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

We study the boundedness of singular Calderón–Zygmund type operators in the spaces Lp(·)(Ω, ρ) over a bounded open set in Rn with the weight ρ(x) = Qm k=1 wk(|x − xk|), xk ∈ Ω, where wk has the property that r n p(xk)wk(r) ∈ Φ0n, where Φ0n is a certain Zygmund-type class. The boundedness of the singular Cauchy integral operator SΓ along a Carleson curve Γ is also considered in the spaces Lp(·)(Γ, ρ) with similar weights. The weight functions wk 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)