A note on the local Weyl formula on compact Lie groups
In this note we reformulate the spectral side of the Weyl law in the language of the matrix-valued quantisation on compact Lie groups.
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arXiv subjects
Duván Cardona
Publications and source records attributed to Duván Cardona.
Boundedness of the dyadic maximal function on graded Lie groups
Let $1<p\leq \infty$ and let $n\geq 2.$ It was proved independently by C. Calderón, R. Coifman and G. Weiss that the dyadic maximal function \begin{equation*} \mathcal{M}^{dσ}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)dσ(y)\right| \end{equation*} is a bounded operator on $L^p(\mathbb{R}^n)$ where $dσ(y)$ is the surface measure on $\mathbb{S}^{n-1}.$ In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure $dσ$ with compact support on a graded Lie group $G,$ we associate the corresponding dyadic maximal function $\mathcal{M}_D^{dσ}$ using the homogeneous structure of the group. Then, we prove a criterion in terms of the order (at zero and at infinity) of the group Fourier transform $\widehat{dσ}$ of $dσ$ with respect to a fixed Rockland operator $\mathcal{R}$ on $G$ that assures the boundedness of $\mathcal{M}_D^{dσ}$ on $L^p(G)$ for all $1<p\leq \infty.$
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On the sharpness of Strichartz estimates and spectrum of compact Lie groups
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 Schrödinger 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 $p=2.$
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Estimates for the full maximal function on graded Lie groups
On $\mathbb{R}^n,$ a classical result due to Bourgain establishes the restricted weak $(\frac{n}{n-1},1)$ inequality for the full maximal function $M_F^{dσ}$ associated to the spherical averages. In this work we present an extension to Bourgain's result on graded Lie groups for a family of full maximal operators. We formulate this extension using the group Fourier transform of the measures under consideration and the symbols of (positive Rockland operators which are) positive left-invariant hypoelliptic partial differential operators on graded Lie groups.
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$L^p$-$L^q$ estimates for subelliptic pseudo-differential operators on compact Lie groups
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 Hörmander 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 Hörmander classes on $G$ are given in the complete range $0\leq δ\leq ρ\leq 1,$ $δ<1.$ 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.
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$L^p$-$L^q$ boundedness of pseudo-differential operators on graded Lie groups
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 Hörmander symbol classes on graded Lie groups, within the range $1<p\leq 2 \leq q<\infty$. Additionally, we present a sufficient condition for the $L^p$-$L^q$ estimates of pseudo-differential operators within the range $1<p\leq q\leq 2$ or $2\leq p\leq q<\infty$. The proofs rely on estimates of the Riesz and Bessel potentials associated with Rockland operators, along with previously established results on $L^p$-boundedness of global pseudo-differential operators on graded Lie groups. Notably, as a byproduct, we also establish the sharpness of the Sobolev embedding theorem for the inhomogeneous Sobolev spaces on graded Lie groups.
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Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups
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 Hörmander 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 Hörmander classes. Between the obtained estimates we prove subelliptic versions of the celebrated sharp Fefferman $L^p$-theorem and the Calderón-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 Hörmander classes (by proving that the definition of these classes is independent of certain parameters) will be also applied in order to characterise the global Hörmander classes on arbitrary graded Lie groups developed in [90].
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Degenerate Schrödinger equations with irregular potentials
In this work we investigate a class of degenerate Schrödinger equations associated to degenerate elliptic operators with irregular potentials on $\Ran$ by introducing a suitable Hörmander metric $g$ and a $g$-weight $m$. We establish the well-posedness for the corresponding degenerate Schrödinger 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 $S(m^{-β},g)$ classes in the spirit of classical Fefferman's $L^p$-bounds for the $(ρ, δ)$ calculus. Finally, within our $S(m,g)$-classes, sharp $L^p$-estimates and Schatten properties for Schrödinger operators for Hörmander sums of squares are also investigated.
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Control of the Cauchy problem on Hilbert spaces: A global approach via symbol criteria
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 $σ_A$ and $σ_B$ 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 Hörmander sub-Laplacians on compact Lie groups. We also give conditions for the controllibility of wave and Schrödinger equations in these settings.
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Schatten-von Neumann properties for Hörmander classes on compact Lie groups
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 $(ρ,δ)$-type for the large range $0\leq δ<ρ\leq 1.$ 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 $(ρ,δ),$ on the torus $\mathbb{T}^n,$ we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class $Ψ^{-\varkappa}_{0,0}(\mathbb{T}^n),$ $\varkappa>0,$ 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.
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Spectral inequalities for elliptic pseudo-differential operators on closed manifolds
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 $ν>0,$ in the Hörmander class $Ψ^ν_{ρ,δ}(M).$ 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 $ω\subset M$ 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.
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Estimates for sums of eigenfunctions of elliptic pseudo-differential operators on compact Lie groups
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.
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Oscillating singular integral operators on compact Lie groups revisited
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 $Δ,$ namely, operators of the form \begin{equation} T_θ(-Δ):= (1-Δ)^{-\frac{nθ}{4}}e^{i (1-Δ)^{\fracθ{2}}},\,0\leq θ<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.
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Global pseudo-differential operators on the Lie group $G= (-1,1)^n$
In this work we characterise the Hörmander classes $\symbClassOn{m}ρδ{\group,\textnormal{Hör}}$ 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 Hörmander classes $\symbClassOn{m}ρδ{\group \times \R^n}$ recovers the Hörmander classes $\symbClassOn{m}ρδ{\group,\textnormal{loc}}$ defined by local coordinate systems. The analytic and qualitative properties of the classes $\symbClassOn{m}ρδ{\group \times \R^n}$ are presented in terms of the corresponding global symbols. In particular, $L^p$-Fefferman type estimates and Calderón-Vaillancourt theorems are analysed, as well as the spectral properties of the operators.
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Björk-Sjölin condition for strongly singular convolution operators on graded Lie groups
In this work we extend the $L^1$-Björk-Sjölin theory of strongly singular convolution operators to arbitrary graded Lie groups. Our criteria are presented in terms of the oscillating Hörmander condition due to Björk and Sjölin 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 Björk and Sjölin is re-obtained in the case of the Euclidean space.
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Boundedness of oscillating singular integrals on Lie groups of polynomial growth
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 Hörmander 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.
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Drift diffusion equations with fractional diffusion on compact Lie groups
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 Hörmander classes associated to a sub-Riemannian structure on the group (encoded by a Hörmander 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.
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The Wodzicki residue for pseudo-differential operators on compact Lie groups
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 Hörmander classes on $G$ defined by local coordinate systems.
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