Advertisement
Published in:
2016 | OriginalPaper | Chapter
\(L^p\)-Boundedness of Two Singular Integral Operators of Convolution Type
Authors : Sten Kaijser, John Musonda
Published in: Engineering Mathematics II
Publisher: Springer International Publishing
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
Abstract
We investigate boundedness properties of two singular integral operators defined on \(L^p\)-spaces \((1<p<\infty )\) on the real line, both as convolution operators on \(L^p(\mathbb R)\) and on the spaces \(L^p(\omega )\), where \(\omega (x)=1/(2\cosh \frac{\pi }{2}x)\). It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for \(p=2\) and weak boundedness for \(p=1\), and then using interpolation to obtain boundedness for \(1<p\le 2\). To obtain boundedness also for \(2\le p<\infty \), we use duality in the translation invariant case, while the weighted case is partly based on the expositions on the conjugate function operator in (M. Riesz, Mathematische Zeitschrift, 27, 218–244, 1928) [7].