Advanced Search

Soit $L$ un opérateur parabolique sur ${\mathbf{R}}_{n+1}$ écrit sous forme divergence et à coefficients lipschitziens relativement à une métrique adaptée. Nous cherchons à comparer près de la frontière le comportement relatif des $L$-solutions positives sur un domaine “lipschitzien”. Dans un premier temps, nous démontrons un principe de Harnack uniforme pour certaines $L$-solutions positives. Ce principe nous permet alors de démontrer une inégalité de Harnack forte à la frontière pour certains couples de $L$-solutions positives....

Various tools can be used to calculate or estimate the dimension of measures. Using a probabilistic interpretation, we propose very simple proofs for the main inequalities related to this notion. We also discuss the case of quasi-Bernoulli measures and point out the deep link existing between the calculation of the dimension of auxiliary measures and the multifractal analysis.

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to this definition....