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Publications (7 of 7) Show all publications
Daghighi, A. (2018). On A Uniqueness Condition For Cr Functions On Hypersurfaces. Bulletin of Mathematical Analysis and Applications, 10(1), 68-79
Open this publication in new window or tab >>On A Uniqueness Condition For Cr Functions On Hypersurfaces
2018 (English)In: Bulletin of Mathematical Analysis and Applications, ISSN 1821-1291, E-ISSN 1821-1291, Vol. 10, no 1, p. 68-79Article in journal (Refereed) Published
Abstract [en]

Let f be a smooth CR function on a smooth hypersurface M subset of C-n, such that f vanishes to infinite order along a C-infinity-smooth curve gamma subset of M. Assume that for each q is an element of gamma there exists a truncated double cone C at q in M, such that at least one of the following three conditions holds true: (a) There is a constant theta is an element of R, such that C subset of {|Re(e(i theta)f)| <= |Im(e(i theta)f)|}. (b) C subset of {Ref >= 0}. (c) |f(z)|(|z-q|) -> 0, z -> q, z subset of C. Then f vanishes on an M-open neighborhood of gamma.

Keywords
Unique continuation, CR manifold, CR functions
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-33738 (URN)000432520000006 ()
Available from: 2018-06-10 Created: 2018-06-10 Last updated: 2020-02-17Bibliographically approved
Daghighi, A. (2017). A local maximum principle for locally integrable structures. Communications in Contemporary Mathematics, 19(1), Article ID 1550091.
Open this publication in new window or tab >>A local maximum principle for locally integrable structures
2017 (English)In: Communications in Contemporary Mathematics, ISSN 0219-1997, Vol. 19, no 1, article id 1550091Article in journal (Refereed) Published
Abstract [en]

Let Omega subset of R-N be an open subset, for a positive integer N, and let L subset of C circle times T Omega be a C-infinity -smooth locally integrable subbundle. We give a proof of the following result: If (Omega, L) is nowhere strictly hypoanalytically pseudoconvex (as defined in the paper) then for any sufficiently small domain omega (sic) Omega, and any f C-0(omega) which is continuous up to the boundary such that f is a solution with respect to L on., it holds true that max(z is an element of partial derivative omega) |f(z)| = max(z is an element of(omega) over bar) |f(z)|. We also point out a relation to Levi curvature.

Keywords
Hypoanalytic structure, locally integrable structures, local maximum principle
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-29803 (URN)10.1142/S0219199715500911 (DOI)000389231700003 ()2-s2.0-84947730597 (Scopus ID)
Available from: 2017-01-02 Created: 2017-01-02 Last updated: 2017-09-14Bibliographically approved
Daghighi, A. & Wikstrom, F. (2017). Level Sets of Certain Subclasses of alpha-Analytic Functions. Journal of Partial Differential Equations, 30(4), 281-298
Open this publication in new window or tab >>Level Sets of Certain Subclasses of alpha-Analytic Functions
2017 (English)In: Journal of Partial Differential Equations, ISSN 1000-940X, E-ISSN 2079-732X, Vol. 30, no 4, p. 281-298Article in journal (Refereed) Published
Abstract [en]

For an open set V subset of C-n, denote by M-alpha(V) the family of a-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded "harmonically fat" domain Omega subset of C-n, a function f is an element of M-alpha (Omega\f(-1)(0)) automatically satisfies f is an element of M-alpha(Omega), if it is C alpha j-1-smooth in the z(j) variable, alpha is an element of Z(+)(n) up to the boundary. For a submanifold U subset of C-n, denote by M-alpha(U), the set of functions locally approximable by a-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a C-3-smooth hypersurface, Omega, a member of M-alpha(Omega), cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form. The result can be partially generalized to C-4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.

Keywords
Polyanalytic functions, q-analytic functions, zero sets, level sets, alpha-analytic functions
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-33393 (URN)10.4208/jpde.v30.n4.1 (DOI)000426542400001 ()
Available from: 2018-04-03 Created: 2018-04-03 Last updated: 2018-04-03Bibliographically approved
Daghighi, A. & Krantz, S. G. (2015). A note on a conjecture concerning boundary uniqueness. Complex Variables and Elliptic Equations, 60(7), 945-950
Open this publication in new window or tab >>A note on a conjecture concerning boundary uniqueness
2015 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 60, no 7, p. 945-950Article in journal (Refereed) Published
Keywords
unique continuation, boundary uniqueness
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-25641 (URN)10.1080/17476933.2014.984608 (DOI)000355108000004 ()2-s2.0-84929951717 (Scopus ID)
Note

Erratum published online 13th Dec 2015.

Correction published apr 2016 61(4) p587

DOI: 10.1080/17476933.2015.1086761

ISI: 000372019700012

Scopus: 2-s2.0-84961214448

Available from: 2015-08-28 Created: 2015-08-18 Last updated: 2017-12-04Bibliographically approved
Daghighi, A. (2014). Regularity and uniqueness-related properties of solutions with respect to locally integrable structures. (Doctoral dissertation). Sundsvall: Mid Sweden University
Open this publication in new window or tab >>Regularity and uniqueness-related properties of solutions with respect to locally integrable structures
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We prove that a smooth generic embedded CR submanifold of C^n obeys the maximum principle for continuous CR functions if and only if it is weakly 1-concave. The proof of the maximum principle in the original manuscript has later been generalized to embedded weakly q-concave CR submanifolds of certain complex manifolds. We give a generalization of a known result regarding automatic smoothness of solutions to the homogeneous problem for the tangential CR vector fields given local holomorphic extension. This generalization ensures that a given locally integrable structure is hypocomplex at the origin if and only if it does not allow solutions near the origin which cannot be represented by a smooth function near the origin. We give a sufficient condition under which it holds true that if a smooth CR function f on a smooth generic embedded CR submanifold, M, of C^n, vanishes to infinite order along a C^infty-smooth curve  \gamma in M, then f vanishes on an M-neighborhood of \gamma. We prove a local maximum principle for certain locally integrable structures.

