arXiv subjects
Publications and source records attributed to Michael Ruzhansky.
In this work, we study Fourier multipliers on noncommutative spaces. In particluar, we show a simple proof of $L^p$-$L^q$ estimate of Fourier multipliers on general noncommutative spaces associated with semi-finite von Neumann algebras. This includes the case of Fourier multipliers on general locally compact unimodular groups.
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 $σ\in \mathbb{R}$. Among others, it is shown that these spaces are independent of the choice of the Littlewood-Paley decomposition and that they admit characterizations in terms of continuous maximal functions and molecular frame decompositions. The defined spaces include as special cases various classical function spaces, such as Hardy spaces on homogeneous groups and homogeneous Sobolev spaces and Lipschitz spaces associated to sub-Laplacians on stratified groups.
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 $0<p\leq q<\infty$. Furthermore, for the case $p=q$, we prove the sharp inequalities with power weights and derive some other new inequalities.
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 example, $G(u)=u^α$ for $0<α<1$). In contrast with the case of convex $G$, where the Fujita exponent appears, and only existence of a positive solution for special data is achieved, in this concave case we obtain the existence and uniqueness for any positive data. The method relies on proving the existence and uniqueness of solutions for a certain class of Hammerstein-Volterra-type integral equations with a concave non-linear term, in the very general setting of arbitrary measure spaces. As a consequence, we obtain results for the existence and uniqueness of mild solutions to semilinear parabolic equations with concave nonlinearities under rather general assumptions on $f, h$ and $G$, in a variety of settings: e.g. for $\mathcal{L}$ being the Laplacian on complete Riemannian manifolds (e.g., symmetric spaces of any rank) or Hörmander sum of squares on general Lie groups. We also study the corresponding equations in the presence of a damping term, in which case assumptions can be relaxed even further.
Our primary objective in this article is to establish Hörmander type $L^p \rightarrow L^q$ Fourier multiplier theorems in the context of noncompact type Riemannian symmetric spaces $\mathbb{X}$ of arbitrary rank for the range $1 < p \leq 2 \leq q < \infty$. As a consequence of the Fourier multiplier theorem, we also derive a spectral multiplier theorem on $\mathbb{X}$. We then apply this theorem to prove $L^p \rightarrow L^q$ boundedness for functions of the Laplace-Beltrami operator and to obtain embedding theorems and operator estimates for the potentials and heat semigroups. Additionally, we provide mixed-norm versions of the Hausdorff-Young and Paley inequalities. In this context, where the Fourier transform is holomorphic, and its domain consists of various strips, we present two versions of these inequalities and explore their interrelation. Furthermore, our findings and methods are also applicable to harmonic $NA$ groups, also known as Damek-Ricci spaces.
In this paper, we obtain a fractional Hardy inequality in the case $Q sp$ was extensively studied in the literature, while here we are dealing with the complementary range $Q<sp$.
We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on $L^p(\mathbb{G})$ $(1<p<+\infty$, $\mathbb{G}$ is a graded Lie group). We show the explicit solution of the considered equation by using the Fourier analysis of the group. The equation involves a nonlocal in time operator (with a general kernel) and a positive Rockland operator acting on $\mathbb{G}.$ Also, we provide $L^p(\mathbb{G})-L^q(\mathbb{G})$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates and time decay rate for the solutions. In fact, by using some contemporary results, one can translate the latter regularity problem to the study of boundedness of its propagator which strongly depends on the traces of the spectral projections of the Rockland operator. Moreover, in many cases, we can obtain time asymptotic decay for the solutions which depends intrinsically on the considered kernel. As a complement, we give some norm estimates for the solutions in terms of a homogeneous Sobolev space in $L^{2}(\mathbb{G})$ that involves the Rockland operator. We also give a counterpart of our results in the setting of compact Lie groups. Illustrative examples are also given.
In this article, we establish three fundamental Fourier inequalities: the Hausdorff-Young inequality, the Paley inequality, and the Hausdorff-Young-Paley inequality for $(l, n)$-type functions on $\mathrm{SL}(2,\mathbb{R})$. Utilizing these inequalities, we demonstrate the $L^p$-$L^q$ boundedness of $(l, n)$-type Fourier multipliers on $\mathrm{SL}(2,\mathbb{R})$. Furthermore, we explore applications related to the $L^p$-$L^q$ estimates of the heat kernel of the Casimir element on $\mathrm{SL}(2,\mathbb{R})$ and address the global well-posedness of certain parabolic and hyperbolic nonlinear equations.
We study existence, uniqueness, norm estimates and asymptotic time behaviour (in some cases can be claimed to be sharp) for the solution of a general evolutionary integral (differential) equation of scalar type on a locally compact separable unimodular group $G$ governed by any positive left invariant operator (unbounded and either with discrete or continuous spectrum) on $G$. We complement our studies by proving some time-space (Strichartz type) estimates for the classical heat equation, and its time-fractional counterpart, as well as for the time-fractional wave equation. The latter estimates allow us to give some results about the well-posedness of nonlinear partial integro-differential equations. We provide many examples of the results by considering particular equations, operators and groups through the whole paper.
