Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE NORM OF THE OPERATOR aI + bH ON Lp(ℝ)

  • Ding, Yong (Laboratory of Mathematics and Complex Systems School of Mathematical Sciences Beijing Normal University Ministry of Education of China) ;
  • Grafakos, Loukas (Department of Mathematics University of Missouri) ;
  • Zhu, Kai (School of Mathematical Sciences Beijing Normal University)
  • Received : 2017.08.01
  • Accepted : 2018.06.12
  • Published : 2018.07.31
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Abstract

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.

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References

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