Consider a finite group $G$ acting on a Riemann surface $S$, and the associated branched Galois cover $\pi_G:S \to Y=S/G$. We introduce the concept of \emph{geometric signature} for the action of $G$, and we show that it captures much information: the geometric structure of the lattice of intermediate covers, the isotypical decomposition of the rational representation of the group $G$ acting on the Jacobian variety $JS$ of $S$, and the dimension of the subvarieties of the isogeny decomposition of $JS$. We also give a version of Riemann's existence theorem, adjusted to the present setting.

We study operators of the form $T_{g}f\left( z\right) =\int\nolimits_{0}^{z}f\left( \xi \right) \,g^{\prime }\left( \xi \right) \,d\left( \xi \right) $ ($g$ is an analytic function unity disc) on weighted Bergman spaces $L_{a}^{p}\left( w\right) $ of the unit disc where symbol $g$ is analytic function on the disc. For the case of $$ w(r) =\exp \Big( \frac{-a}{( 1-r)^{\beta }}\Big)\qquad \left( a>0, 0<\beta \leq 1\right) $$ it is shown that operator $T_{g}$ is bounded (compact) on $L_{a}^{2}\left( w\right) $ if and only if $\left( 1-\left\vert z\right\vert \right)^{\beta +1}\left\vert g^{\prime }\left( z\right) \right\vert =O\left( 1\right) \left( =o\left( 1\right) \right) $ as $\left\vert z\right\vert \rightarrow 1-$, thus solving a problem formulated in [Aleman, A. and Siskakis, A.G.: Integration Operators on Bergman Spaces. Indiana Univ. Math. J. 46 (1997), no. 2, 337-356.].

In the Heisenberg group ${\mathbb H}$ (endowed with its Carnot-Carathéodory structure), we prove that a compact set $E \subset {\mathbb H}$ which satisfies an analog of Peter Jones' geometric lemma is contained in a rectifiable curve. This quantitative condition is given in terms of Heisenberg $\beta$ numbers which measure how well the set $E$ is approximated by Heisenberg straight lines.

We study the smoothing effect of averaging over spheres for solutions of kinetic transport equations in hyperbolic Sobolev spaces.

In this paper we prove a symmetry theorem for the Green function associated to the heat equation in a certain class of bounded domains $\Omega\subset\mathbb{R}^{n+1}$. For $T>0$, let $\Omega_T=\Omega\cap[\mathbb{R}^n\times (0,T)]$ and let $G$ be the Green function of $\Omega_T$ with pole at $(0,0)\in\partial_p\Omega_T$. Assume that the adjoint caloric measure in $\Omega_T$ defined with respect to $(0,0)$, $\hat\omega$, is absolutely continuous with respect to a certain surface measure, $\sigma$, on $\partial_p\Omega_T$. Our main result states that if $$\frac {d\hat\omega}{d\sigma}(X,t)=\lambda\frac {|X|}{2t}$$ for all $(X,t)\in \partial_p\Omega_T\setminus\{(X,t): t=0\}$ and for some $\lambda>0$, then $\partial_p\Omega_T\subseteq\{(X,t):W(X,t)=\lambda\}$ where $W(X,t)$ is the heat kernel and $G=W-\lambda$ in $\Omega_T$. This result has previously been proven by Lewis and Vogel under stronger assumptions on $\Omega$.

In this article, we show that the Navier-Stokes system with variable density and viscosity is locally well-posed in the Besov space $$ \dot B^{\frac{N}{p}}_{p\,1}(\R^N)\times\big(\dot B^{\frac{N}{p}-1}_{p\,1}(\R^N)\big)^N, $$ for $1 < p\leq N$ when the initial density approaches a strictly positive constant. This result generalizes the work by R. Danchin for the case where the viscosity is constant and $p=2$ (see [Danchin, R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 1311-1334.]). Moreover, we prove existence and uniqueness in the Sobolev space\arriba{2} $$ H^{\frac{N}{2}+\alpha}(\R^N)\times\big(H^{\frac{N}{2}-1+\alpha}(\R^N)\big)^N $$ for $\alpha>0,$ generalizing R. Danchin's result for the case where viscosity is constant (see [Danchin, R.: Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9 (2004), 353-386.]).

The basic aim of this paper is to study asymptotic properties of the convolution powers $K^{(n)} = K*K* \cdots *K$ of a possibly non-symmetric probability density $K$ on a locally compact, compactly generated group $G$. If $K$ is centered, we show that the Markov operator $T$ associated with $K$ is analytic in $L^p(G)$ for $1 < p < \infty$, and establish Davies-Gaffney estimates in $L^2$ for the iterated operators $T^n$. These results enable us to obtain various Gaussian bounds on $K^{(n)}$. In particular, when $G$ is a Lie group we recover and extend some estimates of Alexopoulos and of Varopoulos for convolution powers of centered densities and for the heat kernels of centered sublaplacians. Finally, in case $G$ is amenable, we discover that the properties of analyticity or Davies-Gaffney estimates hold only if $K$ is centered.

Given an arbitrary set $E \subset \mathbb R^n$, $n\ge 2$, and a function $f: E \rightarrow \mathbb R$, consider the problem of extending $f$ to a $C^1$ function defined on the entire $\mathbb R^n$. A procedure for determining whether such an extension exists was suggested in 1958 by G. Glaeser. In 2004 C. Fefferman proposed a related procedure for dealing with the much more difficult cases of higher smoothness. The procedures in question require iterated computations of some bundles until the bundles stabilize. How many iterations are needed? We give a sharp estimate for the number of iterations that could be required in the $C^1$ case. Some related questions are discussed.

Let $G$ be a planar graph with polynomial growth and isoperimetric dimension bigger than $1$. Then the critical $p$ for Bernoulli percolation on $G$ satisfies $p_{c} < 1$.

Let $T_1 \in \mathscr B( \mathscr H_1)$ be a completely non-unitary contraction having a non-zero characteristic function $\Theta_1$ which is a $2 \times 1$ column vector of functions in $H^\infty$. As it is well-known, such a function $\Theta_1$ can be written as $ \Theta_1=w_1 m_1 \left[ {a_1} \atop {b_1} \right] $ where $w_1, m_1, a_1, b_1 \in H^\infty$ are such that $w_1$ is an outer function with $|w_1|\leq 1$, $m_1$ is an inner function, $|a_1|^2 + |b_1|^2 =1$, and $a_1 \wedge b_1 = 1$ (here $\wedge$ stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction $T_2 \in \mathscr B( \mathscr H_2)$ having also a $2 \times 1$ characteristic function $ \Theta_2=w_2 m_2 \left[ {a_2} \atop {b_2} \right] $. We prove that $T_1$ is quasi-similar to $T_2$ if, and only if, the following conditions hold: \begin{enumerate} \item $m_1=m_2$, \item $\left\{ z \in \T : \abs{w_1(z)} < 1 \right\} = \left\{ z \in \T : \left\vert w_2(z)\right\vert < 1 \right\}$ a.e., and \item the ideal generated by $a_1$ and $b_1$ in the Smirnov class $\mathscr N^+$ equals the corresponding ideal generated by $a_2$ and $b_2$. \end{enumerate}

In this paper we prove $L^p$ estimates for a tri-linear operator, whose symbol is given by the product of two standard symbols, satisfying the well known Marcinkiewicz-Hörmander-Mihlin condition. Our main result contains in particular the classical Coifman-Meyer theorem. This tri-linear operator is the simplest example of a large class of multi-linear operators, which we called \textit{paraproducts with flag singularities}.