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• 1.
Université du Luxembourg.
SUNY Stony Brook, New York. Universitá Roma Tor Vergata. Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Complex vector fields and hypolleptic partial differential operators2010In: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 60, no 3, p. 39p. 987-1034Article in journal (Refereed)

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for CR manifolds, that was first introduced by two of the authors; and the Hormander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of (0,1) vector fields satisfies a subelliptic estimate.

• 2. Altomani, Andrea
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Holomorphic Extension from Weakly Pseudoconcave CR Manifolds2010In: Rendiconti del Seminario Matematico della Universita di Padova, ISSN 0041-8994, E-ISSN 2240-2926, Vol. 123, p. 69-90Article in journal (Refereed)

Let M be a smooth locally embeddable CR manifold, having some CR dimension M and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition, but which are not pseudoconcave in the strong sense. These results demonstrate that for codimension d > 1 there are additional phenomena, which are invisible when d = 1.

• 3.
Sapienza Università di Roma, Rome, Italy.
Sapienza Università di Roma, Rome, Italy. Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education. Uniwersytet Jana Kochanowskiego W Kielcach, Kielce, Poland. Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education. Uniwersytet Jana Kochanowskiego W Kielcach, Kielce, Poland.
A novel noncommutative KdV-type equation, its recursion operator, and solitons2018In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 59, no 4, article id 043501Article in journal (Refereed)

A noncommutative KdV-type equation is introduced extending the Bäcklund chart in Carillo et al. [Symmetry Integrability Geom.: Methods Appl. 12, 087 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in Olver and Sokolov [Commun. Math. Phys. 193, 245 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies, and an explicit solution class are derived.

• 4.
Institut Fourier Grenoble.
Analytic Extension from Non-Pseudoconvex Boundaries and A(D)-Convexity2003In: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 53, no 3, p. 847-857Article in journal (Refereed)
• 5.
SUNY Stony Brook, New York.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
The H-Principle and Pseudoconcave CR Manifolds2005In: Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627, Vol. 50, p. 117-125Article in journal (Refereed)
• 6.
Uppsala universitet.
Hulls and analytic extension from non-pseudoconvex boundaries2002Report (Other academic)
• 7.
Uppsala universitet.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
On the Continuity Principle2007In: Asian Journal of Mathematics, ISSN 1093-6106, E-ISSN 1945-0036, Vol. 11, no 1, p. 167-178Article in journal (Refereed)
• 8.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics. Uniwersytet Jana Kochanowskiego Kielcach, Inst Matemat, Kielce, Poland.
On thickening of holomorphic hulls and envelopes of holomorphy on Stein spaces2016In: International Journal of Mathematics, ISSN 0129-167X, Vol. 27, no 6, article id 1650051Article in journal (Refereed)

On a normal Stein variety X, we study the thickening problem, i.e. the problem whether the assumption that a compact set K is contained in the interior of another compact set, L implies that the same inclusion holds for their holomorphic hulls. An affirmative answer is given for X with isolated quotient singularities. On any Stein space X with isolated singularities, we prove thickening for those hulls which have analytic structure at the singular points, obtaining a limitation for possible counter-examples. In dimension 2, we finally relate the holomorphic hulls to analytic extension from parts of strictly pseudoconvex boundaries.

• 9. Merker, Joel
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
A morse-theoretical proof of the Hartogs extension theorem2007In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 17, no 3, p. 513-546Article in journal (Refereed)

One hundered years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was understood later: Holomorphic functions in a connected neighborhood nu(partial derivative Omega) of a connected boundary a partial derivative Omega C subset of C-n (n >= 2) do extend holomorphically and uniquely to the domain Omega. Martinelli, in the early 1940's, and Ehrenpreis in 1961 obtained a rigorous proof using a new multidimensional integral kernel or a short argument, but it remained unclear how to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906), and E E. Levi (1911) in some special, model cases. In fact, known attempts (e.g., Osgood, 1929, Brown, 1936) struggled for monodromy against multivaluations, but failed to get the general global theorem.

