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 & Sscr;(1)(L-2(G)) , as well as in the setting of the Schatten ideals & Sscr;(r) (L-2(G)) , for all r > 0. In particular, we deduce Schatten characterizations of elliptic pseudo-differential operators of (rho,delta)-type for the large range 0 <= delta < rho <= 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 x G, with the momentum variables belonging to the unitary dual G of G. In terms of the parameters ( rho , delta ), on the torus T (n), we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class Psi(-& varkappa;)(0,0)(T-n), kappa > 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.

This paper addresses the existence of nontrivial solutions to a class of mixed local-nonlocal problems involving a mixed interpolated Hardy potential. We first establish a concentration-compactness principle for mixed local and nonlocal operators. This result is combined with Ricceri's variational principle to obtain an existence result for quasilinear elliptic problems under different growth assumptions on the nonlinearity. Furthermore, we apply the classical mountain pass theorem to obtain a second existence result in the superlinear case.

Let 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 ℛ of homogeneous degree ν≥ 2 on 𝔾 with power type nonlinearity |u|^p and initial data taken from negative order homogeneous Sobolev space Ḣ^-γ(𝔾), γ>0, for the critical exponent case p=1+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 ℝ^n, and more generally, on stratified Lie groups.

Given a compact Lie group G and its unitary dual G, we establish the weak (1,1) continuity for pseudo-differential operators in the global Hörmander classes of order -n(1-ρ)/2 on G×G. Our approach consists of proving suitable estimates for the kernel of such operators. Furthermore, we use these kernel estimates to give an alternative proof for the H^1(G)-L^1(G)-continuity of these classes now allowing the full range 0≤1, ρ≠0, δ≠1. The conditions for the operators are formulated using the Hörmander classes S^m_ρ,δ(G):=S^m_ρ,δ(G×G) of symbols in the non-commutative phase space G×G, which are extensions of the well-known (ρ,δ)-classes in the Euclidean space. Our results are formulated in the complete range 0≤ δ≤ ρ≤ 1, ρ≠0,δ≠ 1. As an application of this boundedness result we provide end-point a-priori L^1-estimates for the sub-Laplacian ℒ_sub=X^2+Y^2, and for the heat type operator T=Z-X^2-Y^2 on SU(2)≅𝕊^3 that cannot be obtained by application of the standard pseudo-differential calculus due to Hörmander. More precisely, we prove that if one considers the subelliptic problem, Tu=f , u,f∈𝒟'(SU(2)):=(C^∞(SU(2)))', then, for f∈ W^1,-1/4(SU(2)), one has that u∈ L^1,∞(SU(2)).

In the case when d<2s, where d is the space dimension and s is the fractional power of the Laplacian, we study the well-posedness for a cubic nonlinear Schrödinger equation (CNLSE) generated by the fractional Laplacian and involving distributional, or less regular, coefficients. We formulate our problem in the setting of the concept of so-called very weak solutions and prove that it has a very weak solution. Moreover, we prove the uniqueness in some adequate sense as well as the compatibility of the very weak solution with the classical one when the latter exists. Our results cover the classical case when: d=1, s=1. A second task in this paper is to conduct some numerical experiments where interesting behaviours of the very weak solution are observed. The obtained result is the first example of the very weak well-posedness in the setting of nonlinear partial differential equations.

In this paper, we study the Cauchy problem for a heat equation governed by a mixed local–nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded, measurable coefficients that satisfy uniform positivity, and we derive an a priori estimate ensuring uniqueness and continuous dependence on the initial data. We then extend the notion of solution to distributional coefficients and initial data by a Friedrichs-type regularisation procedure. Within this very weak framework, we establish the existence and uniqueness of solution nets and prove consistency with the classical weak solution whenever the coefficients are regular.

We study heat and wave type equations on a separable Hilbert space $\mathcal{H}$ by considering non-local operators in time with any positive densely defined linear operator with discrete spectrum. We show the explicit representation of the solution and analyse the time-decay rate in a scale of suitable Sobolev space. We perform similar analysis on multi-term heat and multi-wave type equations. The main tool here is the Fourier analysis which can be developed in a separable Hilbert space based on the linear operator involved. As an application, the same Cauchy problems are considered and analysed in the setting of a graded Lie group. In this case our analysis relies on the group Fourier analysis. An extra ingredient in this framework allows, in the case of heat type equations, to establish $L^p$-$L^q$ estimates for $1\leqslant p\leqslant 2\leqslant q<+\infty$ for the solutions on graded Lie group groups. Examples and applications of the developed theory are given, either in terms of self-adjoint operators on compact or non-compact manifolds, or in the case of particular settings of graded Lie groups. The results of this paper significantly extend in different directions the results of Part I, where operators on compact Lie groups were considered. We note that the results obtained in this paper are also new already in the Euclidean setting of $\mathbb{R}^n$.

In this article, we investigate the semiclassical version of the wave equation for the discrete Schrödinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $\mathcal{L}_{\hbar}$ is the discrete Laplacian, and $V$ is a non-negative multiplication operator. We prove that $\mathcal{H}_{\hbar,V}$ has a purely discrete spectrum when the potential $V$ satisfies the condition $|V(k)|\to \infty$ as $|k|\to\infty$. We also show that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev type spaces and very weakly well-posed for distributional coefficients. Finally, we recover the classical solution as well as the very weak solution in certain Sobolev type spaces as the limit of the semiclassical parameter $\hbar\to 0$.

