A parameter estimation problem is considered for a stochastic parabolic equation driven by additive Gaussian noise that is white in time and space. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed. A necessary and sufficient condition for consistency and asymptotic normality of the estimator is derived in terms of the eigenvalues of the operators in the equation, and a detailed proof is provided. Other estimation problems are briefly surveyed.

By means of Steiner symmetrization we get some estimates for the first eigenfunction of a class of linear problems, having as prototype the Laplacian with Dirichlet boundary conditions.

In 1973, Macbeath found a general formula for the number of points fixed by an arbitrary orientation preserving automorphism of a Riemann surface $X$. It was given in terms of a group $G$ of conformal automorphisms of $X$ and the ramification data of the covering $X\to X/G$, which corresponds to the so called universal covering transformation group. In these terms, for the case of a cyclic group of automorphisms of an unbordered non-orientable Klein surface, the formula was given later by Izquierdo and Singerman and here we find formulas valid for an arbitrary (finite) group $G$ of automorphisms.

In this paper we show that, in order to check Gromov hyperbolicity of any surface with curvature $K \le -k^2<0$, we just need to verify the Rips condition on a very small class of triangles, namely, those contained in simple closed geodesics. This result is, in fact, a new characterization of Gromov hyperbolicity for this kind of surfaces.

A cusp is the image of the unit disk under a proper holomorphic map into ${\mathbb C}^n$ that is one-to-one and whose derivative vanishes at exactly one point. It is simple if not all the second derivatives vanish. We characterize when two simple cusps are isomorphic, and show that they can all be realized in ${\mathbb C}^2$.

A band is a semigroup consisting of idempotents. It is proved that for any field $K$ and any band $S$ with finitely many components, the semigroup algebra $K[S]$ can be embedded in upper triangular matrices over a commutative $K$-algebra. The proof of a theorem of Malcev on embeddability of algebras into matrix algebras over a field is corrected and it is proved that if $S=F\cup E$ is a band with two components $E$, $F$ such that $F$ is an ideal of $S$ and $E$ is finite, then $S$ is a linear semigroup. Certain sufficient conditions for linearity of a band $S$, expressed in terms of annihilators associated to $S$, are also obtained.

This paper studies Besov $p$-capacities as well as their relationship to Hausdorff measures in Ahlfors regular metric spaces of dimension $Q$ for $1<Q<p<\infty$. Lower estimates of the Besov $p$-capacities are obtained in terms of the Hausdorff content associated with gauge functions $h$ satisfying the decay condition $\int_0^1 h(t)^{1/(p-1)} \frac{dt}{t}<\infty$.

We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on $L^p$, $1<p<\infty$, and of a perfect dyadic singular integral operator. This allows to deduce a local $T(b)$ theorem for singular integral operators from the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Tao, Thiele and the first author.

We study the removable singularities for solutions to the Beltrami equation $\overline\partial f=\mu\, \partial f$, where $\mu$ is a bounded function, $\|\mu\|_\infty\leq\frac{K-1}{K+1}<1$, and such that $\mu\in W^{1,p}$ for some $p\leq 2$. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported Beltrami coefficient $\mu\in W^{1,p}$, $\frac{2K^2}{K^2+1}<p\leq 2$, preserve compact sets of $\sigma$-finite length and vanishing analytic capacity, even though they need not be bilipschitz.

If $E$ is a Banach space, $b\in \mathit{BMO}({\mathbb R}^n,\mathcal{L}(E))$ and $T$ is a $\mathcal{L}(E)$-valued Calderón-Zygmund type operator with operator-valued kernel $k$, we show the boundedness of the commutator $T_b(f)= b T(f)- T(bf)$ on $L^p({\mathbb R}^n,E)$ for $1<p<\infty$ whenever $b$ and $k$ verify some commuting properties. Some endpoint estimates are also provided.

We construct a class of singular integral operators associated with homogeneous Calderón-Zygmund standard kernels on $d$-dimensional, $d <1$, Sierpinski gaskets $E_d$. These operators are bounded in $L^2(\mu_d)$ and their principal values diverge $\mu_d$ almost everywhere, where $\mu_d$ is the natural ($d$-dimensional) measure on $E_d$.