The notion of indicator of an analytic function, that describes the function’s growth along rays, was introduced by Phragmen and Lindelöf. Trigonometric convexity is a defining property of the indicator. For multivariate cases, an analogous property of trigonometric convexity was not known so far. We prove the property of trigonometric convexity for the indicator of multivariate analytic functions, introduced by Ivanov. The results that we obtain are sharp. Derivation of a multidimensional analogue of the inverse Fourier transform in a sector and obtaining estimates on its decay is an important step of our proof.

Let $(\mathbb {D}^2,\mathscr {F},\{0\})$ be a singular holomorphic foliation on the unit bidisc $\mathbb {D}^2$ defined by the linear vector field

$$ \begin{align*} z \frac{\partial}{\partial z}+ \unicode{x3bb} w \frac{\partial}{\partial w}, \end{align*} $$

where $\unicode{x3bb} \in \mathbb {C}^*$ . Such a foliation has a non-degenerate singularity at the origin ${0:=(0,0) \in \mathbb {C}^2}$ . Let T be a harmonic current directed by $\mathscr {F}$ which does not give mass to any of the two separatrices $(z=0)$ and $(w=0)$ . Assume $T\neq 0$ . The Lelong number of T at $0$ describes the mass distribution on the foliated space. In 2014 Nguyên (see [16]) proved that when $\unicode{x3bb} \notin \mathbb {R}$ , that is, when $0$ is a hyperbolic singularity, the Lelong number at $0$ vanishes. Suppose the trivial extension $\tilde {T}$ across $0$ is $dd^c$ -closed. For the non-hyperbolic case $\unicode{x3bb} \in \mathbb {R}^*$ , we prove that the Lelong number at $0$ :

1. (1) is strictly positive if $\unicode{x3bb}>0$ ;

2. (2) vanishes if $\unicode{x3bb} \in \mathbb {Q}_{<0}$ ;

3. (3) vanishes if $\unicode{x3bb} <0$ and T is invariant under the action of some cofinite subgroup of the monodromy group.

This paper deals with the analysis of the singularities arising from the solutions of the problem ${-}\,{\rm Curl\ } F=\mu $, where F is a 3 × 3 matrix-valued Lp-function ($1\les p<2$) and μ a 3 × 3 matrix-valued Radon measure concentrated in a closed loop in Ω ⊂ ℝ3, or in a network of such loops (as, for instance, dislocation clusters as observed in single crystals). In particular, we study the topological nature of such dislocation singularities. It is shown that $F=\nabla u$, the absolutely continuous part of the distributional gradient Du of a vector-valued function u of special bounded variation. Furthermore, u can also be seen as a multi-valued field, that is, can be redefined with values in the three-dimensional flat torus 𝕋3 and hence is Sobolev-regular away from the singular loops. We then analyse the graphs of such maps represented as currents in Ω × 𝕋3 and show that their boundaries can be written in term of the measure μ. Readapting some well-known results for Cartesian currents, we recover closure and compactness properties of the class of maps with bounded curl concentrated on dislocation networks. In the spirit of previous work, we finally give some examples of variational problems where such results provide existence of solutions.

In this paper, we study a problem of extension of holomorphic functions given on a complex hypersurface with singularities on the boundary of a strictly pseudoconvex domain.