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Séverine Biard

Publications and source records attributed to Séverine Biard.

7 recordsLinked to original sources
arXiv2025

Equivalence between VMO functions and plurisubharmonic functions with zero Lelong numbers

We prove that a plurisubharmonic function on a domain in the complex Euclidean space is a locally VMO (Vanishing Mean Oscillation) function if and only if its Lelong number at each point vanishes. We also give a global version of this result when the boundary of the domain satisfies the \textit{interior sphere condition}. An example emphasizes the importance of this condition. These equivalences contribute to a better understanding of the behavior of singular plurisubharmonic functions. We end the paper by discussing the link between the residual Monge-Ampère mass and VMO functions, by providing examples.

math.CV↗
arXiv2022

A residue formula for meromorphic connections and applications to stable sets of foliations

We discuss residue formulae that localize the first Chern class of a line bundle to the singular locus of a given holomorphic connection. As an application, we explain a proof for Brunella's conjecture about exceptional minimal sets of codimension one holomorphic foliations with ample normal bundle and for a nonexistence theorem of Levi flat hypersurfaces with transversely affine Levi foliation in compact Kähler surfaces.

math.CV↗
arXiv2020

Weighted $L^2$ version of Mergelyan and Carleman approximation

We study the density of polynomials in $H^2(E,φ)$, the space of square integrable functions with respect to $e^{-φ}dm$ and holomorphic on the interior of $E$ in $\mathbb{C}$, where $φ$ is a subharmonic function and $dm$ is a measure on $E$. We give a result where $E$ is the union of a Lipschitz graph and a Carathéodory domain, that we state as a weighted $L^2$-version of the Mergelyan theorem. We also prove a weighted $L^2$-version of the Carleman theorem. Keywords: Mergelyan theorem, Carleman theorem, Weighted $L^2$- spaces, Rectifiable non-Lipschitz arc

math.CV↗
arXiv2019

Weighted-$L^2$ polynomial approximation in $\mathbb{C}$

We study the density of polynomials in $H^2(Ω,e^{-φ})$, the space of square integrable holomorphic functions in a bounded domain $Ω$ in $\mathbb{C}$, where $φ$ is a subharmonic function. In particular, we prove that the density holds in Carathéodory domains for any subharmonic function $φ$ in a neighborhood of $\overlineΩ$. In non-Carathéodory domains, we prove that the density depends on the weight function, giving examples.

math.CV↗
arXiv2017

Estimates for the complex Green operator: symmetry, percolation, and interpolation

Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. We show that Sobolev estimates for the complex Green operator hold simultaneously for forms of symmetric bidegrees, that is, they hold for $(p,q)$--forms if and only if they hold for $(m-p,m-1-q)$--forms. Here $m$ equals the CR dimension of $M$ plus one. Symmetries of this type are known to hold for compactness estimates. We further show that with the usual microlocalization, compactness estimates for the positive part percolate up the complex, i.e. if they hold for $(p,q)$--forms, they also hold for $(p,q+1)$--forms. Similarly, compactness estimates for the negative part percolate down the complex. As a result, if the complex Green operator is compact on $(p,q_{1})$--forms and on $(p,q_{2})$--forms ($q_{1}\leq q_{2}$), then it is compact on $(p,q)$--forms for $q_{1}\leq q\leq q_{2}$. It is interesting to contrast this behavior of the complex Green operator with that of the $\overline{\partial}$--Neumann operator on a pseudoconvex domain.

math.CV↗
arXiv2017

$L^{2}$-Sobolev theory for the complex Green operator

These notes are concerned with the $L^{2}$-Sobolev theory of the complex Green operator on pseudoconvex, oriented, bounded and closed CR--submanifolds of $\mathbb{C}^{n}$ of hypersurface type. This class of submanifolds generalizes that of boundaries of pseudoconvex domains. We first discuss briefly the CR--geometry of general CR--submanifolds and then specialize to this class. Next, we review the basic $L^{2}$-theory of the tangential Cauchy-Riemann operator and the associated complex Green operator(s) on these submanifolds. After these preparations, we discuss recent results on compactness and regularity in Sobolev spaces of the complex Green operator(s).

math.CV↗