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  • 1.
    Altomani, Andrea
    et al.
    Université du Luxembourg.
    Hill, C. Denson
    SUNY Stony Brook, New York.
    Nacinovich, Mauro
    Universitá Roma Tor Vergata.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap, teknik och matematik.
    Complex vector fields and hypolleptic partial differential operators2010Ingår i: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 60, nr 3, s. 39s. 987-1034Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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
    et al.
    Hill, C. Denson
    Nacinovich, Mauro
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap, teknik och matematik.
    Holomorphic Extension from Weakly Pseudoconcave CR Manifolds2010Ingår i: Rendiconti del Seminario Matematico della Universita di Padova, ISSN 0041-8994, E-ISSN 2240-2926, Vol. 123, s. 69-90Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Carillo, Sandra
    et al.
    Sapienza Università di Roma, Rome, Italy.
    Lo Schiavo, Mauro
    Sapienza Università di Roma, Rome, Italy.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för matematik och ämnesdidaktik. Uniwersytet Jana Kochanowskiego W Kielcach, Kielce, Poland.
    Schiebold, Cornelia
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för matematik och ämnesdidaktik. Uniwersytet Jana Kochanowskiego W Kielcach, Kielce, Poland.
    A novel noncommutative KdV-type equation, its recursion operator, and solitons2018Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 59, nr 4, artikel-id 043501Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Christine, Laurent-Thiébaut
    et al.
    Institut Fourier Grenoble.
    Porten, Egmont
    Analytic Extension from Non-Pseudoconvex Boundaries and A(D)-Convexity2003Ingår i: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 53, nr 3, s. 847-857Artikel i tidskrift (Refereegranskat)
  • 5.
    Denson Hill, C
    et al.
    SUNY Stony Brook, New York.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap.
    The H-Principle and Pseudoconcave CR Manifolds2005Ingår i: Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627, Vol. 50, s. 117-125Artikel i tidskrift (Refereegranskat)
  • 6.
    Jöricke, Burglind
    et al.
    Uppsala universitet.
    Porten, Egmont
    Hulls and analytic extension from non-pseudoconvex boundaries2002Rapport (Övrigt vetenskapligt)
  • 7.
    Jöricke, Burglind
    et al.
    Uppsala universitet.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap.
    On the Continuity Principle2007Ingår i: Asian Journal of Mathematics, ISSN 1093-6106, E-ISSN 1945-0036, Vol. 11, nr 1, s. 167-178Artikel i tidskrift (Refereegranskat)
  • 8.
    Lind, Andreas
    et al.
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik. Uniwersytet Jana Kochanowskiego Kielcach, Inst Matemat, Kielce, Poland.
    On thickening of holomorphic hulls and envelopes of holomorphy on Stein spaces2016Ingår i: International Journal of Mathematics, ISSN 0129-167X, Vol. 27, nr 6, artikel-id 1650051Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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
    et al.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för teknik, fysik och matematik.
    A morse-theoretical proof of the Hartogs extension theorem2007Ingår i: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 17, nr 3, s. 513-546Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Merker, Joël
    et al.
    Ecole Normale Superiore Paris.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap.
    Characteristic Foliations on Maximally Real Submanifolds of C-n and removable singularities for CR functions2006Ingår i: International Mathematics Research Papers, ISSN 1687-3017, E-ISSN 1687-3009, Vol. 2006, s. 1-131Artikel i tidskrift (Refereegranskat)
  • 11.
    Merker, Joël
    et al.
    Ecole Normale Superiore Paris.
    Porten, Egmont
    Enveloppe d'Holomorphie Locale des Variétés CR et l'Élimination des Singularités pour les Fonctions CR Intégrables1999Ingår i: Comptes rendus de l'Académie des sciences. Série 1, Mathématique, ISSN 0764-4442, E-ISSN 1778-3577, Vol. 328, nr 10, s. 853-858Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Merker, Joël
    et al.
    Ecole Normale Superiore Paris.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap.
    Holomorphic Extension of CR Functions, Envelopes of Holomorphy, and Removable Singularities2006Bok (Övrigt vetenskapligt)
  • 13.
    Merker, Joël
    et al.
    Ecole Normale Superiore Paris.
    Porten, Egmont
    Metrically Thin Singularities of Integrable CR Functions2000Ingår i: International Journal of Mathematics, ISSN 0129-167X, Vol. 11, nr 7, s. 857-872Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Merker, Joël
    et al.
    Ecole Normale Superiore Paris.
    Porten, Egmont
    On Removable singularities for integrable CR functions1999Ingår i: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 48, nr 3, s. 805-856Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Merker, Joël
    et al.
    Ecole Normale Superiore Paris.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik.
    On the Local Meromorphic Extension of CR Meromorphic Mappings: Complex analysis and applications1998Ingår i: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 70, nr 1/3, s. 163-193Artikel i tidskrift (Refereegranskat)
  • 16.
    Merker, Joël
    et al.
    Ecole Normale Superiore Paris.
    Porten, Egmont
    On Wedge Extendability of CR-Meromorphic Functions2002Ingår i: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 241, nr 3, s. 485-512Artikel i tidskrift (Refereegranskat)
  • 17.
    Merker, Joël
    et al.
    Ecole Normale Superiore Paris.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap, teknik och matematik.
    The Hartogs Extension Theorem on (n-1)-Complete Complex Spaces2009Ingår i: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 637, s. 23-29Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Nacinovich, Mauro
    et al.
    II Univ Roma Tor Vergata, Dipartimento Matemat, I-00133 Rome, Italy.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik.
    C-hypoellipticity and extension of CR functions2015Ingår i: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 14, nr 3, s. 677-703Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Nacinovich, Mauro
    et al.
    Dipartimento di Matematica, II Universita` di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma (Italy).
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik. Instytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Kielce, Poland.
    Weak q-concavity conditions for CR-manifolds2017Ingår i: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 196, nr 5, s. 1779-1817Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap, teknik och matematik.
    Cinfinity-hypoellipticity and extension of CR functions2011Rapport (Övrigt vetenskapligt)
  • 21. Porten, Egmont
    Ein Dualitätssatz für Hardyräume auf streng pseudokonvexen Gebieten1993Licentiatavhandling, monografi (Övrigt vetenskapligt)
  • 22. Porten, Egmont
    Geometric methods in the study of CR functions and their singularities2005Doktorsavhandling, monografi (Övrigt vetenskapligt)
  • 23. Porten, Egmont
    Hebbare Singularitäten von CR-Funktionen und Analytische Fortsetzung von Teilen Nicht-Pseudokonvexer Ränder1997Doktorsavhandling, monografi (Övrigt vetenskapligt)
  • 24.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap, teknik och matematik.
    Local envelopes on CR manifolds2008Ingår i: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, s. 240-244Artikel i tidskrift (Refereegranskat)
  • 25.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap.
    On generalized tube domains over C^n2005Ingår i: Complex Variables, Theory & Application, ISSN 0278-1077, E-ISSN 1563-5066, Vol. 50, nr 1, s. 1-5Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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 domains2002Ingår i: Complex Variables, Theory & Application, ISSN 0278-1077, E-ISSN 1563-5066, Vol. 47, nr 4, s. 325-332Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik. Uniwersytet Jana Kochanowskiego Kielcach, Inst Matemat, Kielce, Poland.
    Polynomial hulls and analytic discs2017Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 145, nr 10, s. 4443-4448Artikel i tidskrift (Övrigt vetenskapligt)
    Abstract [en]

