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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

We analyse the sharpness of the Sobolev order for vector fields on compact Riemannian manifolds. As an application, we then prove ill-posedness for a certain class of Cauchy problems., Comment: 7 Pages

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 H\"ormander 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 Calder\'on-Vaillancourt theorem, Fefferman theorem and also, of the G\r{a}rding inequality. We present applications of the obtained G\r{a}rding inequality to the wellposedness of evolution problems. We characterise the classes of pseudo-differential operators on homogeneous vector bundles in the sense of H\"ormander (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., Comment: 101 Pages

Let $T$ be a Fourier integral operator of order $-(n-1)/2$ associated with a canonical relation locally parametrised by a real-phase function. A fundamental result due to Seeger, Sogge, and Stein proved in the 90's, gives the boundedness of $T$ from the Hardy space $H^1$ into $L^1.$ Additionally, it was shown by T. Tao the weak (1,1) type of $T$. In this work, we establish the weak (1,1) boundedness of a Fourier integral operator $T$ of order $-(n-1)/2$ when it has associated a canonical relation parametrised by a complex phase function., Comment: 44 Pages; 4 Figures. The proof of Theorem 3.1 has been updated

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

Bourgain in his seminal paper [2] about the analysis of maximal functions associated to convex bodies, has estimated in a sharp way the $L^2$-operator norm of the maximal function associated to a kernel $K\in L^1,$ with differentiable Fourier transform $\widehat{K}.$ We formulate the extension to Bourgain's $L^2$-estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the $L^p$-boundedness of maximal functions on graded Lie groups., Comment: 16 Pages

We establish necessary and sufficient conditions for the global hypoellipticity of $G$-invariant operators on homogeneous vector bundles. These criteria are established in terms of the corresponding matrix-valued symbols as developed by Ruzhansky and Turunen and extended in [7] to homogeneous vector-bundles., Comment: 15 pages

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 paper we establish the $L^p$-$L^q$ estimates for global pseudo-differential operators on graded Lie groups. We provide both necessary and sufficient conditions for the $L^p$-$L^q$ boundedness of pseudo-differential operators associated with the global H\"ormander symbol classes on graded Lie groups, within the range $1

We prove Strichartz estimates on any compact connected simple Lie group. In the diagonal case of Bourgain's exponents $p=q,$ we provide a new regularity order $s_{0}^{R}(p)$ in the sense that our (reverse) Strichartz estimates are valid when $s> s_{0}^{R}(p)$ and when $p\rightarrow 2^{+}.$ As expected our Sobolev index satisfies the estimate $ s_{0}^{R}(p)>s_{0}(d)=\frac{d}{2}-\frac{d+2}{p}.$ Motivated by the recent progress in the field, in the spirit of the analytical number theory methods developed by Bourgain in the analysis of periodic Schr\"odinger equations, we link the problem of finding Strichartz estimates on compact Lie groups with the problem of counting the number of representations $r_{s,2}(R)$ of an integer number $R$ as a sum of $s$ squares, and then, we implicitly use the very well known bounds for $r_{s,2}(R)$ from the Hardy-Littlewood-Ramanujan circle method. In our analysis, we explicitly compute the parametrisation of the spectrum of the Laplacian (modulo a factor depending on the geometry of the group) in terms of sums of squares. As a byproduct, our approach also yields explicit expressions for the spectrum of the Laplacian on all compact connected simple Lie groups, providing also a number of results for Strichartz estimates in the borderline case $p=2.$, Comment: 34 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

Let $1

Let $G$ be a compact Lie group of dimension $n.$ In this work we characterise the membership of classical pseudo-differential operators on $G$ in the trace class ideal $S_{1}(L^2(G)),$ as well as in the setting of the Schatten ideals $S_{r}(L^2(G)),$ for all $r>0.$ In particular, we deduce Schatten characterisations of elliptic pseudo-differential operators of $(\rho,\delta)$-type for the large range $0\leq \delta<\rho\leq 1.$ Additional necessary and sufficient conditions are given in terms of the matrix-valued symbols of the operators, which are global functions on the phase space $G\times \widehat{G},$ with the momentum variables belonging to the unitary dual $\widehat{G}$ of $G$. In terms of the parameters $(\rho,\delta),$ on the torus $\mathbb{T}^n,$ we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class $\Psi^{-\varkappa}_{0,0}(\mathbb{T}^n),$ $\varkappa>0,$ belonging to all the Schatten ideals. Additional order criteria are given in the setting of classical pseudo-differential operators. We present also some open problems in this setting., Comment: 33 pages; 1 figure

