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

✍ Scribed by Vakhtang Kokilashvili; Natasha Samko; Stefan Samko

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 189 KB
• Volume: 280
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200510542

✦ Synopsis

Abstract

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)