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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 $I^\mu_{\rho,1-\rho}(X,Y;C)$ 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

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

In this work we consider the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^{\gamma}}$ for $\gamma>0$ associated to an anharmonic oscillator of the form $ \mathcal{A}_{k,\,\ell}=(-\Delta)^{\ell}+|x|^{2k}$ where $k,\ell$ are integers $\geq 1$. By introducing a suitable H\"ormander metric on the phase-space we analyse the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^{\gamma}}$ within the framework of H\"ormander $S(M,g)$ classes and obtain mapping properties in the scale of modulation spaces $M^{p,q},\, 0

We establish the $L^p$-$L^q$-boundedness of subelliptic pseudo-differential operators on a compact Lie group $G$. Effectively, we deal with the $L^p$-$L^q$-bounds for operators in the sub-Riemmanian setting because the subelliptic classes are associated to a H\"ormander sub-Laplacian. The Riemannian case associated with the Laplacian is also included as a special case. Then, applications to the $L^p$-$L^q$-boundedness of pseudo-differential operators in the H\"ormander classes on $G$ are given in the complete range $0\leq \delta\leq \rho\leq 1,$ $\delta<1.$ This also gives the $L^p$-$L^q$-bounds in the Riemannian setting, because the later classes are associated with the Laplacian on $G$. In both cases, in the Riemannian and the sub-Riemannian settings, necessary and sufficient conditions for the $L^p$-$L^q$-boundedness of operators are also anaysed., Comment: 22 Pages

In this work we investigate a class of degenerate Schr\"odinger equations associated to degenerate elliptic operators with irregular potentials on $\Ran$ by introducing a suitable H\"ormander metric $g$ and a $g$-weight $m$. We establish the well-posedness for the corresponding degenerate Schr\"odinger and degenerate parabolic equations. When the subelliticity is available on the degenerate elliptic operator we deduce spectral properties for a class of degenerate Hamiltonians. We also study the $L^p$ mapping properties for operators with symbols in the $S(m^{-\beta},g)$ classes in the spirit of classical Fefferman's $L^p$-bounds for the $(\rho, \delta)$ calculus. Finally, within our $S(m,g)$-classes, sharp $L^p$-estimates and Schatten properties for Schr\"odinger operators for H\"ormander sums of squares are also investigated., Comment: 45 Pages

Let $A$ and $B$ be invariant linear operators with respect to a decomposition $\{H_{j}\}_{j\in \mathbb{N}}$ of a Hilbert space $\mathcal{H}$ in subspaces of finite dimension. We give necessary and sufficient conditions for the controllability of the Cauchy problem $$ u_t=Au+Bv,\,\,u(0)=u_0,$$ in terms of the (global) matrix-valued symbols $\sigma_A$ and $\sigma_B$ of $A$ and $B,$ respectively, associated to the decomposition $\{H_{j}\}_{j\in \mathbb{N}}$. Then, we present some applications including the controllability of the Cauchy problem on compact manifolds for elliptic operators and the controllability of fractional diffusion models for H\"ormander sub-Laplacians on compact Lie groups. We also give conditions for the controllibility of wave and Schr\"odinger equations in these settings., Comment: 38 Pages

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In this note we reformulate the spectral side of the Weyl law in the language of the matrix-valued quantisation on compact Lie groups., Comment: 17 Pages. To appear in J. Lie Theory

We extend the estimates proved by Donnelly and Fefferman and by Lebeau and Robbiano for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in the control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups., Comment: 40 Pages, 5 Figures, New subsubsection 3.9. arXiv admin note: text overlap with arXiv:2209.10690

In this work we introduce a Poincar\'e determinant type for operators on the torus $\To^n$. As an application we establish the existence of nontrivial solutions for elliptic equations of the form $(-\Delta)^{\frac{\nu}{2}}u+Qu=0$ on $\To^n$ by using the Hill's method., Comment: arXiv admin note: text overlap with arXiv:2012.13216