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

Let $(M,g)$ be a closed Riemannian manifold. The aim of this work is to prove the Lebeau-Robbiano spectral inequality for a positive elliptic pseudo-differential operator $E(x,D)$ on $M,$ of order $\nu>0,$ in the H\"ormander class $\Psi^\nu_{\rho,\delta}(M).$ In control theory this has been an open problem prior to this work. As an application of this fundamental result, we establish the null-controllability of the (fractional) heat equation associated with $E(x,D).$ The sensor $\omega\subset M$ in the observability inequality is an open subset of $M.$ The obtained results (that are, the corresponding spectral inequality for an elliptic operator and the null-controllability for its diffusion model) extend in the setting of closed manifolds, classical results of the control theory, as the spectral inequality due to Lebeau and Robbiano and their result on the null-controllability of the heat equation giving a complete picture of the subject in the setting of closed manifolds. For the proof of the spectral inequality we introduce a periodization approach in time inspired by the global pseudo-differential calculus due to Ruzhansky and Turunen., Comment: 31 Pages

In this work we characterise the H\"ormander classes $\symbClassOn{m}{\rho}{\delta}{\group,\textnormal{H\"or}}$ on the open manifold $\group = (-1,1)^n$. We show that by endowing the open manifold $\group = (-1,1)^n$ with a group structure, the corresponding global Fourier analysis on the group allows one to define a global notion of symbol on the phase space $\group \times \R^n$. Then, the class of pseudo-differential operators associated to the global H\"ormander classes $\symbClassOn{m}{\rho}{\delta}{\group \times \R^n}$ recovers the H\"ormander classes $\symbClassOn{m}{\rho}{\delta}{\group,\textnormal{loc}}$ defined by local coordinate systems. The analytic and qualitative properties of the classes $\symbClassOn{m}{\rho}{\delta}{\group \times \R^n}$ are presented in terms of the corresponding global symbols. In particular, $L^p$-Fefferman type estimates and Calder\'on-Vaillancourt theorems are analysed, as well as the spectral properties of the operators., Comment: 34 Pages; 1 Figure

In this work we extend the $L^1$-Bj\"ork-Sj\"olin theory of strongly singular convolution operators to arbitrary graded Lie groups. Our criteria are presented in terms of the oscillating H\"ormander condition due to Bj\"ork and Sj\"olin of the kernel of the operator, and the decay of its group Fourier transform is measured in terms of the infinitesimal representation of an arbitrary Rockland operator. The historical result by Bj\"ork and Sj\"olin is re-obtained in the case of the Euclidean space., Comment: 22 Pages. arXiv admin note: text overlap with arXiv:2201.12881

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

In [24, Theorem 2'] Charles Fefferman has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian $\Delta,$ namely, operators of the form \begin{equation} T_{\theta}(-\Delta):= (1-\Delta)^{-\frac{n\theta}{4}}e^{i (1-\Delta)^{\frac{\theta}{2}}},\,0\leq \theta <1. \end{equation} The aim of this work is to extend Fefferman's result to oscillating singular integrals on any arbitrary compact Lie group. Applications to oscillating spectral multipliers of the Laplace-Beltrami operator are also considered. The proof of our main theorem illustrates the delicate relationship between the condition on the kernel of the operator, its Fourier transform (defined in terms of the representation theory of the group) and the microlocal/geometric properties of the group., Comment: 21 pages. The proof of the main theorem has been simplified. Final version to appear in Math. Z

We investigate the boundedness of oscillating singular integrals on Lie groups of polynomial growth in order to extend the classical oscillating conditions due to Fefferman and Stein for the boundedness of oscillating convolution operators. Kernel criteria are presented in terms of a fixed sub-Riemannian structure on the group induced by a sub-Laplacian associated to a H\"ormander system of vector fields. In the case where the group is graded, kernel criteria are presented in terms of the Fourier analysis associated to an arbitrary Rockland operator., Comment: 27 Pages, 1 Figure, editions of style were made

