Luther, Uwe : Cauchy Singular Integral Operators in Weighted Spaces of Continuous Functions

Author(s):

Luther, Uwe

Title:

Cauchy Singular Integral Operators in Weighted Spaces of Continuous Functions

Electronic source:

application/postscript
application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 9, 2002

Mathematics Subject Classification:

44A15			[ Special transforms ]
46E15			[ Banach spaces of continuous, differentiable or analytic functions ]

Abstract:

We study the Cauchy singular integral operator $SwI$ on $(-1,1)$, where $|w|$ is a generalized Jacobi weight. This operator is considered in pairs of weighted spaces of continuous functions, where the weights $u$ and $v$ are generalized Jacobi weights with nonnegative exponents such that $|w|=u/v$. We introduce a certain polynomial approximation space which is well appropriated to serve as domain of definition of $SwI$. A description of this space in terms of smoothness properties shows that it can be viewed as a limit case of weighted Besov spaces of continuous functions. We use our results to characterize those of the operators $awI+SbwI$ and $\varrho^{-1}(aw\varrho I+bSw\varrho I)$, $\varrho^{-1}\in b^{-1}\Pi$, which act in certain pairs of Ditzian-Totik type Besov spaces.

Keywords:

Cauchy singular integral operators, Weighted spaces of continuous functions, Approximation spaces

Language:

English

Publication time:

8 / 2002