In this paper, we prove the anisotropic Shannon inequality for the Rényi entropy with the best constant on Folland-Stein homogeneous Lie groups. As a consequence, we also prove the optimal Shannon inequality in the same setting. Using a logarithmic Sobolev inequality in the setting of stratified groups, we prove a Heisenberg-type uncertainty principle in the latter setting.

In this paper, we show global existence and non-existence results for the heat equation with sum of the squares of smooth vector fields on ℝ^n satisfying Hörmander’s rank condition with a non-linearity of the form f(u), where f is a suitable function and u is the solution. In particular, when f(u)=u^p , we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type φ (t)f(u) .

In this paper, we establish a family of weighted logarithmic Hardy–Rellich inequalities on connected Lie groups equipped with Hörmander systems of left-invariant vector fields. In the setting of graded Lie groups, we obtain refinements in terms of homogeneous Sobolev norms associated with Rockland operators. As a consequence, on stratified (Carnot) groups we derive Gross-type logarithmic Hardy inequalities with respect to a product measure that is Gaussian on the first stratum and Lebesgue on the higher strata. We also present a logarithmic Poincaré inequality on stratified groups and a fractional logarithmic Hardy inequality for the fractional p-sub-Laplacian on homogeneous groups. Several of the resulting weighted logarithmic Hardy–Rellich inequalities appear to be new already in ℝ^n .

In this paper, we obtain bounds for the best constants in two inequalities which can be seen as analogues of the Lieb-Thirring inequality, but with the Dirac operator, on the n-sphere. We then apply these results in order to improve the known upper bounds on the classical Lieb-Thirring constant on the n-sphere for n≥ 5.

On a compact Lie group G, we consider the reproducing kernel Hilbert space ℋ_K associated with the integral kernel K of a left-invariant, positive, symmetric, trace class integral operator on L^2(G). We present lower and upper asymptotic estimates for the entropy Kolmogorov numbers (also called covering numbers) for the embedding of ℋ_K into the space C(G) of continuous functions on G.

We investigate compactness and spectral properties of multiplier operators associated with the Walsh system in the spaces L^p[0,1], 1<p<∞. Building upon previously established criteria for boundedness of Walsh multipliers, we prove an exact compactness criterion in the L^p→ L^p regime for all 1<p<∞(assuming boundedness of the multiplier), and also in the L^p→ L^2 regime for 2<p<∞. The key result states that compactness is equivalent to the condition a_n→ 0 for the multiplier symbol. We also examine in detail the point spectrum and derive strict spectral inclusions; in the Hilbert space case p=2 we obtain a complete description of the spectrum. For p≠ 2, we emphasize the limitations of transferring "diagonal" arguments and formulate results in a form that does not admit incorrect generalizations.

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. This result in the complex-valued setting, cannot be derived from its counterpart in the real-valued case.

Given a smooth, closed Riemannian manifold $(M,g)$ equipped with a linear connection $\nabla$ (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes $Ψ_{ρ, δ}^m\left(Ω^κ, \nabla, τ\right)$ introduced by Safarov. As a consequence of our main result, we establish a Szegö type-theorem, derive asymptotic expansion of the heat kernel trace, and calculate some associated spectral $ζ$-functions.

In this paper, we study the heat equation with an irregular spatially dependent thermal conductivity coefficient. We prove that it has a solution in an appropriate very weak sense. Moreover, the uniqueness result and consistency with the classical solution if the latter exists are shown. Indeed, we allow the coefficient to be a distribution with a toy example of a Delta-function.

The aim of this paper is to establish a pseudo-differential Weyl calculus on graded nilpotent Lie groups $G$ which extends the celebrated Weyl calculus on $\mathbb{R}^n$. To reach this goal, we develop a symbolic calculus for a very general class of quantization schemes, following [Doc. Math., 22, 1539--1592, 2017], using the Hörmander symbol classes $S^m_{\rho, \delta}(G)$ introduced in [Progress in Mathematics, 314. Birkhäuser/Springer, 2016]. We particularly focus on the so-called symmetric calculi, for which quantizing and taking the adjoint commute, among them the Euclidean Weyl calculus, but we also recover the (non-symmetric) Kohn-Nirenberg calculus, on $\mathbb{R}^n$ and on general graded groups [Progress in Mathematics, 314. Birkhäuser/Springer, 2016]. Several interesting applications follow directly from our calculus: expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the G\r{a}rding inequality for elliptic operators, and a generalization of the Poisson bracket for symmetric quantizations on stratified groups. In the particular case of the Heisenberg group $\mathbb{H}_n$, we are able to answer the fundamental questions of this paper: which, among all the admissible quantizations, is the natural Weyl quantization on $\mathbb{H}_n$? And which are the criteria that determine it uniquely? The surprisingly simple but compelling answers raise the question whether what is true for $\mathbb{R}^n$ and $\mathbb{H}_n$ also extends to general graded groups, which we answer in the affirmative in this paper. Among other things, we discuss and investigate an analogue of the symplectic invariance property of the Weyl quantization in the setting of graded groups, as well as the notion of the Poisson bracket for symbols in the setting of stratified groups, linking it to the symbolic properties of the commutators.

Let Ω⋐ℍ^N, N∈{3,4,5}, be a bounded connected C^2 domain. We prove that the pure critical Dirichlet problem -Δ_ℍu=u^N+2/N-2 in Ω, u=0 on admits a positive solution whenever H_d(Ω;𝔽_2)≠0 for some 1≤ d≤ N-1. This gives a hyperbolic Bahri-Coron theorem for 3≤ N≤5 under C^2 boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.

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.