Pl\"unnecke inequalities for measure graphs with applications

We generalize Petridis's new proof of Pl\"unnecke's graph inequality to graphs whose vertex set is a measure space. Consequently, this gives new Pl\"unnecke inequalities for measure preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that improve on those given by Jin.Comment: 24 pages, 1 figur

Orbital measures on SU(2)/SO(2)

We let U=SU(2) and K=SO(2) and denote N_{U}(K) the normalizer of K in U. For a an element of U\ N_{U} (K), we let \mu_{a} be the normalized singular measure supported in KaK. For p a positive integer, it was proved that \mu_{a}^{( p)}, the convolution of p copies of \mu_{a}, is absolutely continuous with respect to the Haar measure of the group U as soon as p>=2. The aim of this paper is to go a step further by proving the following two results : (i) for every a in U\ N_{U} (K) and every integer p >=3, the Radon-Nikodym derivative of \mu_{a}^{(p)} with respect to the Haar measure m_{U} on U, namely d\mu_{a}^{(p)}/d m_{U}, is in L^{2}(U), and (ii) there exist a in U\ N_{U} (K) for which d\mu_{a}^{(2)}/ dm_{U} is not in L^{2}(U), hence a counter example to the dichotomy conjecture. Since L^{2} (G) \subseteq L^{1} (G), our result gives in particular a new proof of the result when p>2

Spectral measures of small index principal graphs

The principal graph $X$ of a subfactor with finite Jones index is one of the important algebraic invariants of the subfactor. If $\Delta$ is the adjacency matrix of $X$ we consider the equation $\Delta=U+U^{-1}$. When $X$ has square norm $\leq 4$ the spectral measure of $U$ can be averaged by using the map $u\to u^{-1}$, and we get a probability measure $\epsilon$ on the unit circle which does not depend on $U$. We find explicit formulae for this measure $\epsilon$ for the principal graphs of subfactors with index $\le 4$, the (extended) Coxeter-Dynkin graphs of type $A$, $D$ and $E$. The moment generating function of $\epsilon$ is closely related to Jones' $\Theta$-series.Comment: 23 page

Quasilinear Lane-Emden equations with absorption and measure data

We study the existence of solutions to the equation -\Gd_pu+g(x,u)=\mu when $g(x,.)$ is a nondecreasing function and \gm a measure. We characterize the good measures, i.e. the ones for which the problem as a renormalized solution. We study particularly the cases where g(x,u)=\abs x^{\beta}\abs u^{q-1}u and g(x,u)=\abs x^{\tau}\rm{sgn}(u)(e^{\tau\abs u^\lambda}-1). The results state that a measure is good if it is absolutely continuous with respect to an appropriate Lorentz-Bessel capacities.Comment: 28 page

Entropy, Invertibility and Variational Calculus of the Adapted Shifts on Wiener space

In this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the Wiener measure is one, to be represented as the Radon-Nikodym derivative of the image of the Wiener measure under an adapted perturbation of identity with the help of the associated innovation process. We prove that the innovation conjecture holds if and only if the original process is almost surely invertible. We also give variational characterizations of the invertibility of the perturbations of identity and the representability of a positive random variable whose total mass is equal to unity. We prove in particular that an adapted perturbation of identity $U=I_W+u$ satisfying the Girsanov theorem, is invertible if and only if the kinetic energy of $u$ is equal to the entropy of the measure induced with the action of $U$ on the Wiener measure $\mu$, in other words $U$ is invertible iff \half \int_W|u|_H^2d\mu=\int_W \frac{dU\mu}{d\mu}\log\frac{dU\mu}{d\mu}d\mu >. otherwise the l.h.s. is always strictly greater than the r.h.s. The relations with the Monge-Kantorovitch measure transportation are also studied. An application of these results to a variational problem related to large deviations is also given