In this work we extend the Euclidean theory of oscillating singular integrals due to Fefferman and Stein to arbitrary graded Lie groups. Our approach reveals the strong compatibility between the geometric measure theory of a graded Lie group and the Fourier analysis associated with Rockland operators. Our criteria are presented in terms of the H\"ormander condition of the kernel of the operator, and its group Fourier transform. One of the novelties of this work, is that we use the infinitesimal representation of a Rockland operator to measure the decay of the Fourier transform of the kernel., Comment: 19 Pages. The proof of the main theorem has been simplified

Let $G$ be an arbitrary compact Lie group. In this work we apply the method of the analytic continuation of traces in order to compute the Wodzicki residue for a classical pseudo-differential operator on $G$ in terms of its matrix-valued symbol (which is globally defined on the non-commutative phase space $G\times \widehat{G},$ with $\widehat{G}$ being the unitary dual of $G$). Our main theorem is complementary to the results in [2], where we remove the ellipticity hypothesis when the operators belong to the H\"ormander classes on $G$ defined by local coordinate systems., Comment: 12 Pages. arXiv admin note: text overlap with arXiv:2110.00838

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

In this work we establish a subelliptic sharp G\r{a}rding inequality on compact Lie groups for pseudo-differential operators with symbols belonging to global subelliptic H\"ormander 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 $\mathscr{S}^m_{\rho,\delta}(G)$-classes, $0\leq \delta<\rho$, the one in [26] about the validity of the sharp G\r{a}rding 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.

\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 $X$ 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 $I^{\mu}_1(X,Y;\Lambda),$ $\mu\leq -(n-1)/2,$ 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 $I^{\mu}_1(X,Y;\Lambda),$ $\mu\leq -(n-1)/2,$ 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., Comment: 8 Pages. This note is a corrected version of an old one submitted previously to the ArXiv

In this work we provide a criterion for the global weak (1,1) type of integral operators which are known to be locally uniformly of weak (1,1) type. As an application, we establish the global weak (1,1) type for a class of Fourier integral operators. While the local result is known from the work of Tao [33] for Fourier integral operators of order $-(n-1)/2,$ we give natural sufficient conditions in order to extend it to the corresponding global estimate, Comment: Unfortunately, we have found a mistake in the proof of Theorem 1.2, that is the main theorem of this work. We claim in page 10, below of (2.5) that the uniform estimate of the weak L^1 norm of T at any atom is enough for concluding the proof, but this is not the case, because the weak L^1 space is not normable

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the H\"ormander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp H\"ormander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject., Comment: 26 Pages. arXiv admin note: text overlap with arXiv:2101.05856

We consider the inhomogeneous Dirichlet initial boundary value problem for the Benjamin-Ono equation formulated on the half line. We study the global in time existence of solutions to the initial-boundary value problem. This work is a continuation of the ones [14,15] by Hayashi and Kaikina where the global in time existence and the asymptotic behaviour of solutions for large time were considered., Comment: 24 Pages

In this work we investigate the boundedness of Fourier multipliers on Triebel-Lizorkin spaces associated to positive Rockland operators on a graded Lie group. The found criterion is expressed in terms of the H\"ormander-Mihlin condition on the global symbol of a Fourier multiplier., Comment: 25 Pages. arXiv admin note: text overlap with arXiv:1610.04701

Given a smooth manifold $M$ (with or without boundary), in this paper we study the regularisation of traces for the global pseudo-differential calculus in the context of non-harmonic analysis. Indeed, using the global pseudo-differential calculus on manifolds (with or without boundary) developed in [30], the Calder\'on-Vaillancourt Theorem and the global functional calculus in [6], we determine the singularity orders in the regularisation of traces and the sharp regularity orders for the Dixmier traceability of the global H\"ormander classes. Our analysis (free of coordinate systems) allows us to obtain non-harmonic analogues of several classical results arising from the microlocal analysis of regularised traces for pseudo-differential operators with symbols defined by localisations., Comment: 34 Pages. arXiv admin note: text overlap with arXiv:2101.02519

