Singular operators in variable spaces L p (·) (Ω, ρ ) with oscillating weights

We study the boundedness of singular Calderón–Zygmund type operators in the spaces L p (·) (Ω, ρ ) over a bounded open set in ℝ n with the weight ρ ( x ) = $ \prod ^m_{k=1} $ w k (| x – x k |), x k ∈ $ \bar \Omega $ , where w k has the property that $ r^{ {n \over {p(x_k)}} } $ w k ( r ) ∈ $ \Phi ^0_n $ , where $ \Phi ^0_n $ is a certain Zygmund‐type class. The boundedness of the singular Cauchy integral operator S Γ along a Carleson curve Γ is also considered in the spaces L p (·) (Γ, ρ ) with similar weights. The weight functions w k may oscillate between two power functions with different exponents. It is assumed that the exponent p (·) satisfies the Dini–Lipschitz condition. The final statement on the boundedness is given in terms of the index numbers of the functions wk (similar in a sense to the Boyd indices for the Young functions defining Orlicz spaces). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)