Place, publisher, year, edition, pages
Sundsvall: Mid Sweden University, 2014. p. 145
Series
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 183
Keywords
Maximum principle, hypocomplexity, locally integrable structure, hypoanalytic structure, weak pseudoconcavity, uniqueness, CR functions
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-21641 (URN)978-91-87557-44-6 (ISBN)
Public defence
2014-05-21, O102, Mid Sweden University, Sundsvall, 10:15 (English)
Opponent
Supervisors
Note

Funding  by FMB, based at Uppsala University.

Available from: 2014-04-24 Created: 2014-03-29 Last updated: 2015-09-17Bibliographically approved
Daghighi, A. (2012). Approach regions of Lebesgue measurable, locally bounded, quasi-continuous functions. International Journal of Mathematical Analysis, 6(13), 659-680
Open this publication in new window or tab >>Approach regions of Lebesgue measurable, locally bounded, quasi-continuous functions
2012 (English)In: International Journal of Mathematical Analysis, ISSN 1312-8876, E-ISSN 1314-7579, Vol. 6, no 13, p. 659-680Article in journal (Refereed) Published
Abstract [en]

Quasi-continuity (in the sense of Kempisty) generalizes directional continuity of complex-valued functions on open subsets of ℝ n or ℂ n, and in particular provides certain approach regions at every point. We show that these can be used as a proof tool for proving several properties forLebesgue measurable, locally bounded, quasi-continuous functions e.g. that for such a function f the polynomial ring C(M,K)[f] (where K = ℝ or ℂ) satisfies that the equivalence classes under identification a.e. have cardinality one and an asymptotic maximum principle.

Keywords
Approach regions; Locally bounded lebesgue measurable quasi-continuous functions
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:miun:diva-15999 (URN)2-s2.0-84865142026 (Scopus ID)
Available from: 2012-03-15 Created: 2012-03-15 Last updated: 2017-12-07Bibliographically approved
Daghighi, A. (2012). The Maximum Principle for Cauchy-Riemann Functions and Hypocomplexity. (Licentiate dissertation). Sundsvall: Mittuniversitetet
Open this publication in new window or tab >>The Maximum Principle for Cauchy-Riemann Functions and Hypocomplexity
2012 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This licentiate thesis contains results on the maximum principle forCauchy–Riemann functions (CR functions) on weakly 1-concave CRmanifolds and hypocomplexity of locally integrable structures. Themaximum principle does not hold true in general for smooth CR functions,and basic counterexamples can be constructed in the presenceof strictly pseudoconvex points. We prove a maximum principle forcontinuous CR functions on smooth weakly 1-concave CR submanifolds.Because weak 1-concavity is also necessary for the maximumprinciple, a consequence is that a smooth generic CR submanifold ofCn obeys the maximum principle for continuous CR functions if andonly if it is weakly 1-concave. The proof is then generalized to embeddedweakly p-concave CR submanifolds of p-complete complexmanifolds. The second part concerns hypocomplexity and hypoanalyticstructures. We give a generalization of a known result regardingautomatic smoothness of solutions to the homogeneous problemfor the tangential CR vector fields given local holomorphic extension.This generalization ensures that a given locally integrable structureis hypocomplex at the origin if and only if it does not allow solutionsnear the origin which cannot be represented by a smooth function nearthe origin.

Abstract [sv]

Uppsatsen innehåller resultat om maximumprincipen för kontinuerligaCauchy–Riemann funktioner (CR-funktioner) på svagt 1-konkava CRmångfalder,samt hypokomplexitet för lokalt integrerbara strukturer.Maximumprincipen gäller inte generellt för släta CR funktioner ochmotexempel kan konstrueras givet strängt pseudokonvexa punkter.Vi bevisar en maximumprincip för kontinuerliga CR-funktioner påsläta inbäddade svagt 1-konkava CR-mångfalder. Eftersom svagt 1-konkavitet också är nödvändigt får vi som konsekvens att för slätageneriska inbäddade CR-mångfalder i Cn gäller att maximum-principenför kontinuerliga CR-funktioner håller om och endast om CR-mångfaldenär svagt 1-konkav. Vi generaliserar satsen till svagt p-konkava CRmångfalderi p-kompletta mångfalder. Den andra delen behandlarhypokomplexitet och hypoanalytiska strukturer. Vi generaliserar enkänd sats om automatisk släthet för lösningar till de tangentiella CRekvationerna,givet existensen av lokal holomorf utvidgning. Generaliseringenger att en lokalt integrerbar struktur är hypokomplex iorigo om och endast om den inte tillåter lösningar nära origo som inteär släta nära origo.

Place, publisher, year, edition, pages
Sundsvall: Mittuniversitetet, 2012. p. 75
Series
Mid Sweden University licentiate thesis, ISSN 1652-8948 ; 94
Keywords
Hypocomplexity, hypoanalytic structure, CR functions, maximum principle, weak pseudoconcavity
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-17701 (URN)978-91-87103-49-0 (ISBN)
Presentation
2012-12-18, R106, Holmgatan 10, Sundsvall, 13:15 (English)
Opponent
Supervisors
Note

Forskning finansierad av Forskarskolan i Matematik och Beräkningsvetenskap (FMB), baserad i Uppsala.

Available from: 2012-12-14 Created: 2012-12-13 Last updated: 2015-09-17Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-7488-8004

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