Given a smooth manifold $M$ (with or without boundary), in this paper we establish a global functional calculus (without the standard assumption that the operators are classical pseudo-differential operators) and the G\r{a}rding inequality for global pseudo-differential operators associated with boundary value problems. The analysis that we follow is free of local coordinate systems. Applications of the G\r{a}rding inequality to the global solvability for a class of evolution problems are also considered., Comment: 35 pages. arXiv admin note: text overlap with arXiv:1504.00777

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 note, we consider a non-commutative analogy of the classical Fefferman multiplier problem for the ball. More precisely, if $\chi$ is the characteristic function of the unit interval $I=[0,1],$ we investigate a family of differential operators $\mathcal{R}$ on a graded Lie group $G,$ for which the multipliers $\chi(\mathcal{R})$ are bounded on $L^p(G),$ if and only if $p=2.$, Comment: 10 pages

In this memoir we extend the theory of global pseudo-differential operators to the setting of arbitrary sub-Riemannian structures on a compact Lie group. More precisely, given a compact Lie group $G$, and the sub-Laplacian $\mathcal{L}$ associated to a system of vector fields $X=\{X_1,\cdots,X_k\}$ satisfying the H\"ormander condition, we introduce a (subelliptic) pseudo-differential calculus associated to $\mathcal{L},$ based on the matrix-valued quantisation process developed in [138]. This theory will be developed as follows. First, we will investigate the singular kernels of this calculus, estimates of $L^p$-$L^p$, $H^1$-$L^1$, $L^\infty$-$BMO$ type and also the weak (1,1) boundedness of these subelliptic H\"ormander classes. Between the obtained estimates we prove subelliptic versions of the celebrated sharp Fefferman $L^p$-theorem and the Calder\'on-Vaillancourt theorem. The obtained estimates will be used to establish the boundedness of subelliptic operators on subelliptic Sobolev and Besov spaces. We will investigate the ellipticity, the construction of parametrices, the heat traces and the regularisation of traces for the developed subelliptic calculus. A subelliptic global functional calculus will be established as well as a subelliptic version of Hulanicki theorem. The approach established in characterising our subelliptic H\"ormander classes (by proving that the definition of these classes is independent of certain parameters) will be also applied in order to characterise the global H\"ormander classes on arbitrary graded Lie groups developed in [90]., Comment: 163 pages. The final version of this monograph will appear in MSJ Memoirs (Mathematical Society of Japan Memoirs), 180 pages approx

In this contribution we investigate the Schr\"ordinger equation associated to the Laplacian on the sphere in the form of sharp Strichartz estimates. We will provided simple proofs for our main theorems using purely the $L^2\rightarrow L^p$ spectral estimates for the operator norm of the spectral projections (associated to the spherical harmonics) proved in [8]. A sharp index of regularity is established for the initial data in spheres of arbitrary dimension $d\geq 2$., Comment: 12 pages

In this paper we provide sharp results for the Dixmier traceability of discrete pseudo-differential operators on $\ell^2(\mathbb{Z}^n)$. In this setting, we introduce a suitable notion of a class of classical symbols which provide a class of Dixmier traceable discrete pseudo-differential operators. We also present a formula for the Dixmier trace of a Dixmier traceable discrete pseudo-differential operator by using the Connes equivalence between the Wodzicki residue and the Dixmier trace., Comment: 10 pages. Dedicated to Professor L\'azaro Recht on his 79 birthday

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 note we investigate some conditions of H\"ormander-Mihlin type in order to assure the $L^\infty$-${\textnormal{BMO}}$ boundedness for pseudo-multipliers of the harmonic oscillator. The $\textnormal{H}^1$-$L^1$ continuity for Hermite multipliers also is investigated. The final version of this paper will appear in Rev. Colombiana Mat., Comment: 13 pages

In this paper, we investigate the nuclear trace of vector-valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo-differential operators. First, we characterise the nuclearity of pseudo-differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector-valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector-valued Fourier multiplier in terms of its operator-valued symbol and then we use this formula for expressing the index of certain elliptic operators belonging to periodic H\"ormander classes., Comment: 31 pages