The logarithmic derivative of the $\Gamma$-function, namely the $\psi$-function, has numerous applications. We define analogous functions in a four dimensional space. We cut lattices and obtain Clifford-valued functions. These functions are holomorphic cliffordian and have similar properties as the $\psi$-function. These new functions show links between well-known constants: the Euler gamma constant and some generalisations, $\zeta^R(2)$, $\zeta^R(3)$. We get also the Riemann zeta function and the Epstein zeta functions.

Consider the group $G:=PSL_2(\mathbb R)$ and its subgroups $\Gamma:= PSL_2(\mathbb{Z})$ and $\Gamma':= DSL_2(\mathbb{Z})$. $G/\Gamma$ is a canonical realization (up to an homeomorphism) of the complement $\mathbb S^3\setminus T$ of the trefoil knot $T$, and $G/\Gamma'$ is a canonical realization of the 6-fold branched cyclic cover of $\mathbb S^3\setminus T$, which has 3-dimensional cohomology of 1-forms. Putting natural left-invariant Riemannian metrics on $G$, it makes sense to ask which is the asymptotic homology performed by the Brownian motion in $G/\Gamma'$, describing thereby in an intrinsic way part of the asymptotic Brownian behavior in the fundamental group of the complement of the trefoil knot. A good basis of the cohomology of $ G/\Gamma'$, made of harmonic 1-forms, is calculated, and then the asymptotic Brownian behavior is obtained, by means of the joint asymptotic law of the integrals of the above basis along the Brownian paths. Finally the geodesics of $G$ are determined, a natural class of ergodic measures for the geodesic flow is exhibited, and the asymptotic geodesic behavior in $G/\Gamma'$ is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).

We aim here at analyzing the fundamental properties of positive semidefinite Schrödinger operators on networks. We show that such operators correspond to perturbations of the combinatorial Laplacian through 0-order terms that can be totally negative on a proper subset of the network. In addition, we prove that these discrete operators have analogous properties to the ones of elliptic second order operators on Riemannian manifolds, namely the monotonicity, the minimum principle, the variational treatment of Dirichlet problems and the condenser principle. Unlike the continuous case, a discrete Schrödinger operator can be interpreted as an integral operator and therefore a discrete Potential Theory with respect to its associated kernel can be built. We prove that the Schrödinger kernel satisfies enough principles to assure the existence of equilibrium measures for any proper subset. These measures are used to obtain systematic expressions of the Green and Poisson kernels associated with Dirichlet problems.

This paper deals with explicit spectral gap estimates for the linearized Boltzmann operator with hard potentials (and hard spheres). We prove that it can be reduced to the Maxwellian case, for which explicit estimates are already known. Such a method is constructive, does not rely on Weyl's Theorem and thus does not require Grad's splitting. The more physical idea of the proof is to use geometrical properties of the whole collision operator. In a second part, we use the fact that the Landau operator can be expressed as the limit of the Boltzmann operator as collisions become grazing in order to deduce explicit spectral gap estimates for the linearized Landau operator with hard potentials.

This paper studies the relationship between the bidual of the (projective) tensor product of Banach spaces and the tensor product of their biduals.

This paper aims at giving an overview of estimates in general Besov spaces for the Cauchy problem on $t=0$ related to the vector field $\partial_t+v\cdot\nabla$. The emphasis is on the conservation or loss of regularity for the initial data. When $\nabla v$ belongs to $L^1(0,T;L^\infty)$ (plus some convenient conditions depending on the functional space considered for the data), the initial regularity is preserved. On the other hand, if $\nabla v$ is slightly less regular (e.g. $\nabla v$ belongs to some limit space for which the embedding in $L^\infty$ fails), the regularity may coarsen with time. Different scenarios are possible going from linear to arbitrarily small loss of regularity. This latter result will be used in a forthcoming paper to prove global well-posedness for two-dimensional incompressible density-dependent viscous fluids (see [Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm. Partial Differential Equations 26 (2001), 1183-1233, and Erratum, 27 (2002), 2531-2532.]). Besides, our techniques enable us to get estimates uniformly in $\nu\geq0$ when adding a diffusion term $-\nu\Delta u$ to the transport equation.

In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set $G\subset \mathbb{R}^{2}$ we construct a differentiable function $f:G\to\mathbb{R}$ for which there exists an open set $\Omega_{1}\subset\mathbb{R}^{2}$ such that $\nabla f(\mathbf{p})\in \Omega_{1}$ for a $\mathbf{p}\in G$ but $\nabla f(\mathbf{q})\not\in\Omega_{1}$ for almost every $\mathbf{q}\in G$. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.

In this paper we characterize the semigroups of analytic functions in the unit disk which lead to semigroups of operators in the disk algebra. These characterizations involve analytic as well as geometric aspects of the iterates and they are strongly related to the classical theorem of Carath\'eodory about local connection and boundary behaviour of univalent functions.

Let $G$ be a Lie group of polynomial volume growth, with Lie algebra $\mbox{\gothic g}$. Consider a second-order, right-invariant, subelliptic differential operator $H$ on $G$, and the associated semigroup $S_t = e^{-tH}$. We identify an ideal $\mbox{\gothic n}'$ of $\mbox{\gothic g}$ such that $H$ satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of $\mbox{\gothic n}'$. The regularity is expressed as $L_2$ estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We obtain the boundedness in $L_p$, $1<p<\infty$, of some associated Riesz transform operators. Finally, we show that $\mbox{\gothic n}'$ is the largest ideal of $\mbox{\gothic g}$ for which the regularity results hold. Various algebraic characterizations of $\mbox{\gothic n}'$ are given. In particular, $\mbox{\gothic n}'= \mbox{\gothic s}\oplus \mbox{\gothic n}$ where $\mbox{\gothic n}$ is the nilradical of $\mbox{\gothic g}$ and $\mbox{\gothic s}$ is the largest semisimple ideal of $\mbox{\gothic g}$. Additional features of this article include an exposition of the structure theory for $G$ in Section 2, and a concept of twisted multiplications on Lie groups which includes semidirect products in the Appendix.

We establish that a quasiconformal group is of compact type if and only if its limits set is purely conical and find that the limit set of a quasiconformal group of compact type is uniformly perfect. A key tool is the result of Bowditch-Tukia on compact-type convergence groups. These results provide crucial tools for studying the deformations of quasiconformal groups and in establishing isomorphisms between such groups and conformal groups.

We prove that every homogeneous Carnot group can be lifted to a free homogeneous Carnot group. Though following the ideas of Rothschild and Stein, we give simple and self-contained arguments, providing a constructive proof, as shown in the examples.

We show a result of approximation in law of the $d$-parameter fractional Brownian sheet in the space of the continuous functions on $[0,T]^d$. The construction of these approximations is based on the functional invariance principle.

We study the family of closed Riemannian $n$-manifolds with holonomy group isomorphic to $\mathbb{Z}_2^{n-1}$, which we call generalized Hantzsche-Wendt manifolds. We prove results on their structure, compute some invariants, and find relations between them, illustrated in a graph connecting the family.

In this paper, we formulate necessary conditions for decay rates of $L^p$ operator norms of weighted oscillatory integral operators on $\mathbb{R}$ and give sharp $L^2$ estimates and nearly sharp $L^p$ estimates.