$L^p$-Boundedness of Two Singular Integral Operators of Convolution Type

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].