Marianna Chatzakou

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.

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 ) ) {\mathscr{S}_{1}(L^{2}(G))} , as well as in the setting of the Schatten ideals 𝒮 r ⁢ ( L 2 ⁢ ( G ) ) {\mathscr{S}_{r}(L^{2}(G))} , for all r > 0 {r>0} . In particular, we deduce Schatten characterizations of elliptic pseudo-differential operators of ( ρ , δ ) {(\rho,\delta)} -type for the large range 0 ≤ δ < ρ ≤ 1 {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 × G ^ {G\times\widehat{G}} , with the momentum variables belonging to the unitary dual G ^ {\widehat{G}} of G . In terms of the parameters ( ρ , δ ) {(\rho,\delta)} , on the torus 𝕋 n {\mathbb{T}^{n}} , we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class Ψ 0 , 0 - ϰ ⁢ ( 𝕋 n ) {\Psi^{-\varkappa}_{0,0}(\mathbb{T}^{n})} , ϰ > 0 {\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.

We develop a unified strategy to obtain the geometric logarithmic Hardy inequality on any open set of a stratified group , provided the validity of the Hardy inequality in this setting, where the so‐called “weight” is regarded to be any measurable nonnegative function on . Provided the legitimacy of the latter for some , we also show an inequality that is an extension of the ‘generalized Poincaré inequality’ introduced by Beckner with the addition of the weight , and this is referred to as the “geometric Hardy‐Poincaré inequality.” The aforesaid inequalities become explicit in the case where , the half‐space of , when , and in the case where , when is the “horizontal norm” on the first stratum of . For the second case, the semi‐Gaussian analog of the derived inequalities is proved, when the Gaussian measure is regarded with respect to the first stratum of . Applying our results to the case where (abelian case), we generalize the classical probabilistic Poincaré inequality by adding weights.

In this work we consider the semigroup e−tAk,ℓγ for γ > 0 associated to an anharmonic oscillator of the form Ak,ℓ=(−Δ)ℓ+|x|2k where k, ℓ are integers ≥1. By introducing a suitable Hörmander metric on the phase-space we analyze the semigroup e−tAk,ℓγ within the framework of Hörmander S(M, g) classes and obtain mapping properties in the scale of modulation spaces Mp,q,0<p,q≤∞, with respect to an anharmonic modulation weight. As an application, we apply the obtained bounds to establish the well-posedness for the nonlinear heat equation associated with Ak,ℓγ. It is worth noting that the results presented in this paper are novel, even in the case where γ = 1.