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Singular integral operators in weighted spaces of continuous functions with non‐equilibrated continuity modulus
Author(s) -
Samko Natasha
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410425
Subject(s) - mathematics , modulus of continuity , pure mathematics , piecewise , moduli , type (biology) , lp space , mathematical analysis , modulus , absolute continuity , banach space , geometry , ecology , physics , quantum mechanics , biology
We show that singular integral operators with piecewise continuous coefficients may gain massive spectra when considered in weighted spaces of continuous functions with a prescribed continuity modulus (generalized Hölder spaces H ω (Γ, ρ )), a fact known for example for Lebesgue spaces L p (Γ, ρ ) in the case of general Muckenhoupt weights ρ or bad‐behaved curves Γ. In the case under consideration the appearance of “lunes” generating massivity of the spectra is due to the presence of a general (non‐equilibrated) continuity modulus ω . These lunes arise when the Boyd‐type indices of the function ω ( h ) do not coincide. Thus, the massive spectra may appear in the non‐weighted case and on nice curves, a situation similar to Orlicz spaces. The main problems arising in the investigation are the nature of non‐equilibrated continuity moduli ω and the failure of the density of “nice” functions in Hölder‐type spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)