Singular integral operators in weighted spaces of continuous functions with non‐equilibrated continuity modulus

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)