共查询到20条相似文献,搜索用时 93 毫秒
1.
Roman Lávička 《Advances in Applied Clifford Algebras》2007,17(3):549-554
In [7], B. Fuglede has proved that finely holomorphic functions on a finely open subset U of the complex plane C are finely locally extendable to usual continuously differentiable functions. We shall adopt B. Fuglede’s approach to show
that the same remains true even for functions which have only finely continuous fine differential on U. In higher dimensions, an analogous result may be obtained and the result can be applied to finely monogenic functions which
were introduced recently as a higher dimensional analogue of finely holomorphic functions.
I acknowledge the financial support from the grant GA 201/05/2117. This work is also a part of the research plan MSM 0021620839,
which is financed by the Ministry of Education of the Czech Republic. 相似文献
2.
B. Fuglede 《Aequationes Mathematicae》1990,39(2-3):198-203
Summary It is shown that a finely superharmonic function in a planar fine domainU is greater than or equal to its lower integral with respect to harmonic measure associated with any bounded finely open setV with fine closure contained inU. Examples are given showing that this result does not extend to dimension 3 or more (unlessf is supposed to be, e.g., lower bounded onV) and also that the integral need not exist. 相似文献
3.
Harmonic morphisms as unit normal bundles¶of minimal surfaces 总被引:2,自引:0,他引:2
Let be an isometric immersion between Riemannian manifolds and be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space and necessary and sufficient conditions on f for the projection map to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian
manifold is a harmonic morphism with totally geodesic fibres.
Received: 6 February 1999 相似文献
4.
In this paper, we study the composition operator CΦ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for CΦ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cn (respectively Rn) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces. 相似文献
5.
T.J Lyons 《Journal of Functional Analysis》1980,37(1):19-26
Consider the map from the fine interior of a compact set to the measures on the fine boundary given by Balayage of the unit point mass onto the fine boundary (the Keldych measure). It is shown that for any point in the domain there is a compact fine neighborhood of the point on which the map is continuous from the initial topology on the compact set to the norm topology on measures. In this paper we only prove a rather special case, the method could easily be generalized to more abstract potential spaces. One consequence of this result is a Hartog-type theorem for finely harmonic functions. We use the Hartog theorem, rational approximation theory, and results proved in a previous paper by the author to prove that the derivative of a finely holomorphic function exists everywhere and is finely holomorphic. 相似文献
6.
We show that if the graph of an analytic function in the unit disk D is not complete pluripolar in C2 then the projection of its pluripolar hull contains a fine neighborhood of a point
. Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function
f in D extends to a function ℱ which is defined on a fine neighborhood of a point
and is finely analytic at p then the pluripolar hull of the graph of f contains the graph of ℱ over a smaller fine neighborhood of p. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes
of analytic functions in the disk which have non-trivial pluripolar hulls, among them C∞ functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls.
Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation. 相似文献
7.
It is known that iff is a continuous function on the complex plane which extends holomorphically from each circle surrounding the origin, thef is not necessarily holomorphic. In the paper we prove that if, in addition,f extends holomorphically from each circle belonging to an open family of circles which do not surround the origin, thef is holomorphic. 相似文献
8.
Let f be a generalized holomorphic function on a connected open set
W ì \Bbb C\Omega\subset {\Bbb C}
. It is proved that f equals zero if and only if there exists a smooth curve and a set A of positive (one-dimensional) measure such that f takes zero value on A. Also, a holomorphic generalized function different from zero on the disc, which takes zero values on a dense G
δ-set of the disc, is constructed. The generalized zero set of a holomorphic function is introduced and studied in an analogous
way. 相似文献
9.
Mark L. Agranovsky 《Advances in Mathematics》2007,211(1):284-326
We prove that homologically nontrivial generic smooth (2n−1)-parameter families of analytic discs in Cn, n?2, attached by their boundaries to a CR-manifold Ω, test CR-functions in the following sense: if a smooth function on Ω analytically extends into any analytic discs from the family, then the function satisfies tangential CR-equations on Ω. In particular, we give an answer (Theorem 1) to the following long standing open question, so called strip-problem, earlier solved only for special families (mainly for circles): given a smooth one-parameter family of Jordan curves in the plane and a function f admitting holomorphic extension inside each curve, must f be holomorphic on the union of the curves? We prove, for real-analytic functions and arbitrary generic real-analytic families of curves, that the answer is “yes,” if no point is surrounded by all curves from the family. The latter condition is essential. We generalize this result to characterization of complex curves in C2 as real 2-manifolds admitting nontrivial families of attached analytic discs (Theorem 4). The main result implies fairly general Morera type characterization of CR-functions on hypersurfaces in C2 in terms of holomorphic extensions into three-parameter families of attached analytic discs (Theorem 2). One of the applications is confirming, in real-analytic category, the Globevnik-Stout conjecture (Theorem 3) on boundary values of holomorphic functions. It is proved that a smooth function on the boundary of a smooth strictly convex domain in Cn extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain. The proofs are based on a universal approach, namely, on the reduction to a problem of propagation, from the boundary to the interior, of degeneracy of CR-foliations of solid torus type manifolds (Theorem 2.2). 相似文献
10.
