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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.

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball $$\Delta u = \lambda_{+\chi_{\{ u>0 \}}} - \lambda_{-\chi_{\{ u<0 \}}},\quad\lambda_{\pm}>0. $$ We prove that the free boundary touches the fixed boundary (uniformly) tangentially if the boundary data$f$and its first and second derivatives vanish at the touch-point.

We study the problem of density of polynomials in the de Branges spaces ℋ($E$) of entire functions and obtain conditions (in terms of the distribution of the zeros of the generating function$E$) ensuring that the polynomials belong to the space ℋ($E$) or are dense in this space. We discuss the relation of these results with the recent paper of V. P. Havin and J. Mashreghi on majorants for the shift-coinvariant subspaces. Also, it is shown that the density of polynomials implies the hypercyclicity of translation operators in ℋ($E$).

We find a real analytic Levi-flat hypersurface in$C$^{2}containing a bounded contractible domain which is a determining set for pluriharmonic functions.

Let $\mathcal{D}$ be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We give a new sufficient condition, not far from the known necessary condition, for a function$f$∈ $\mathcal{D}$ to be$cyclic$, i.e. for {$pf$:$p$is a polynomial} to be dense in $\mathcal{D}$ .

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We study spectral and scattering properties of the Laplacian$H$^{(σ)}=-Δ in $L_2(\mathbf{R}^{d+1}_+)$ corresponding to the boundary condition $\frac{\partial u}{\partial\nu} + \sigma u = 0$ with a periodic function σ. For non-negative σ we prove that$H$^{(σ)}is unitarily equivalent to the Neumann Laplacian$H$^{(0)}. In general, there appear additional channels of scattering due to surface states. We prove absolute continuity of the spectrum of$H$^{(σ)}under mild assumptions on σ.

The spectral order on$R$^{$n$}induces a natural partial ordering on the manifold $\mathcal{H}_{n}$ of monic hyperbolic polynomials of degree$n$. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space ℓ^{∞}of real bounded sequences. As a consequence, we deduce that the monoid $\mathcal{A}^{\prime}$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\mathcal{A}^{\prime}$ -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.

Exact propagators are obtained for the degenerate second order hyperbolic operators ∂^{2}_{$t$}-$t$^{2$l$}Δ_{$x$},$l$=1,2,..., by analytic continuation from the degenerate elliptic operators ∂^{2}_{$t$}+$t$^{2$l$}Δ_{$x$}. The partial Fourier transforms are also obtained in closed form, leading to integral transform formulas for certain combinations of Bessel functions and modified Bessel functions.

We give sharp estimates for the fractional maximal function in terms of Hausdorff capacity. At the same time we identify the real interpolation spaces between$L$_{1}and the Morrey space $\mathcal{L}^{1,\lambda}$ . The result can be viewed as an analogue of the Hardy–Littlewood maximal theorem for the fractional maximal function.