Measures on the unit circle are well studied from the view of Fourier analysis. In this paper, we investigate measures from the view of Poisson integrals and of divisibility of singular inner functions in $H^\infty + C$. Especially, we study singular measures which have outer and inner vanishing measures. It is given two decompositions of a singular positive measure. As applications, it is studied division theorems in $H^\infty + C$.

Let $(B_t ; t \ge 0)$, $\big(\mbox{resp. }((X_t, Y_t) ; t \ge 0)\big)$ be a one (resp. two) dimensional Brownian motion started at 0. Let $T$ be a stopping time such that $(B_{t \wedge T} ; t \ge 0)$ \big(resp. $(X_{t \wedge T} ; t \ge 0) ; (Y_{t \wedge T} ; t \ge 0)\big)$ is uniformly integrable. The main results obtained in the paper are: \begin{itemize} \item[1)] if $T$ and $B_T$ are independent and $T$ has all exponential moments, then $T$ is constant. \item[2)] If $X_T$ and $Y_T$ are independent and have all exponential moments, then $X_T$ and $Y_T$ are Gaussian. \end{itemize} We also give a number of examples of stopping times $T$, with only some exponential moments, such that $T$ and $B_T$ are independent, and similarly for $X_T$ and $Y_T$. We also exhibit bounded non-constant stopping times $T$ such that $X_T$ and $Y_T$ are independent and Gaussian.

We characterize the set of pointwise multipliers of the Besov spaces $B^0_{\infty,1}$ and $B^0_{\infty,\infty} $. These characterizations are used to obtain regularity results for elliptic partial differential equations. In addition several counterexamples are provided and the relation of various spaces of continuous functions to these multiplier classes are studied.

In this paper we construct quasiconformal mappings between Y-pieces so that the corresponding Beltrami coefficient has exponential decay away from the boundary. These maps are used in a companion paper to construct quasiFuchsian groups whose limit sets are non-rectifiable curves of dimension 1.

We construct quasiconformal deformations of convergence type Fuchsian groups such that the resulting limit set is a Jordan curve of Hausdorff dimension 1, but having tangents almost nowhere. It is known that no divergence type group has such a deformation. The main tools in this construction are (1) a characterization of tangent points in terms of Peter Jones' $\beta$'s, (2) a result of Stephen Semmes that gives a Carleson type condition on a Beltrami coefficient which implies rectifiability and (3) a construction of quasiconformal deformations of a surface which shrink a given geodesic and whose dilatations satisfy an exponential decay estimate away from the geodesic.

Our main objective is to study the pointwise behaviour of Sobolev functions on a metric measure space. We prove that a Sobolev function has Lebesgue points outside a set of capacity zero if the measure is doubling. This result seems to be new even for the weighted Sobolev spaces on Euclidean spaces. The crucial ingredient of our argument is a maximal function related to discrete convolution approximations. In particular, we do not use the Besicovitch covering theorem, extension theorems or representation formulas for Sobolev functions.

We establish the $L^p$ boundedness of singular integrals with kernels which belong to block spaces and are supported by subvarieties.

Existence of global weak solutions to the discrete coagulation-fragmentation equations with diffusion is proved under general assumptions on the coagulation and fragmentation coefficients. Unlike previous works requiring $L^\infty$-estimates, an $L^1$-approach is developed here which relies on weak and strong compactness methods in $L^1$.

In this paper we will prove that if $G$ is a finite group, $X$ a subnormal subgroup of $ X F^*(G)$ such that $X F^*(G)$ is quasinilpotent and $Y$ is a quasinilpotent subgroup of $N_G(X)$, then $Y F^*(N_G(X})$ is quasinilpotent if and only if $Y F^*(G)$ is quasinilpotent. Also we will obtain that $F^*{G}$ controls its own fusion in $G$ if and only if $G=F^*{G}$.