Moreover, quite unexpectedly, in 1998, Fornaess exhibited a topologically strange (nonpseudoconvex) domain Omega(F) subset of C-2 that cannot befilled in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside Omega F. However one should point out that the standard, unrestricted disc method usually allows discs to go outside the domain (just think of Levi pseudoconcavity).

Using the method of analytic discs for local extensional steps and Morse-theoretical tools for the global topological control of monodromy, we show that the Hartogs extension theorem can be established in such a way.

• 10.
Ecole Normale Superiore Paris.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
Characteristic Foliations on Maximally Real Submanifolds of C-n and removable singularities for CR functions2006In: International Mathematics Research Papers, ISSN 1687-3017, E-ISSN 1687-3009, Vol. 2006, p. 1-131Article in journal (Refereed)
• 11.
Ecole Normale Superiore Paris.
Enveloppe d'Holomorphie Locale des Variétés CR et l'Élimination des Singularités pour les Fonctions CR Intégrables1999In: Comptes rendus de l'Académie des sciences. Série 1, Mathématique, ISSN 0764-4442, E-ISSN 1778-3577, Vol. 328, no 10, p. 853-858Article in journal (Refereed)

Let M be a locally embeddable CR manifold and Phi subset of M be a closed set. We give sufficient conditions in order that L-loc(1) functions on M which are CR on M \ Phi are CR on M. (C) Academie des Sciences/Elsevier, Paris.

• 12.
Ecole Normale Superiore Paris.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
Holomorphic Extension of CR Functions, Envelopes of Holomorphy, and Removable Singularities2006Book (Other academic)
• 13.
Ecole Normale Superiore Paris.
Metrically Thin Singularities of Integrable CR Functions2000In: International Journal of Mathematics, ISSN 0129-167X, Vol. 11, no 7, p. 857-872Article in journal (Refereed)

In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a smoothly embedded Cauchy-Riemann manifold M (CR functions on M \ E). Our main result establishes the removability of E within the space of locally integrable functions on M, which are CR on M\E, under the hypothesis that the (dim M - 2)-dimensional Hausdorff volume of E is zero and that the CR-orbits of M and of M \ E are comparable. [ABSTRACT FROM AUTHOR

• 14.
Ecole Normale Superiore Paris.
On Removable singularities for integrable CR functions1999In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 48, no 3, p. 805-856Article in journal (Refereed)

Let $M$ be a locally embeddable CR manifold. One defines, for $p\geq 1$, $$L^p_{\rm loc,CR}(M)=\{f\in L^p_{\rm loc}(M)| f {\rm is CR on}\ M\},$$ $$L^p_{\rm loc,CR}(M-E)=\{f\in L^p_{\rm loc}(M)| f {\rm is CR on}\ M-E\},$$ with $E$ a closed subset of $M$. This paper gives conditions under which $L^p_{\rm loc,CR}(M)=L^p_{\rm loc,CR}(M-E)$. In order to state results, let us recall the following definitions. Definition 1: The CR orbit of $p\in M$ is defined as $O_{\rm CR}(M,p)=\{q\in M|$ there exists a piecewise smooth integral curve of $T^cM$ with origin $p$ and target $q\}$. By Sussmann's theorem, $O_{\rm CR}(M,p)$ has the structure of a smooth immersed submanifold of $M$ [see M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Univ. Press, Princeton, NJ, 1999; MR1668103 (2000b:32066)]. Definition 2: $M$ is globally minimal if $M$ is a single orbit. Let $H^k$ denote $k$-dimensional Hausdorff measure. The main general assumption throughout the paper is that $M$ and $M-E$ are globally minimal ($M$ globally minimal does not imply $M-E$ globally minimal). Among the results of the paper, we state the following theorem: Let $M$ be $C^{2,\alpha}, 0<\alpha<1$, of real dimension $d$ and CR dimension $\geq 1$. Let $E\subset M$ be a closed subset with $H^{d-3}(E)<\infty$. Assume that $M$ and $M-E$ are globally minimal. Then $L^p_{\rm loc,CR}(M)=L^p_{\rm loc,CR}(M-E)$.

