This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathcal%7BC%7D%5E1" />-smoothness) well-defined even for functions on metric spaces, often suffices as a regularity condition. Thus, Paper III concentrates on the question when Lipschitz functions provide good approximations of Newtonian functions. As shown in the paper, it suffices that the function lattice quasi-norm is absolutely continuous and a fractional sharp maximal operator satisfies a weak norm estimate, which it does, e.g., in doubling Poincaré spaces if a non-centered maximal operator of Hardy–Littlewood type is locally weakly bounded. Therefore, such a local weak boundedness on rearrangement-invariant spaces is explored as well. Finer qualitative properties of Newtonian functions and the Sobolev capacity get into focus in Paper IV. Under certain hypotheses, Newtonian functions are proven to be quasi-continuous, which yields that the capacity is an outer capacity. Various sufficient conditions for local boundedness and continuity of Newtonian functions are established. Finally, quasi-continuity is applied to discuss density of locally Lipschitz functions in Newtonian spaces on open subsets of doubling Poincaré spaces.

Le but de cette thèse est d’étudier le spectre du sous-laplacien sur les variétés CR strictement pseudoconvexe. Nous prouvons que le spectre du sous-laplacien $\Delta_b$ est discret sur un domaine borné $\Omega \subset M$ d’une variété CR strictement pseudoconvexe qui satisfait l’inégalité de Poincaré, sous les conditions de Dirichlet au bord. Nous étudions le comportement des valeurs propres du sous-laplacien $\Delta_b$ sur une variété C] strictement pseudoconvexe compacte $M$, en tait que fonctionnelle sur l’espace ${\mathcal P)_+$ de formes de contact positivement orientées sur $M$ en dotant $(\matheal P}_+$ d’une topologie métrique naturelle. Nous établissons des inégalités pour les valeurs propres de $\Delta_b$ sur des variétés CR strictement pseudoconvexes (éventuellement à bord non vide). Nos estimations prolongent les résultats d,tenus par P-C. Niu \& H. Zhang \cite{NiZh) pour les valeurs propres du sous-laplacien avec conditions de Dirichlet au bord sur un domaine borné du groupe de Heisenberg, et sont dans l’esprit des inégalités de Payne-PV(o)lya-Weinberger et Yang. Nous obtenons une nouvelle borne inférieure sur la première valeur propre non nulle $\lambda_l theta )$ du sous-laplacien $\Delta_b$ sur une variété CR strictement pseudoconvexe compacte $M$ munie d’une forme de contact S\theta$ dont la connexion de Tanaka-Webster est à courbure de Ricci minorée. / The purpose of this thesis is to study the spectrum of sublaplacians on compact strictly pseudoconvex CR manifolds. We prove the discreteness of the Dirichiet spectrum of the sublaplacian $\Delta_b$ on a smoothly bounded domain $\Omega \subset M$ in a strictly pseudoconvex CR manifold M satisfying Poincaré inequality. We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifol as functions on the set ${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P)_+$ with a natural metric topology. We establish inequalities for the eigenvalues of $Delta_b$ on compact strictly pseudoconvex CR manifolds (possibly with nonempty boundary) %$C^2$ semi-isometric maps into a Euclidean space or a Heisenberg group. Our estimates extend those obtained by P-C. Niu \& H. Zhang \cite{NiZh} for the Dirichlet eigenvalues 0f the sublaplacian on a bounded domain in the Heisenberg group, in the spirit of Payne-P\’{o)lya -Weinberger and Yang inequalities. We establish a new lower bound on the first nonzero eigenvalue$\lambda_t (\theta )$ of the sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$ carrying a contact form $\theta$ whose Tanaka-Webster connection has Ricci curvature bounded from below.