systems on $\mathbb{R}^n$, and the aim of this paper is to extend this property to systems on compact Lie groups and compact homogeneous manifolds. Our results extend previous works by Lax and Nirenberg [Comm. Pure Appl. Math., Vol. 8, 129-209, (1966)], and by Ruzhansky and Turunen [J. Funct. Anal., Vol. 267, 144-172, (2011)]. As an application, we establish existence and uniqueness of solution to a class of systems of initial value problems of pseudo-differential equations on compact Lie groups and compact homogeneous manifolds.

math.AP↗

arXiv2024

$L^p$-bounds in Safarov pseudo-differential calculus on manifolds with bounded geometry

Given a smooth complete Riemannian manifold with bounded geometry $(M,g)$ and a linear connection $\nabla$ on it (not necessarily a metric one), we prove the $L^p$-boundedness of operators belonging to the global pseudo-differential classes $Ψ_{ρ, δ}^m\left(Ω^κ, \nabla, τ\right)$ constructed by Safarov. Our result recovers classical Fefferman's theorem, and extends it to the following two situations: $ρ>1/3$ and $\nabla$ symmetric; and $\nabla$ flat with any values of $ρ$ and $δ$. Moreover, as a consequence of our main result, we obtain boundedness on Sobolev and Besov spaces and some $L^p-L^q$ boundedness. Different examples and applications are presented.

math.AP↗

arXiv2024

Pseudo-differential operators on Homogeneous vector bundles over compact homogeneous manifolds

In this work, we introduce a global theory of subelliptic pseudo-differential operators on arbitrary homogeneous vector bundles over orientable compact homogeneous manifolds. We will show that a global pseudo-differential calculus can be associated to the operators acting on any pair of homogeneous vector-bundles with base space $M,$ if the compact Lie group $G$ that acts on $M=G/K$ is endowed with a (Riemannian or) sub-Riemannian structure. This is always possible if we choose on $G$ a sub-Laplacian associated to a Hörmander system of vector-fields or we fix the Laplace-Beltrami operator on $G$. We begin with developing a global subelliptic symbolic calculus for vector-valued pseudo-differential operators on $G$ and then, we show that this vector-valued calculus induces a pseudo-differential calculus on homogeneous vector bundles, which, among other things, is stable under the action of the complex functional calculus. We prove global versions of the Calderón-Vaillancourt theorem, Fefferman theorem and also, of the Gårding inequality. We present applications of the obtained Gårding inequality to the wellposedness of evolution problems. We characterise the classes of pseudo-differential operators on homogeneous vector bundles in the sense of Hörmander (which are defined by using local coordinate systems) in terms of their global symbols. Finally, using this formalism, we compute the global symbol of the exterior derivative, its adjoint, and the symbol of the Dirac operator on the vector bundle of differential forms. We hope that this work will provide a solid foundation for further research using the global quantisation of operators on (vector-bundles over) compact homogeneous manifolds.

math.AP↗

arXiv2024

Heat equation for Sturm-Liouville operator with singular propagation and potential

This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm-Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm-Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

math.AP↗

arXiv2024

Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups

In this paper we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy-Littlewood-Sobolev, Caffarelli-Kohn-Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles.

math.FA↗

arXiv2024

Higher-order evolution inequalities with Hardy potential on the Korányi ball

We consider a higher order in (time) semilinear evolution inequality posed on the Korányi ball under an inhomogeneous Dirichlet-type boundary condition. The problem involves an inverse-square potential $λ/|ξ|_\mathbb{H}^2$, where $λ\geq -(Q-2)^2/4$ and a general weight function $V$ depending on the space variable in front of the power nonlinearity. We first establish a general nonexistence result for the considered problem. Next, in the special case $V(ξ):=|ξ|_\mathbb{H}^a$, $a\in \mathbb{R}$, we prove the sharpness of our nonexistence result and show that the problem admits three different critical behaviors according to the value of the parameter $λ$.

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

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↗