    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.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för tillämpad naturvetenskap och design.
    The Hartogs phenomenon on weakly pseudoconcave hypersurfaces2012Ingår i: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 354, nr 2, s. 659-683Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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 n ≥ 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. © 2011 Springer-Verlag.

  • 29. Porten, Egmont
    Totally real discs in non-pseudoconvex boundaries2003Ingår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 41, nr 1, s. 133-150Artikel i tidskrift (Refereegranskat)
  • 30.
    Porten, Egmont
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik.
    Two-dimensional slices of nonpseudoconvex domains with rough boundary2014Ingår i: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 278, nr 1-2, s. 19-23Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.
    Porten, Egmont
    et al.
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för matematik och ämnesdidaktik.
    Boggess, Al
    Arizona State University, Tempe, AZ, U.S.A.
    Dwilewicz, Roman
    Cardinal Stefan Wyszy´nski University, Warsaw, Poland.
    On the Hartogs extension theorem for unbounded domains in Cn2018Rapport (Övrigt vetenskapligt)
    Abstract [en]

    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.
    Porten, Egmont
    et al.
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap, teknik och matematik.
    Lind, Andreas
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Institutionen för naturvetenskap, teknik och matematik.
    On thickening of holomorphic hulls and envelopes of holomorphy on Stein spaces2011Rapport (Övrigt vetenskapligt)
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