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Michael Ruzhansky

Publications and source records attributed to Michael Ruzhansky.

At least 271 records · Page 15Linked to original sources
arXiv2017

Wave equation for operators with discrete spectrum and irregular propagation speed

Given a Hilbert space, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with discrete non-negative spectrum acting on it. We consider the cases when the time-dependent propagation speed is regular, Hölder, and distributional. We also consider cases when it it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of "very weak solutions" to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic oscillator and the Landau Hamiltonian on $\mathbb R^n$, uniformly elliptic operators of different orders on domains, Hörmander's sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.

math.AP↗
arXiv2017

Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I

In this paper we prove the Hardy inequalities for the quadratic form of the Laplacian with the Landau Hamiltonian magnetic field. Moreover, we obtain Poincaré type inequality and inequalities with more general families of weights, all with estimates for the remainder terms of these inequalities. Furthermore, we establish weighted Hardy inequalities for the quadratic form of the magnetic Baouendi-Grushin operator for the magnetic field of Aharonov-Bohm type. For these, we show refinements of the known Hardy inequalities for the Baouendi-Grushin operator involving radial derivatives in some of the variables. The corresponding uncertainty type principles are also obtained.

math.FA↗
arXiv2017

Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations

In this paper the dependence of the best constants in Sobolev and Gagliardo-Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of $\mathbb R^n$, Heisenberg, and more general stratified Lie groups. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The orders of these equations can be high depending on the Sobolev space order in the Sobolev or Gagliardo-Nirenberg inequalities, or may be fractional. Applications are obtained also to equations with lower order terms given by different hypoelliptic operators. Already in the case of $\mathbb R^n$, the obtained results extend the classical relations by Weinstein to a wide range of nonlinear elliptic equations of high orders with elliptic low order terms and a wide range of interpolation inequalities of Gagliardo-Nirenberg type. However, the proofs are different from those ivy Weinstein because of the impossibility of using the rearrangement inequalities already in the setting of the Heisenberg group. The considered class of graded groups is the most general class of nilpotent Lie groups where one can still consider hypoelliptic homogeneous invariant differential operators and the corresponding subelliptic differential equations.

math.AP↗
arXiv2017

$L^p$-$L^q$ multipliers on locally compact groups

In this paper we discuss the $L^p$-$L^q$ boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups $G$ for the range $1<p\leq q<\infty$. We prove a Lizorkin type multiplier theorem for $1<p\leq q<\infty$, and then refine it as a Hörmander type multiplier theorem for $1<p\leq 2\leq q<\infty$. In the process, we establish versions of Paley and Hausdorff-Young-Paley inequalities on general locally compact separable unimodular groups. As a consequence of the Hörmander type multiplier theorem we derive a spectral multiplier theorem on general locally compact separable unimodular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the $L^p$-$L^q$ norms of the heat kernels for general positive unbounded invariant operators on $G$. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group. We show that our results imply the known results for $L^p$-$L^q$ multipliers such as Hörmander's Fourier multiplier theorem on $\mathbb{R}^{n}$ or known results for Fourier multipliers on compact Lie groups. The new approach developed in this paper relies on the analysis in the group von Neumann algebra for the derivation of the desired multiplier theorems.

math.RT↗
arXiv2017

Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups

In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like nonlinearities. We also obtain similar well-posedness results for the wave equations for Rockland operators on general graded Lie groups. In particular, this includes higher order operators on $\mathbb R^n$ and on the Heisenberg group, such as powers of the Laplacian or of the sub-Laplacian. In addition, we establish a new family of Gagliardo-Nirenberg inequalities on graded Lie groups that play a crucial role in the proof but which are also of interest on their own: if $G$ is a graded Lie group of homogeneous dimension $Q$ and $a>0$, $1<r<\frac{Q}{a},$ and $1\leq p\leq q\leq \frac{rQ}{Q-ar},$ then we have the following Gagliardo-Nirenberg type inequality $$ \|u\|_{L^{q}(G)}\lesssim \|u\|_{\dot{L}_{a}^{r}(G)}^{s} \|u\|_{L^{p}(G)}^{1-s} $$ for $s=\left(\frac1p-\frac1q\right) \left(\frac{a}Q+\frac1p-\frac1r\right)^{-1}\in [0,1]$ provided that $\frac{a}Q+\frac1p-\frac1r\not=0$, where $\dot{L}_{a}^{r}$ is the homogeneous Sobolev space of order $a$ over $L^r$. If $\frac{a}Q+\frac1p-\frac1r=0$, we have $p=q=\frac{rQ}{Q-ar}$, and then the above inequality holds for any $0\leq s\leq 1$.

