Searcharxiv⌕ Search

arXiv subjects

Stefano Pinton

Publications and source records attributed to Stefano Pinton.

35 records · Page 2Linked to original sources
arXiv2020

Infinite order differential operators acting on entire hyperholomorphic functions

Infinite order differential operators appear in different fields of Mathematics and Physics and in the last decades they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger equation. Inspired by the operators arising in quantum mechanics, in this paper we investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions. The two classes of hyperholomorphic functions, that constitute a natural extension of functions ofone complex variable to functions of paravector variables are illustrated by the Fueter-Sce-Qian mapping theorem. We show that, even though the two notions of hyperholomorphic functions are quite different from each other, entire hyperholomorphic functions with exponential bounds play a crucial role in the continuity of infinite order differential operators acting on these two classes of entire hyperholomorphic functions. We point out the remarkable fact that the exponential function of a paravector variable is not in the kernel of the Dirac operator but entire monogenic functions with exponential bounds play an important role in the theory.

math.FA↗
arXiv2019

An extension theorem for regular functions of two quaternionic variables

For functions of two quaternionic variables that are regular in the sense of Fueter, we establish a result similar in spirit to the Hanges and Trèves theorem. Namely, we show that a ball contained in the boundary of a domain is a propagator of regular extendability across the boundary.

math.CV↗
arXiv2019

On the Ekeland-Hofer symplectic capacities of the real bidisc

In $\mathbb{C}^2$ with the standard symplectic structure we consider the bidisc $D^2\times D^2$ constructed as the product of two open real discs of radius $1$. We compute explicit values for the first, second and third Ekeland-Hofer symplectic capacity of $D^2\times D^2$. We discuss some applications to questions of symplectic rigidity.

math.CV↗
arXiv2017

Hölder regularity of the solution to the complex Monge-Ampère equation with $L^p$ density

On a smooth domain $Ω\subset\subset\mathbb C^n$, we consider the Dirichlet problem for the complex Monge-Ampère equation $((dd^cu)^n=fdV,\,u|_{bΩ}\equivϕ)$. We state the Hölder regularity of the solution $u$ when the boundary value $ϕ$ is Hölder continuous and the density $f$ is only $L^p$, $p>1$. Note that in former literature (Guedj-Kolodziej-Zeriahi) the weakness of the assumption $f\in L^p$ was balanced by taking $ϕ\in C^{1,1}$ (in addition to assuming $Ω$ strongly pseudoconvex).

math.CV↗
arXiv2017

The complex Monge-Ampère equation on weakly pseudoconvex domains

We show here a "weak" Hölder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equation with data in the $L^p$ space and the boundary of the domain satisfying an $f$-property. The $f$-property is a potential-theoretical condition which holds for all pseudoconvex domains of finite type and many examples of infinite type.

math.CV↗
arXiv2015

Extension of L^2, di-bar-closed, forms

We prove extension of a di-bar-closed, smooth, form from the intersection of a pseudoconvex domain with a complex hyperplane to the whole domain. The extension form is di-bar-closed, has harmonic coefficients and its L^2-norm is estimated by the L^2-norm of the trace. For holomorphic functions this is proved by Ohsawa-Takegoshi [12]. For forms of higher degree, this is stated by Manivel [9]. It seems, however, that the proof contains a gap because of the use of a a singular weight and the failure of regularity for the solution of the related di-bar-equation. There is a rich literature on the subject (cf. among otheres [7], [14]) but it does not seem to contain complete answer to the question. Also, the problem of extending cohomology classes of di-bar of higher degree in a compact Kahler space is addressed in [8] and [3]. Apart from the formal analogy, this has little in common with our problem in which these classes are 0. We also believe, comparing to the preceding literature, that our approach is original and, somewhat, simpler.

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↗
arXiv2013

Propagation at the vertex of a sector

We discuss holomorphic extension across a boundary point in terms of sector property. The point is of infinite type and the sector is accordingly "cusped" at the vertex.

math.CV↗
arXiv2012

Complex Manifolds In $Q$-Convex Boundaries

We consider a smooth boundary bΩwhich is q-convex in the sense that its Levi-form has positive trace on every complex q-plane. We prove that bΩis tangent of infinite order to the complexification of each of its submanifolds which is complex tangential and of finite bracket type. This generalizes Diederich-Fornaess [Annals 1978] from pseudoconvex to q-convex domains. We also readily prove that the rows of the Levi-form are (1/2)-subelliptic multipliers for the di-bar-Neumann problem on q-forms (cf. Ho [Math. Ann. 1991]). This allows to run the Kohn algorithm of [Acta Math. 1979] in the chain of ideals of subelliptic multipliers for q-forms. If bΩis real analytic and the algorithm stucks on q-forms, then it produces a variety of holomorphic dimension q, and in fact, by our result above, a complex q-manifold which is not only tangent but indeed contained in bΩ. Altogether, the absence of complex q-manifolds in bΩproduces a subelliptic estimate on q-forms.

math.CV↗
arXiv2011

The Diederich-Fornaess index and the global regularity of the di-bar-Neumann problem

We describe along the guidelines of Kohn "Quantitative estimates..." (1999), the constant E_s which is needed to control the commutator of a totally real vector field T with di-bar* in order to have Sobolev s-regularity of the Bergman projection in any degree of forms, on a smooth pseudoconvex domain D of the complex space. This statement, not explicit in Kohn's paper, yields Straube's Theorem in "A sufficient condition..." (2008). Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function r of D, the operators (T^+)^{-delta/2} and (-r)^{delta/2}. We are thus able to extend to general degree of forms the conclusion of Kohn which only holds for functions: if for the Diederich-Fornaess index delta of D, we have that (1-δ)^{1/2} < E_s, then the Bergman projection is s-regular.

math.CV↗
arXiv2010

Compactness of $\Box_b$ in a CR manifold

This note is aimed at simplifying current literature about compactness estimates for the Kohn-Laplacian on CR manifolds. The approach consists in a tangential basic estimate in the formulation given by the first author in \cite{Kh10} which refines former work by Nicoara \cite{N06}. It has been proved by Raich \cite{R10} that on a CR manifold of dimension $2n-1$ which is compact pseudoconvex of hypersurface type embedded in $\C^n$ and orientable, the property named "$(CR-P_q)$" for $1\leq q\leq \frac{n-1}2$, a generalization of the one introduced by Catlin in \cite{C84}, implies compactness estimates for the Kohn-Laplacian $\Box_b$ in degree $k$ for any $k$ satisfying $q\leq k\leq n-1-q$. The same result is stated by Straube in \cite{S10} without the assumption of orientability. We regain these results by a simplified method and extend the conclusions in two directions. First, the CR manifold is no longer required to be embedded. Second, when $(CR-P_q)$ holds for $q=1$ (and, in case $n=1$, under the additional hypothesis that $\dib_b$ has closed range on functions) we prove compactness also in the critical degrees $k=0$ and $k=n-1$.

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↗