Mihai Cristea 《Complex Analysis and Operator Theory》2011,5(3):863-880
The paper continues the work of Royster (Duke Math J 19:447–457, 1952), Mocanu [Mathematica (Cluj) 22(1):77–83, 1980; Mathematica
(Cluj) 29:49–55, 1987], Cristea [Mathematica (Cluj) 36(2):137–144, 1994; Complex Var 42:333–345, 2000; Mathematica (Cluj)
43(1):23–34, 2001; Mathematica (Cluj), 2010, to appear; Teoria Topologica a Functiilor Analitice, Editura Universitatii Bucuresti,
Romania, 1999] of extending univalence criteria for complex mappings to C
1 mappings. We improve now the method of Loewner chains which is usually used in complex univalence theory for proving univalence
criteria or for proving quasiconformal extensions of holomorphic mappings f : B → C
n
to C
n
. The results are surprisingly strong. We show that the usual results from the theory, like Becker’s univalence criteria remain
true for C
1 mappings and since we use a stronger form of Loewner’s theory, we obtain results which are stronger even for holomorphic
mappings f : B → C
n
. In our main result (Theorem 4.1) we end the researches dedicated to quasiconformal extensions of K-quasiregular and holomorphic mappings f : B → C
n
to C
n
. We show that a C
1 quasiconformal map f : B → C
n
can be extended to a quasiconformal map F : C
n
→ C
n
, without any metric condition imposed to the map f. 相似文献
11.
Let M be a two-dimensional complex manifold and let be a holomorphic map that fixes pointwise a (possibly) singular, compact, reduced and globally irreducible curve . We give a notion of degeneracy of f at a point of C. It turns out that f is non-degenerate at one point if and only if it is non-degenerate at every point of C. When f is non-degenerate on C, we define a residual index for f at each point of C. Then we prove that the sum of the indices is equal to the self-intersection number of C.
Received: 15 May 2000; in final form: 10 July 2001 / Published online: 1 February 2002 相似文献
13.
Ning Zhong 《Mathematical Logic Quarterly》1998,44(3):304-316
As is well known the derivative of a computable and C1 function may not be computable. For a computable and C∞ function f, the sequence {f(n)} of its derivatives may fail to be computable as a sequence, even though its derivative of any order is computable. In this paper we present a necessary and sufficient condition for the sequence {f(n)} of derivatives of a computable and C∞ function f to be computable. We also give a sharp regularity condition on an initial computable function f which insures the computability of its derivative f′. 相似文献
14.
Mats Andersson 《Journal of Functional Analysis》2004,212(1):76-88
Let be a holomorphic mapping in a neighborhood of the origin in . We find sufficient condition, in terms of residue currents, for a smooth function to belong to the ideal in C∞ (or Ck) generated by f. If f is a complete intersection the condition is necessary. More generally we give a sufficient condition for an element of class C∞ (or Ck) in the Koszul complex induced by f to be exact. For the proofs we introduce explicit homotopy formulas for the Koszul complex induced by f. 相似文献
15.
Jie Miao 《Integral Equations and Operator Theory》2007,59(1):53-65
We give a necessary and sufficient condition for Hankel operators Hf on the harmonic Bergman space of the unit ball to be in the Schatten p-class for 2 ≤ p < ∞. A special case when symbol f is a harmonic function is also considered. 相似文献
16.
We study the Bloch constant for Κ-quasiconformal holomorphic mappings of the unit ball B of C
n
. The final result we prove in this paper is: If f is a Κ-quasiconformal holomorphic mappig of B into C
n
such that det(f′(0)) = 1, then f(B) contains a schlicht ball of radius at least
where C
n
> 1 is a constant depending on n only, and as n→∞.
Received June 24, 1998, Accepted January 14, 1999 相似文献
17.
Charles J. K. Batty Mark D. Blake Sachi Srivastava 《Integral Equations and Operator Theory》2003,45(2):125-154
Let
f : \mathbbR+ ? \mathbbC f : \mathbb{R}_{+} \longrightarrow \mathbb{C} be an exponentially bounded, measurable function. We introduce a growth bound z(f) \zeta(f) which measures the extent to which f f can be approximated by holomorphic functions. This growth bound is related to the location of the domain of holomorphy of the Laplace transform of f f far from the real axis. The denition extends to vector and operator-valued cases. For a C0 C_{0} -semigroup T T of operators, z(T) \zeta(T) is closely related to the critical growth bound of T T . 相似文献
18.
Let f and g be two permutable transcendental holomorphic maps in the plane. We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide.
Dedicated to Professor Sheng GONG on the occasion of his 75th birthday 相似文献
19.
We show that there is no immersed compact Levi-flat hypersurface of class C1 in the complex projective plane, if the foliation by holomorphic curves carries a harmonic current which is absolutely continuous with respect to the Lebesgue measure, with a density bounded from above and below. This is a corollary of a rigidity result for immersed compact Levi-flat hypersurfaces in complex surfaces of non-negative curvature. 相似文献
20.
Marco Abate 《Journal d'Analyse Mathématique》1998,74(1):275-306
The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls inC n and by the author to holomorphic maps between strongly (pseudo)convex domains. Here we describe Julia-Wolff-Carathéodory theorems for holomorphic maps defined in a polydisk and with image either in the unit disk, or in another polydisk, or in a strongly convex domain. One of the main tools for the proof is a general version of the Lindelöf principle valid for not necessarily bounded holomorphic functions. 相似文献