• 15.
Ecole Normale Superiore Paris.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
On the Local Meromorphic Extension of CR Meromorphic Mappings: Complex analysis and applications1998In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 70, no 1/3, p. 163-193Article in journal (Refereed)
• 16.
Ecole Normale Superiore Paris.
On Wedge Extendability of CR-Meromorphic Functions2002In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 241, no 3, p. 485-512Article in journal (Refereed)
• 17.
Ecole Normale Superiore Paris.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
The Hartogs Extension Theorem on (n-1)-Complete Complex Spaces2009In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 637, p. 23-29Article in journal (Refereed)

Employing Morse theory and the method of analytic discs but no @ techniques, we establish a version of the Hartogs extension theorem in a singular setting, namely: for every domain of an (n − 1)- complete normal complex space of pure dimension n > 2, and for every compact set K ⊂ such that \K is connected, holomorphic or meromorphic functions in \K extend holomorphically or meromorphically to

• 18.
II Univ Roma Tor Vergata, Dipartimento Matemat, I-00133 Rome, Italy.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
C-hypoellipticity and extension of CR functions2015In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 14, no 3, p. 677-703Article in journal (Refereed)

Let M be a CR submanifold of a complex manifold X. The main result of this article is to show that CR-hypoellipticity at p(0) is an element of M is necessary and sufficient for holomorphic extension of all germs at pc, of CR functions on M to an ambient neighborhood of p(0) in X. As an application, we obtain that CR-hypoellipticity implies the existence of global generic embeddings and prove holomorphic extension for a large class of CR manifolds satisfying a higher order Levi pseudoconcavity condition. We also obtain results on the relationship of holomorphic wedge-extension and the C-infinity-wave front set for CR distributions.

• 19.
Dipartimento di Matematica, II Universita` di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma (Italy).
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics. Instytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Kielce, Poland.
Weak q-concavity conditions for CR-manifolds2017In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 196, no 5, p. 1779-1817Article in journal (Refereed)

We introduce various notions of q-pseudo-concavity for abstract CR manifolds, and we apply these notions to the study of hypoellipticity, maximum modulus principle and Cauchy problems for CR functions.

• 20.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Cinfinity-hypoellipticity and extension of CR functions2011Report (Other academic)
• 21. Porten, Egmont
Ein Dualitätssatz für Hardyräume auf streng pseudokonvexen Gebieten1993Licentiate thesis, monograph (Other academic)
• 22. Porten, Egmont
Geometric methods in the study of CR functions and their singularities2005Doctoral thesis, monograph (Other academic)
• 23. Porten, Egmont
Hebbare Singularitäten von CR-Funktionen und Analytische Fortsetzung von Teilen Nicht-Pseudokonvexer Ränder1997Doctoral thesis, monograph (Other academic)
• 24.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Local envelopes on CR manifolds2008In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, p. 240-244Article in journal (Refereed)
• 25.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
On generalized tube domains over C^n2005In: Complex Variables, Theory & Application, ISSN 0278-1077, E-ISSN 1563-5066, Vol. 50, no 1, p. 1-5Article in journal (Refereed)

If j : Xj !Rn, j ¼ 1, 2, are domains over Rn, then ¼ X1 X2 carries a canonical structure of a generalized tube domain over Cn. In response to a question of Jarnicki and Pflug (M. Jarnicki and P. Pflug (2000). Extension of Holomorphic Functions, Springer-Verlag), we show that is one-sheeted and equivalent to a product of convex domains, if it is pseudoconvex.

