Functional Calculus on Homogeneous Groups

In the first part of the thesis, we consider the following problem. Let G be a homogeneous group, and let (L_1,...,L_n) be a jointly hypoelliptic commutative finite family of formally self-adjoint, homogeneous, left-invariant differential operators without constant terms. Then, the operators L_j are essentially self-adjoint as operators on L^2(G) with domain C^infty_c(G), and their closures commute emph{as self-adjoint operators}. Therefore, one may consider the joint functional calculus associated with the family (L_1,...,L_n). More precisely, for every bounded Borel measurable function $m$ on $R^n$, the corresponding operator m(L_1,...,L_n) commutes with left translations, so that it admits a unique right convolution kernel K(m). The so-defined kernel transform K then maps S(R^n) continuously into S(G), and L^2(eta) isometrically into L^2(G) for some uniquely determined positive Radon measure eta on R^n; this latter property can be considered as an analogue of the Plancherel isomorphism. In addition, K maps L^1(eta) continuously into C_0(G), and this property can be considered as an analogue of the Riemann--Lebesgue lemma. We focus on the following properties of K: (RL) if K(m)in L^1(G), then m can be taken in C_0(R^n): this is again an analogue of the Riemann--Lebesgue lemma; (S) if K(m)in S(G), then m can be taken in S(R^n). We prove that properties (RL) and (S) are compatible with products, and we characterize the Rockland operators which satisfy property (S) when the underlying group G is abelian. We then consider the case of 2-step stratified groups, and families whose elements are either sub-Laplacians or vector fields of homogeneous degree 2. In this setting, we prove several sufficient conditions, as well as some necessary ones, for properties (RL) and (S); we even characterize them in some more specific settings. In addition, we study the case of general (that is, not necessarily homogeneous) sub-Laplacians on 2-step stratified groups, and prove that they always satisfy properties (RL) and (S). We also prove that, under some mild assumptions, a multiplier m can be taken so as to satisfy Mihlin--Hormander conditions of order infinity if and only if the corresponding kernel K(m) satisfies Calderon--Zygmund conditions of order infinity. In the second part of the thesis, we present some results which are joint work with T. Bruno. We fix the standard sub-Laplacian on an H-type group, and consider its heat kernel (p_s)_{s>0}. We provide sharp asymptotic estimates at $infty$ for basically all the derivatives of p_1. Because of the homogeneity of the family (p_s), these estimates can also be considered as short-time asymptotics.