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RISS 인기검색어
- 검색키워드
- 저자 : Loukas Grafakos (검색결과 2 건)
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ON THE NORM OF THE OPERATOR aI + bH ON L<sup>p</sup>(ℝ)
Ding, Yong,Grafakos, Loukas,Zhu, Kai Korean Mathematical Society 2018 대한수학회보 Vol.55 No.4
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}}$$. Our proof avoids passing through the analogous result for the conjugate function on the circle, as in [7], and is given directly on the line. We also provide new approximate extremals for aI + bH in the case p > 2.
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On the norm of the operator $aI+bH$ on $L^p(\mathbb R)$
Yong Ding,Loukas Grafakos,Kai Zhu 대한수학회 2018 대한수학회보 Vol.55 No.4
We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky \cite{HKV}: let $H$ be the Hilbert transform and let $a,b$ be real constants. Then for $1<p<\infty$ the norm of the operator $aI+bH$ from $L^p(\mathbb R)$ to $L^p(\mathbb R)$ is equal to $$ \bigg(\max_{x\in \mathbb R}\frac{|ax-b+(bx+a)\tan \frac{\pi}{2p}|^p+|ax-b-(bx+a)\tan \frac{\pi}{2p}|^p}{|x+\tan \frac{\pi}{2p}|^p+|x-\tan \frac{\pi}{2p}|^p} \bigg)^{\frac 1p}. $$ Our proof avoids passing through the analogous result for the conjugate function on the circle, as in \cite{HKV}, 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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