ON THE NORM OF THE OPERATOR aI + bH ON Lp(ℝ)

We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky [7]: let H be the Hilbert transform and let a, b be real constants. Then for 1 < p < ${\infty}$ the norm of the operator aI + bH from $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$ is equal to $$\({\max_{x{\in}{\mathbb{R}}}}{\frac{{\mid}ax-b+(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}ax-b-(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p}{{\mid}x+{\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}x-{\tan}{\frac{\pi}{2p}}{\mid}^p}}\)^{\frac{1}{p}}$$. Our proof avoids passing through the analogous result for the conjugate function on the circle, as in [7], and is given directly on the line. We also provide new approximate extremals for aI + bH in the case p > 2.