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Luca Baracco

Publications and source records attributed to Luca Baracco.

30 records · Page 2Linked to original sources
arXiv2015

Extension of L^2 holomorphic functions

The purpose of this note is to show that the di-bar-estimate which is needed in the Ohsawa-Takegoshi Extension Theorem [6] is a direct consequence of the Hormander-Kohn-Morrey weigthed inequality. In this inequality, the Donnelly-Fefferman argument is not required and a single 1-parameter family of non-singular weights is used. This paper is the furtherst step of a great deal of work devoted to the simplification of the original proof of Ohsawa-Takegoshi Theorem, among other papers on the subject, we mention [1] and [8] which are based on "twisted" basic estimates and, in recent time, [3] and [9].

math.CV↗
arXiv2014

Gain/Loss of derivatives for complex vector fields

In $\C_z\times\R_t$ we consider the function $g=g(z)$, set $g_1=\di_z g$, $g_{1\bar 1}=\di_z\dib_zg$ and define the operator $L_g=\di_z+ig_1\di_t$. We discuss estimates with loss of derivatives, in the sense of Kohn, for the system $(\bar L_g,f^kL_g)$ where $(\bar L_g,L_g)$ is $\frac1{2m} $ subelliptic at 0 and $f(0)=0,\,\,df(0)\neq0$. We prove estimates with a loss $l=\frac{k-1}{2m} $ if the "multiplier" condition $|f|\simgeq |g_{1\bar 1}|^{\frac1{2(m-1)}}$ is fulfilled. (For estimates without cut-off, subellipticity can be weakened to compactness and this results in a loss of $l=\frac [{2(m-1)}$.) For the choice $(g,f^k)=(|z|^{2m},\bar z^k)$ this result was obtained by Kohn and Bove-Derridj-Kohn-Tartakoff for $m=1$ and $m\geq1$ respectively. Also, the loss $l=\frac{k-1}{2m}$ was proven to be optimal. We show that it remains optimal for the model $(g,f^k)=(x^{2m},x^k)$. Instead, for the model $(g,f^k)=(|z|^{2m},x^k)$, in which the multiplier condition is violated, the loss is not lowered by the type and must be $\geq \frac{k-1}2$.

math.CV↗
arXiv2014

Local regularity of the Green operator in a CR manifold of general "type"

It is here proved that if a pseudoconvex CR manifold $M$ of hypersurface type has a certain "type", that we quantify by a vanishing rate $F$ at a submanifold of CR dimension $0$, then $\Box_b$ "gains $f^2$ derivatives" where $f$ is defined by inversion of $F$. Indeed the estimate is more accurate and it involves the Levi form of $M$ and of additional weights, instead of $\Box_b$. Next a general tangential estimate, "twisted" by a pseudodifferential operator $Ψ$ is established. The combination of the two yields a general "$f$-estimate" twisted by $Ψ$. We apply the twisted estimate for $Ψ$ which is the composition of a cut-off $η$ with a differentiation of order $s$ such as $R^s$ of Section 4. Under the assumption that $[\partial_b,η]$ and $[\partial_b,[\bar\partial_b,η]]$ are superlogarithmic multipliers in a sense inspired to Kohn, we get the local regularity of the Green operator $G=\Box_b^{-1}$. In particular, if $M$ has "infraexponential type" along $S\setminusΓ$ where $S$ is a manifold of CR dimension $0$ and $Γ$ a curve transversal to $T^{\mathbb C} M$, then we have local regularity of $G$. This gives an immediate proof of former work by Baracco, Khanh, Zampieri and by Kohn. The conclusion extends to "block decomposed" domains for whose blocks the above hypotheses hold separately. In the application of Section 4, $Ψ$ is composed by a cut off $η$ and a differentiation of order $s$ such as $Λ^s$ or $R^s$ and $M$ is a decoupled hypersurface which has infraexponential type along the coordinate lines $\mathbb R_{x_j}\setminus\{0\}$ and whose equations have differentials which are superlogarithmic multipliers in the sense of Kohn. In this situation, $\Box_b$ is locally hypoelliptic.

math.CV↗
arXiv2012

Boundaries of analytic varieties

We prove that every smooth CR manifold $M\subset\subset \C^n$, of hypersurface type, has a complex strip-manifold extension in $\C^n$. If $M$ is, in addition, pseudoconvex-oriented, it is the "exterior" boundary of the strip. In turn, the strip extends to a variety with boundary $M$ (Rothstein-Sperling Theorem); in case $M$ is contained in a pseudoconvex boundary with no complex tangencies, the variety is embedded in $\C^n$. Altogether we get: $M$ is the boundary of a variety (Harvey-Lawson Theorem); if $M$ is pseudoconvex oriented the singularities of the variety are isolated in the interior; if $M$ lies in a pseudoconvex boundary, the variety is embedded in $\C^n$ (and is still smooth at $M$)

math.CV↗
arXiv2010

Propagation of CR extendibility at the vertex of a complex sector

We study propagation of CR extendibility at the vertex $p$ of an analytic sector $A$ contained in a CR manifold $M$. Let $k$ be the weighted vanishing order of $M$ and $α$ the complex angle of $A$ at $p$. Propagation takes place if and only if $α>\frac1k$.

math.CV↗
arXiv2005

$CR$ Extension from manifolds of higher type

In this paper, a generalization of the "sector property" theorem first pioneered by Baouendi, Rothschild and Treves is given. The main contribution consists in showing that if a submanifold of $\C^n$ with higher codimension is locally presented in a weighted normal form, similar to that described by Bloom and Graham, then a characterization is given as to when a vector in the tangent space belongs to the analytic wave front set for the set of locally defined CR functions on the submanifold. This characterization is described in terms of the sign of the inner product of this vector with the local graphing functions for the submanifold on sectors of suitable size along complex lines in its tangent space. Examples are given to show that under certain circumstances, the results are sharp. Previous results by Baouendi et. al. contained a semi-rigidity assumption which is not assumed in the present paper. The hypoanalytic wave front set determines the cone of directions in which CR functions extend analytically to the ambient space and thus provides an explicit description of the local hull of holomorphy of a submanifold of $\C^n$.

math.CV↗
arXiv2005

Extremal discs and the holomorphic extension from convex hypersurfaces

Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f holomorphically extends to D. The result partially answers a conjecture by Globevnik and Stout of 1991.

math.CV↗