Let $\Gamma=(G,E)$ be an infinite weighted graph which is Ahlfors $\alpha$-regular, so that there exists a constant $c$ such that $c^{-1} r^\alpha\le V(x,r)\le c r^\alpha$, where $V(x,r)$ is the volume of the ball centre $x$ and radius $r$. Define the escape time $T(x,r)$ to be the mean exit time of a simple random walk on $\Gamma$ starting at $x$ from the ball centre $x$ and radius $r$. We say $\Gamma$ has escape time exponent $\beta>0$ if there exists a constant $c$ such that $c^{-1} r^\beta \le T(x,r) \le c r^\beta$ for $r\ge 1$. Well known estimates for random walks on graphs imply that $\alpha\ge 1$ and $2 \le \beta \le 1+\alpha$. We show that these are the only constraints, by constructing for each $\alpha_0$, $\beta_0$ satisfying the inequalities above a graph $\widetilde{\Gamma}$ which is Ahlfors $\alpha_0$-regular and has escape time exponent $\beta_0$. In addition we can make $\widetilde{\Gamma}$ sufficiently uniform so that it satisfies an elliptic Harnack inequality.

Motivated by a problem arising in astrophysics we study a nonlinear elliptic equation in $\mathbb{R}^{N}$ with cylindrical symmetry and with singularities on a whole subspace of $\mathbb{R}^{N}$. We study the problem in a variational framework and, as the nonlinearity also displays a critical behavior, we use some suitable version of the Concentration-Compactness Principle. We obtain several results on existence and nonexistence of solutions.

We prove some Liouville type theorems for positive solutions of semilinear elliptic equations in the whole space $\mathbb{R}^N$, $N\geq 3$, and in the half space $\mathbb{R}^N_{+}$ with different boundary conditions, using the technique based on the Kelvin transform and the Alexandrov-Serrin method of moving hyperplanes. In particular we get new nonexistence results for elliptic problems in half spaces satisfying mixed (Dirichlet-Neumann) boundary conditions.

Let $A$ and $B$ be C*-algebras and let $T$ be a linear isometry from $A$ \emph{into} $B$. We show that there is a largest projection $p$ in $B^{**}$ such that $T(\cdot)p : A \longrightarrow B^{**}$ is a Jordan triple homomorphism and $$ T(a b^* c + c b^* a) p= T(a) T(b)^* T(c) p + T(c) T(b)^* T(a) p $$ for all $a$, $b$, $c$ in $A$. When $A$ is abelian, we have $\|T(a)p\|=\|a\|$ for all $a$ in $A$. It follows that a (possibly non-surjective) linear isometry between any C*-algebras reduces {\it locally} to a Jordan triple isomorphism, by a projection.

In this paper we characterize manifolds (topological or smooth, compact or not, with or without boundary) which admit flows having a dense orbit (such manifolds and flows are called transitive) thus fully answering some questions by Smith and Thomas. Namely, it is shown that a surface admits a transitive flow (which can be got smooth) if and only if it is connected and it is neither homeomorphic to the sphere nor the projective plane nor embeddable in the Klein bottle (or, alternatively, if it is connected and includes two orientable topological circles intersecting transversally at exactly one point). We also prove that any (connected) manifold with dimension at least 3 admits a transitive flow, which can be got smooth if the manifold admits a smooth structure. In particular, this allows us to characterize $\omega$-limit sets with nonempty interior for flows in a given $n$-manifold (as they can be described by the property of being the closure of its transitive $n$-submanifolds).

Let $T$ be a sublinear operator such that $(Tf)^*(t)\le h(t, \|f\|_1)$ for some positive function $h(t,s)$ and every function $f$ such that $\|f\|_{\infty}\le 1$. Then, we show that $T$ can be extended continuously from a logarithmic type space into a weighted weak Lorentz space. This type of result is connected with the theory of restricted weak type extrapolation and extends a recent result of Arias-de-Reyna concerning the pointwise convergence of Fourier series to a much more general context.

In this paper we are dealing with the boundary problem for Levi flat graphs in the space $\mathbb{R}^4$, endowed with an almost complex structure $J$. This problem can be formalized as a Dirichlet problem for a quasilinear degenerate elliptic equation, called Levi equation. The Levi equation has the form $$D_1^2 + D^2_2 - D_1f = 0,$$ where $D_1$ and $D_2$ are nonlinear vector fields. Under geometrical assumptions on the boundary a lipschitz continuous viscosity solution is found. The regularity of the viscosity solution is studied in suitable anisotropical Sobolev spaces, and it is proved that the solution has derivatives of any order in the direction of the vectors $D_1$ and $D_2$ i.e. it is of class $C^\infty$ in these directions, but not necessary regular in the third direction of the space. Finally, after proving a weak version of the Frobenius theorem, we show that the graph of the solution is foliated in holomorphic curves.

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singularity in the projective space of dimension two. Also we calculate higher Melnikov functions under some generic conditions.

We present a description of isochronous centres of planar vector fields $X$ by means of their groups of symmetries. More precisely, given a normalizer $U$ of $X$ (i.e., $[X,U]=\mu X$, where $\mu$ is a scalar function), we provide a necessary and sufficient isochronicity condition based on $\mu$. This criterion extends the result of Sabatini and Villarini that establishes the equivalence between isochronicity and the existence of commutators ($[X,U]= 0$). We put also special emphasis on the mechanical aspects of isochronicity; this point of view forces a deeper insight into the potential and quadratic-like Hamiltonian systems. For these families we provide new ways to find isochronous centres, alternative to those already known from the literature.

We study the large time behavior of the energy of a class of Navier-Stokes flows in $\mathbb{R}^n$ $(n\ge2)$ with special symmetries. Inside this class, we construct examples of solutions such that the energy norm decays faster than $t^{-(n+2)/4}$.

The accretive operators theory is employed for proving an existence theorem for the evolutive energy equations involving simultaneously conduction, stationary convection (in the sense that the velocity field is assumed to be time independent), and radiation. In doing that we need to use new existence results for elliptic linear problems with mixed boundary conditions and irregular data.

We give a characterization of the two-microlocal Besov spaces in terms of the local Besov type conditions. As an easy consequence, we obtain the inclusions between the two-microlocal Besov spaces and the local Besov spaces. These results are natural extensions of those obtained by Jaffard and Meyer, who treated the pointwise H\"older regularity in terms of two-microlocal estimates. The Daubechies wavelets play a key role throughout the paper.

We prove some results on the zeros of functions of the form $f(z) = \sum_{n=1}^\infty \frac{a_n}{z - z_n}$, with complex $a_n$, using quasiconformal surgery, Fourier series methods, and Baernstein's spread theorem. Our results have applications to fixpoints of entire functions.