In this paper, we give a sharp estimate on the dimension of the space of polynomial growth harmonic functions with fixed degree on a complete Riemannian manifold, under various assumptions.

We study stochastic Hamilton-Jacobi-Bellman equations and the corresponding Hamiltonian systems driven by jump-type Lévy processes. The main objective of the present paper is to show existence, uniqueness and a (locally in time) diffeomorphism property of the solution: the solution trajectory of the system is a diffeomorphism as a function of the initial impulse. This result enables us to implement a stochastic version of the classical method of characteristics for the Hamilton-Jacobi equations. An -in itself interesting- auxiliary result are pointwise a.s. estimates for iterated stochastic integrals driven by a vector of not necessarily independent jump-type semimartingales.

We prove that a set containing translates of every 2-plane must have full Hausdorff dimension.

Let $\{B^{(\alpha)}(t)\}_{t\in\mathbb{R}^{d}}$ be the Fractional Brownian Sheet with multi-index $\alpha=(\alpha_1,\ldots, \alpha_d)$, $0< \alpha_i< 1$. In \cite{Kamont1996}, Kamont has shown that, with probability $1$, the box dimension of the graph of a trajectory of this Gaussian field, over a non-degenerate cube $Q\subset\mathbb{R}^{d}$ is equal to $d+1-\min(\alpha_1,\ldots,\alpha_d)$. In this paper, we prove that this result remains true when the box dimension is replaced by the Hausdorff dimension or the packing dimension.

It is well known that a group $G = AB$ which is the product of two supersoluble subgroups $A$ and $B$ is not supersoluble in general. Under suitable permutability conditions on $A$ and $B$, we show that for any minimal normal subgroup $N$ both $AN$ and $BN$ are supersoluble. We then exploit this to establish some sufficient conditions for $G$ to be supersoluble.

An embedding theorem for the Orlicz-Sobolev space $W^{1,A}_{0}(G)$, $G\subset \mathbb{R}^n$, into a space of Orlicz-Lorentz type is established for any given Young function $A$. Such a space is shown to be the best possible among all rearrangement invariant spaces. A version of the theorem for anisotropic spaces is also exhibited. In particular, our results recover and provide a unified framework for various well-known Sobolev type embeddings, including the classical inequalities for the standard Sobolev space $W^{1,p}_{0}(G)$ by O'Neil and by Peetre ($1\leq p< n$), and by Brezis-Wainger and by Hansson ($p=n$).

The new construction given by Barton and Sudbery of the Freudenthal-Tits magic square, which includes the exceptional classical simple Lie algebras, will be interpreted and extended by using a pair of symmetric composition algebras, instead of the standard unital composition algebras.

We study the boundedness of the maximal operator, potential type operators and operators with fixed singularity (of Hardy and Hankel type) in the spaces $L^{p(\cdot)}(\rho,\Omega)$ over a bounded open set in $\mathbb{R}^n$ with a power weight $\rho(x)=|x-x_0|^\gamma$, $x_0\in \overline{\Omega}$, and an exponent $p(x)$ satisfying the Dini-Lipschitz condition.

We find the minimal real number $k$ such that the $k$th power of the Fourier transform of any continuous, orbital measure on a classical, compact Lie group belongs to $l^{2}$. This results from an investigation of the pointwise behaviour of characters on these groups. An application is given to the study of $L^{p}$-improving measures.

We establish $L^p$-boundedness for a class of product singular integral operators on spaces $\widetilde{M} = M_1 \times M_2\times \cdots \times M_n$. Each factor space $M_i$ is a smooth manifold on which the basic geometry is given by a control, or Carnot-Caratheodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on $M_i$ are non-isotropic smoothing operators of order zero. The boundedness of the product operators is then a consequence of a natural Littlewood-Paley theory on $\widetilde M$. This in turn is a consequence of a corresponding theory on each factor space. The square function for this theory is constructed from the heat kernel for the sub-Laplacian on each factor.

If $f\in L^\infty(\mathbb{D})$ let $T_f$ be the Toeplitz operator on the Bergman space $L^2_a$ of the unit disk $\mathbb{D}$. For a $C^\ast$-algebra $A\subset L^\infty(\mathbb{D})$ let $\mathfrak{T}(A)$ denote the closed operator algebra generated by $\{ T_f : f\in A \}$. We characterize its commutator ideal $\comm(A)$ and the quotient $\mathfrak{T}(A)/ \mathfrak{C}(A)$ for a wide class of algebras $A$. Also, for $n\geq 0$ integer, we define the $n$-Berezin transform $B_nS$ of a bounded operator $S$, and prove that if $f\in L^\infty(\mathbb{D})$ and $f_n = B_n T_f$ then $T_{f_n} \rightarrow T_f$.

Given a $C^{1}$ function $H: \mathbb{R}^3 \to \mathbb{R}$, we look for $H$-bubbles, i.e., surfaces in $\mathbb{R}^3$ parametrized by the sphere $\mathbb{S}^2$ with mean curvature $H$ at every regular point. Here we study the case $H(u)=H_{0}(u)+\epsilon H_{1}(u)$ where $H_{0}$ is some "good" curvature (for which there exist $H_{0}$-bubbles with minimal energy, uniformly bounded in $L^{\infty}$), $\epsilon$ is the smallness parameter, and $H_{1}$ is {\em any} $C^{1}$ function.