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  • 1.
    Daghighi, Abtin
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
    A local maximum principle for locally integrable structures2017In: Communications in Contemporary Mathematics, ISSN 0219-1997, Vol. 19, no 1, article id 1550091Article in journal (Refereed)
    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.

  • 2.
    Daghighi, Abtin
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Approach regions of Lebesgue measurable, locally bounded, quasi-continuous functions2012In: International Journal of Mathematical Analysis, ISSN 1312-8876, E-ISSN 1314-7579, Vol. 6, no 13, p. 659-680Article in journal (Refereed)
    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.

  • 3. Daghighi, Abtin
    On A Uniqueness Condition For Cr Functions On Hypersurfaces2018In: Bulletin of Mathematical Analysis and Applications, ISSN 1821-1291, E-ISSN 1821-1291, Vol. 10, no 1, p. 68-79Article in journal (Refereed)
    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.

  • 4.
    Daghighi, Abtin
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
    Regularity and uniqueness-related properties of solutions with respect to locally integrable structures2014Doctoral 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.

  • 5.
    Daghighi, Abtin
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    The Maximum Principle for Cauchy-Riemann Functions and Hypocomplexity2012Licentiate 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.

  • 6.
    Daghighi, Abtin
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
    Krantz, Steven G.
    Washington Univ, Dept Math, St Louis, MO 63130 USA.
    A note on a conjecture concerning boundary uniqueness2015In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 60, no 7, p. 945-950Article in journal (Refereed)
  • 7.
    Daghighi, Abtin
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
    Wikstrom, Frank
    Lund University, Lund.
    Level Sets of Certain Subclasses of alpha-Analytic Functions2017In: Journal of Partial Differential Equations, ISSN 1000-940X, E-ISSN 2079-732X, Vol. 30, no 4, p. 281-298Article in journal (Refereed)
    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.

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