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 H\"ormander classes associated to a sub-Riemannian structure on the group (encoded by a H\"ormander 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., Comment: 30 Pages. arXiv admin note: text overlap with arXiv:2110.00838

The fundamental solution and the heat semigroup of the Vladimirov-Taibleson operator on constant-order noncommutative Vilenkin groups are obtained, together with some estimates on the associated heat kernel. We also show the existence of a fundamental solution for the "Vladimirov Laplacian" on the $p$-adic Heisenberg group and the $p$-adic Engel group, and discuss possible extensions of our results to more general homogeneous operators on graded $p$-adic Lie groups.

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 $A(\xi)$ and $V(x)$ are $C^{\infty}$ functions enjoying some regularity conditions. Our class includes fractional relativistic Schr\"odinger operators and anharmonic oscillators with fractional potentials. By associating a H\"ormander 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., Comment: 19 pages

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 $(\rho,\delta)$-Euclidean H\"ormander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the G\r{a}rding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953's inequality due to G\r{a}rding [26]., Comment: 31 Pages

\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 $M$., Comment: 31 pages

Let $(\Omega, \mathcal{M}, \mu)$ be a measure space. In this paper we establish the set of G\^ateaux differentiability for the usual norm of $L^{1}(\Omega, \mu)$ and the corresponding derivative formulae at each point in this set.

We establish Plemelj-Smithies formulas for determinants in different algebras of operators. In particular we define a Poincar\'e type determinant for operators on the torus $\Tn$ and deduce formulas for determinants of periodic pseudo-differential operators in terms of the symbol. On the other hand, by applying a recently introduced notion of invariant operators relative to fixed decompositions of Hilbert spaces we also obtain formulae for determinants with respect to the trace class. The analysis makes use of the corresponding notion of full matrix-symbol. We also derive explicit formulas for determinants associated to elliptic operators on compact manifolds, compact Lie groups, and on homogeneous vector bundles over compact homogeneous manifolds., Comment: 22 Pages

In this work we obtain sharp $L^p$-estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to H\"ormander on $\mathbb{R}^n$. The main result extends the classical Fefferman's sharp theorem on the $L^p$-boundedness of pseudo-differential operators for H\"ormander classes on $\mathbb{R}^n$ to general graded Lie groups, also adding the borderline $\rho=\delta$ case., Comment: 40 pages

In this work we study a class of anharmonic oscillators within the framework of the Weyl-H\"ormander calculus. The anharmonic oscillators arise from several applications in mathematical physics as natural extensions of the harmonic oscillator. A prototype is an operator on $\mathbb{R}^n$ of the form $(-\Delta)^{\ell}+|x|^{2k}$ for $k,\ell$ integers $\geq 1$. The simplest case corresponds to Hamiltonians of the form $|\xi|^2+|x|^{2k}$. Here by associating a H\"ormander metric $g$ to a given anharmonic oscillator we investigate several properties of the anharmonic oscillators. We obtain spectral properties in terms of Schatten-von Neumann classes for their negative powers. We also study some examples of anharmonic oscillators arising from the analysis on Lie groups, Comment: This is an updated version with some corrections

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., Comment: 30 pages. The updated and slightly extended final version, to appear in J. Math. Pures Appl

This is a survey on a notion of invariant operators, or Fourier multipliers on Hilbert spaces. This concept is defined with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. In particular this notion can be applied to the important case of $L^2(M)$ where $M$ is a compact manifold $M$ endowed with a positive measure. The partition in this case comes from the spectral properties of a a fixed elliptic operator $E$., Comment: These notes are based on our paper arXiv:1404.6479 (to appear in J. Anal. Math.) and have been prepared for the instructional volume associated to the Summer School on Fourier Integral Operators held in Ouagadougou, Burkina Faso, where the authors took part during 14-26 September 2015

In this paper we extend classical Titchmarsh theorems on the Fourier transform of H\"older-Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for $L^2$-H\"older-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., Comment: 24 pages