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 $\dot H^{-γ}(\mathbb G), γ>0,$ for the critical exponent case $p=1+\frac{2ν}{Q+2γ}.$ We also explore the diffusion phenomenon of the higher-order hypoelliptic damped wave equations on graded Lie groups with initial data belonging to Sobolev spaces of negative order. 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.
In this paper, we investigate the subelliptic nonlocal Brezis-Nirenberg problem on stratified Lie groups involving critical nonlinearities, namely, \begin{align*} (-Δ_{\mathbb{G}, p})^s u&= μ|u|^{p_s^*-2}u+λh(x, u) \quad \text{in}\quad Ω, \\ u&=0\quad \text{in}\quad \mathbb{G}\backslash Ω, \end{align*} where $(-Δ_{\mathbb{G}, p})^s$ is the fractional $p$-sub-Laplacian on a stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $Ω$ is an open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $\frac{Q}{s}>p\geq2,$ $p_s^*:=\frac{pQ}{Q-ps}$ is subelliptic fractional Sobolev critical exponent, $μ, λ>0$ are real parameters and $h$ is a lower order perturbation of the critical power $|u|^{p_s^*-2}u$. Utilising direct methods of the calculus of variation, we establish the existence of at least one weak solution for the above problem under the condition that the real parameter $λ$ is sufficiently small. Additionally, we examine the problem for $μ= 0$, representing subelliptic nonlocal equations on stratified Lie groups depending on one real positive parameter and involving a subcritical nonlinearity. We demonstrate the existence of at least one solution in this scenario as well. We emphasize that the results obtained here are also novel for $p=2$ even for the Heisenberg group.
In this paper, we study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory type integrals appearing in the analysis of time-fractional partial differential equations. Several generalisations of the first and second van der Corput lemmas are proved. Optimal estimates on decay orders for particular cases of the Mittag-Leffler functions are also obtained. As an application of the above results, the generalised Riemann-Lebesgue lemma and the Cauchy problem for the time-fractional Schrödinger equation are considered.
We link Sogge's type $L^p$-estimates for eigenfunctions of the Laplacian on compact manifolds with the problem of providing criteria for the $r$-nuclearity of Fourier integral operators. The classes of Fourier integral operators $I^μ_{ρ,1-ρ}(X,Y;C)$ considered here are associated with complex canonical relations $C$, i.e. they are parametrised by a complex-valued phase function. Our analysis also includes the case of real canonical relations, namely, the class of Fourier integral operators with real-valued phase functions. The nuclear trace in the sense of Grothendieck is investigated for these operators as well as the validity of the Grothendieck-Lidskii formula on Lebesgue spaces. Criteria are presented in terms of the factorisation condition for the complex canonical relation. Necessary and sufficient conditions for the membership of Fourier integral operators in Schatten-von Neumann classes are presented in the case where the Schatten index $r>0$ belongs to the set $\mathbb{N} $ and sharp sufficient conditions are presented in the general case $r>0$. In particular, we establish necessary and sufficient conditions for the membership of Fourier integral operators to the ideal of trace class operators and to the ideal of Hilbert-Schmidt operators on $L^2(X)$. The rate of decay of eigenvalues and the trace of Fourier integral operators is also investigated in both settings, in the Hilbert space case of $L^2(X)$ using Schatten-von Neumann properties and in the context of the Banach spaces $L^p(X),$ $1<p<\infty,$ utilising the notion of $r$-nuclearity.
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 specific test for Schatten class properties of bounded operators on the quantum torus is established.
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, the space-time fractional diffusion equation when diffusion is under the influence of sub-Laplacian on the Heisenberg group, or general stratified Lie groups. We establish the global well-posedness of the Cauchy problem for the general space-time fractional diffusion equation for the Rockland operator on a graded Lie group in the associated Sobolev spaces. More precisely, we establish the existence and uniqueness results for both homogeneous and inhomogeneous fractional diffusion equations. In addition, we also develop some regularity estimates.
In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for $1<p\leq q<\infty$ is playing a key role in the proof. Moreover, we also prove the fractional Hardy-Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy-Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.
This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $V$ is a non-negative multiplication operator and $\mathcal{L}_{\hbar}$ is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit $\hbar\to 0$.
In this paper, we obtain new upper bounds for the Lieb-Thirring inequality on the spheres of any dimension greater than $2$. As far as we have checked, our results improve previous results found in the literature for all dimensions greater than $2$. We also prove and exhibit an explicit new upper bound for the Lieb-Thirring inequality on $SO(3)$. We also discuss these estimates in the case of general compact Lie groups. Originally developed for estimating the sums of moments of negative eigenvalues of the Schrödinger operator in $L^2(\mathbb{R}^n)$, these inequalities have applications in quantum mechanics and other fields.