math.AP↗
arXiv2017

On Green functions for Dirichlet sub-Laplacians on H-type groups

We construct Green functions of Dirichlet boundary value problems for sub-Laplacians on certain unbounded domains of a prototype Heisenberg-type group (prototype H-type group, in short). We also present solutions in an explicit form of the Dirichlet problem for the sub-Laplacian with non-zero boundary datum on half-space, quadrant-space, etc, as well as in a strip.

math.AP↗
arXiv2017

Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for $L^{p}$-weighted Hardy inequalities

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for $1<p,q<\infty$, $0<r<\infty$ with $p+q\geq r$, $δ\in[0,1]\cap\left[\frac{r-q}{r},\frac{p}{r}\right]$ with $\frac{δr}{p}+\frac{(1-δ)r}{q}=1$ and $a$, $b$, $c\in\mathbb{R}$ with $c=δ(a-1)+b(1-δ)$, and for all functions $f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\})$ we have $$ \||x|^{c}f\|_{L^{r}(\mathbb{R}^{n})} \leq \left|\frac{p}{n-p(1-a)}\right|^δ \left\||x|^{a}\nabla f\right\|^δ_{L^{p}(\mathbb{R}^{n})} \left\||x|^{b}f\right\|^{1-δ}_{L^{q}(\mathbb{R}^{n})} $$ for $n\neq p(1-a)$, where the constant $\left|\frac{p}{n-p(1-a)}\right|^δ$ is sharp for $p=q$ with $a-b=1$ or $p\neq q$ with $p(1-a)+bq\neq0$. In the critical case $n=p(1-a)$ we have $$ \left\||x|^{c}f\right\|_{L^{r}(\mathbb{R}^{n})} \leq p^δ \left\||x|^{a}\log|x|\nabla f\right\|^δ_{L^{p}(\mathbb{R}^{n})} \left\||x|^{b}f\right\|^{1-δ}_{L^{q}(\mathbb{R}^{n})}. $$ Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for $L^{p}$-weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of $\mathbb{R}^{n}$. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of $L^{p}$-weighted Hardy inequalities involving a distance and stability estimates. We also establish sharp Hardy type inequalities in $L^{p}$, $1<p<\infty$, with superweights, i.e. with the weights of the form $\frac{(a+b|x|^α)^{\fracβ{p}}}{|x|^{m}}$ allowing for different choices of $α$ and $β$.

math.FA↗
arXiv2017

Titchmarsh theorems for Fourier transforms of Hölder-Lipschitz functions on compact homogeneous manifolds

In this paper we extend classical Titchmarsh theorems on the Fourier transform of Hölder-Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for $L^2$-Hölder-Lipschitz spaces on compact Lie groups. We also derive conditions and a characterisation for Dini-Lipschitz classes on compact homogeneous manifolds in terms of the behaviour of their Fourier coefficients.

math.FA↗
arXiv2017

$L^p$-bounds for pseudo-differential operators on compact Lie groups

Given a compact Lie group $G$, in this paper we establish $L^p$-bounds for pseudo-differential operators in $L^p(G)$. The criteria here are given in terms of the concept of matrix symbols defined on the non-commutative analogue of the phase space $G\times\widehat{G}$, where $\widehat{G}$ is the unitary dual of $G$. We obtain two different types of $L^p$ bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using $\mathscr{S}_{ρ,δ}^m(G)$ classes which are a suitable extension of the well known $(ρ,δ)$ ones on the Euclidean space. The results herein extend classical $L^p$ bounds established by C. Fefferman on $\mathbb R^n$. While Fefferman's results have immediate consequences on general manifolds for $ρ>\max\{δ,1-δ\}$, our results do not require the condition $ρ>1-δ$. Moreover, one of our results also does not require $ρ>δ$. Examples are given for the case of SU(2)$\cong\mathbb S^3$ and vector fields/sub-Laplacian operators when operators in the classes $\mathscr{S}_{0,0}^m$ and $\mathscr{S}_{\frac12,0}^m$ naturally appear, and where conditions $ρ>δ$ and $ρ>1-δ$ fail, respectively.

