We show that if the graph of an analytic function in the unit disk$D$is not complete pluripolar in$C$^{2}then the projection of its pluripolar hull contains a fine neighborhood of a point $p\in\partial\mathbf{D}$ . Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function$f$in$D$extends to a function ℱ which is defined on a fine neighborhood of a point $p\in\partial\mathbf{D}$ and is finely analytic at$p$then the pluripolar hull of the graph of$f$contains the graph of ℱ over a smaller fine neighborhood of$p$. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes of analytic functions in the disk which have non-trivial pluripolar hulls, among them$C$^{∞}functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls. Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation.

We give a sufficient condition on a Lévy measure$μ$which ensures that the generator$L$of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., $\mathop {\mathrm {sing\,supp}}u\subseteq \mathop {\mathrm {sing\,supp}}Lu$ for all admissible$u$. In particular, we assume that $\mu|_{\mathbb {R}^{d}\setminus \{0\}}\in C^{\infty}(\mathbb {R}^{d}\setminus \{0\})$ . We also show that this condition is necessary provided that $\mathop {\mathrm {supp}}\mu$ is compact.

We show that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.

We prove that certain parabolic Kazhdan-Lusztig polynomials calculate the graded decomposition matrices of v-Schur algebras given by the Jantzen filtration of Weyl modules, confirming a conjecture of Leclerc and Thibon.

In this paper the existence of positive 2\pi -periodic solutions to the ordinary differential equation \begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}is studied, where 2\pi is a positive 2\pi -periodic smooth function. By virtue of a new generalized Blaschke鈥揝antal贸 inequality, we obtain a new existence result of solutions.