• 26. Porten, Egmont
On the Hartogs-phenomenon and extension of analytic hypersurfaces in non-separated Riemann domains2002In: Complex Variables, Theory & Application, ISSN 0278-1077, E-ISSN 1563-5066, Vol. 47, no 4, p. 325-332Article in journal (Refereed)

In the present paper, we answer two questions raised by Jarnicki and Pflug: First, we show by a counterexample that the Hartogs-Bochner theorem is no longer true for non-separated Riemann domains. Secondly, we generalize a structure theorem of Dloussky, which examines the extension of singularity sets contained in analytic hypersurfaces, to non-separated Riemann domains. Moreover, our method yields a new proof of Dloussky's original result.

• 27.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics. Uniwersytet Jana Kochanowskiego Kielcach, Inst Matemat, Kielce, Poland.
Polynomial hulls and analytic discs2017In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 145, no 10, p. 4443-4448Article in journal (Other academic)

The goal of the present note is to construct a class of examples for connected compact sets K subset of C-n whose polynomial hull (K) over cap cannot be covered by analytic discs with boundaries contained in an arbitrarily small neighborhood of K. This gives an answer to a recent question raised by B. Drinovec Drnovsek and F. Forstneric.

• 28.
Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
The Hartogs phenomenon on weakly pseudoconcave hypersurfaces2012In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 354, no 2, p. 659-683Article in journal (Refereed)

In 2002, Henkin and Michel proved a local Hartogs phenomenon for real analytic CR functions on real analytic weakly pseudoconcave CR manifolds. The aim of the present article is to remove the assumptions on real analyticity in the case of weakly pseudoconcave hypersurfaces {Mathematical expression} . If M is a graph of class {Mathematical expression} and nÂ â‰¥ 3, a global theorem is proved for the extension of holomorphic germs along M. If the appearing domains have nicely shaped boundary, a Hartogs theorem even holds for continuous CR functions, where the difference to the case of holomorphic germs relies on the possible presence of lower-dimensional CR orbits. Levi flat hypersurfaces in {Mathematical expression} require a separate treatment. Here an affirmative answer is given to the question of Tomassini, whether 2-spheres bound 3-balls in M. Â© 2011 Springer-Verlag.

• 29. Porten, Egmont
Totally real discs in non-pseudoconvex boundaries2003In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 41, no 1, p. 133-150Article in journal (Refereed)
• 30.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
Two-dimensional slices of nonpseudoconvex domains with rough boundary2014In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 278, no 1-2, p. 19-23Article in journal (Refereed)

For a domain , the set is defined as the set of all points for which the intersection of with every complex -plane through is pseudoconvex. For nonpseudoconvex, it is shown that is contained in an affine subspace of codimension . This results solves a problem raised by Nikolov and Pflug.

• 31.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
Arizona State University, Tempe, AZ, U.S.A. Cardinal Stefan Wyszy´nski University, Warsaw, Poland.
On the Hartogs extension theorem for unbounded domains in Cn2018Report (Other academic)

Let Ω ⊂ Cn, n ≥ 2, be a domain with smooth connected boundary. IfΩ is relatively compact, the Hartogs-Bochner theorem ensures that everyCR distribution on ∂Ω has a holomorphic extension to Ω. For unboundeddomains this extension property may fail, for example if Ω contains a complex hypersurface. The main result in this paper tells that the extensionproperty holds if and only if the envelope of holomorphy of Cn\Ω is Cn.It seems that it is a first result in the literature which gives a geometriccharacterization of unbounded domains in Cnfor which the Hartogs phenomenon holds. Comparing this to earlier work by the first two authorsand Z. S lodkowski, one observes that the extension problem sensitively depends on a finer geometry of the contact of a complex hypersurface andthe boundary of the domain.

• 32.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
On thickening of holomorphic hulls and envelopes of holomorphy on Stein spaces2011Report (Other academic)
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