math.AP↗
arXiv2017

Non-local initial problem for second order time-fractional and space-singular equation

In this work, we consider an initial problem for second order partial differential equations with Caputo fractional derivatives in the time-variable and Bessel operator in the space-variable. For non-local boundary conditions, we present a solution of this problem in an explicit form representing it by the Fourier-Bessel series. The obtained solution is written in terms of multinomial Mittag-Leffler functions and first kind Bessel functions.

math.AP↗
arXiv2017

Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces

We analyse Morrey spaces, generalised Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalised Bessel-Riesz operators and generalised fractional integral operators in generalised Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalised fractional integral operator and Olsen type inequalities in Campanato spaces and generalised Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidean settings, however, some of them are new already in the standard Euclidean cases.

math.FA↗
arXiv2016

Layer potentials, Green's formulae, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups

We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's "principle of not feeling the boundary". We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.

math.AP↗
arXiv2016

Anisotropic $L^{2}$-weighted Hardy and $L^{2}$-Caffarelli-Kohn-Nirenberg inequalities

We establish sharp remainder terms of the $L^{2}$-Caffarelli-Kohn-Niren\-berg inequalities on homogeneous groups, yielding the inequalities with best constants. Our methods also give new sharp Caffarelli-Kohn-Nirenberg type inequalities in $\mathbb{R}^{n}$ with arbitrary quasi-norms. We also present explicit examples to illustrate our results for different weights and in abelian cases.

math.FA↗
arXiv2016

Uncertainty relations on nilpotent Lie groups

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg-Kennard type and Heisenberg-Pauli-Weyl type uncertainty inequalities, as well as Caffarelli-Kohn-Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic $\mathbb R^{n}$, and of the Heisenberg group.

math-ph↗
arXiv2016

Critical Hardy inequalities

We prove a range of critical Hardy inequalities and uncertainty type principles on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. Moreover, we establish a new type of critical Hardy inequality and prove Hardy-Sobolev type inequalities. Most of the obtained estimates are new already for the case of $\mathbb R^{n}$. For example, for any $f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\})$ our results imply the range of critical Hardy inequalities of the form $$\qquad \underset{R>0}{\sup}\left\|\frac{f-f_{R}}{|x|^{\frac{n}{p}}{\log}\frac{R}{|x|}}\right\|_{L^{p}(\mathbb{R}^{n})}\leq \frac{p}{p-1}\left\| \frac{1}{|x|^{\frac{n}{p}-1}} \nabla f\right\|_{L^{p}(\mathbb{R}^{n})},\quad 1<p<\infty,$$ where $f_{R}=f(R\frac{x}{|x|})$, with sharp constant $\frac{p}{p-1}$, recovering the known cases of $p=n$ and $p=2$. Moreover, our results also imply a new type of a critical Hardy inequality of the form $$\left\|\frac{f}{|x|}\right\|_{L^{n}(\mathbb{R}^{n})}\leq n\left\|(\log|x|)\nabla f\right\|_{L^{n}\mathbb{R}^{n})}, $$ for all $f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\}),$ where the constant $n$ is sharp. However, homogeneous groups provide a perfect degree of generality to talk about such estimates without using specific properties of $\mathbb R^n$ or of the Euclidean distance.

math.AP↗
arXiv2016

$L^{p}$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups

We prove $L^{p}$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups, which is one of most general subclasses of nilpotent Lie groups, all with sharp constants. We also discuss some of their consequences. Already in the abelian case of $\mathbb{R}^{n}$ our results provide new insights in view of the arbitrariness of the choice of the not necessarily Euclidean quasi-norm.

math.FA↗
arXiv2016

Wave equation for 2D Landau Hamiltonian

This note is devoted to the study of the well-posedness of the Cauchy problem for the Landau Hamiltonian wave equation in the plane, with nonzero constant magnetic field. We show the well-posedness in suitably defined Sobolev spaces taking into account the spectral properties of the operator.

math-ph↗