On removable singularities for the analytic Zygmund class.

Carmona, Juan J.; Donaire, Juan Jesús

doi:10.1307/mmj/1029005389

Cited by 8 publications

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“…In [7], Kaufman proved that in the case of the Zygmund class it is not possible to find a measure function φ with the property M φ (K) = 0 if and only if K is Λ * -removable. Nevertheless, in [1] it was shown that if M Φ (K) = 0 with…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Porosity of Sets and the Zygmund Class

Donaire

2002

Bull. London Math. Soc.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…In [1], it was proved that the compact set used in the proof of Theorem 2 is nonporous in the sense of Kaufman. Consequently, there are nonporous sets with P (K) = 0, as was asserted after the statement of Theorem 1.…”

Section: )mentioning

confidence: 99%

Porosity of Sets and the Zygmund Class

Donaire

2002

Bull. London Math. Soc.

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“…Much attention have been given to find a characterization of the removable singularities for bounded analytic functions, a problem which was recently solved by Tolsa [30]. Other spaces of analytic functions for which removable singularities have been studied include: the Nevanlinna class N (Rudin [28]); the Smirnov class N + (Khavinson [22]); the Smirnov spaces E p (Khavinson [21]); the Dirichlet spaces AD p (Hedberg [17]); the John-Nirenberg class BMO (Král [26], Kaufman [20], Koskela [25] and Björn [9]); the Hölder classes C α (Dolzhenko [12] and Koskela [25]); the Lipschitz space Lip (Nguyen [27] and Khrushchëv [23]); the Zygmund class ZC (Carmona-Donaire [11]); the spaces VMO, lip α and Campanato spaces (Král [26] as special cases of the corresponding problem for more general partial differential operators); the locally Lipschitz classes locLip α and loclip α (Björn [9]); and let us also mention the paper by Ahlfors and Beurling [2]. In a sequence of papers [4], [5], [7], [8] the author built on older work in the study of removable singularities for H p .…”

Section: Introductionmentioning

confidence: 99%

Removable singularities for weighted Bergman spaces

Björn

2006

Czech Math J

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We develop a theory of removable singularities for the weighted Bergman space A p µ (Ω) = {f analytic in Ω :RThe general theory developed is in many ways similar to the theory of removable singularities for Hardy H p spaces, BMO and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable.In the case when weak and strong removability are the same for all sets, in particular if µ is absolutely continuous with respect to the Lebesgue measure m, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable.When dµ = w dm and w is a Muckenhoupt Ap weight, 1 < p < ∞, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent p = p/(p − 1) and the dual weight w = w 1/(1−p) .

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“…Let us just mention: H ∞ (Tolsa [43]); the Nevanlinna class N (Rudin [39]); the Smirnov class N + (Khavinson [28]); the Smirnov spaces E p (Khavinson [27]); the Dirichlet spaces AD p (Hedberg [22]); the Zygmund class ZC (Carmona-Donaire [10]); the Lebesgue spaces L p (Carleson [9] and Hedberg [22]); and let us finally mention the paper by Ahlfors and Beurling [1].…”

Section: Introductionmentioning

confidence: 99%

Removable singularities for analytic functions in BMO and locally Lipschitz spaces

Björn

2003

Mathematische Zeitschrift

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In this paper we study removable singularities for holomorphic functions such that sup z∈Ω |f (n) (z)| dist(z, ∂Ω) s < ∞. Spaces of this type include spaces of holomorphic functions in Campanato classes, BMO and locally Lipschitz classes. Dolzhenko (1963), Král (1976) and Nguyen (1979) characterized removable singularities for some of these spaces. However, they used a different removability concept than in this paper. They assumed the functions to belong to the function space on Ω and be holomorphic on Ω E, whereas we only assume that the functions belong to the function space on Ω E, and are holomorphic there. Koskela (1993) obtained some results for our type of removability, in particular he showed the usefulness of the Minkowski dimension. obtained some results for s = 0.

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On removable singularities for the analytic Zygmund class.

Cited by 8 publications

References 0 publications

Porosity of Sets and the Zygmund Class

Porosity of Sets and the Zygmund Class

Removable singularities for weighted Bergman spaces

Removable singularities for analytic functions in BMO and locally Lipschitz spaces

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