Abstract
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 Hω (Γ, ρ )), a fact known for example for Lebesgue spaces Lp (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)
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