Let M, N be real-valued martingales such that N is differentially subordinate to M. The paper contains the proofs of the following weak-type inequalities:
(i) If M≥0 and 0<p≤1, then
‖N‖p, ∞≤2‖M‖p
and the constant is the best possible.
(ii) If M≥0 and p≥2, then
![\[\Vert N\Vert_{p,\infty}\leq\frac{p}{2}(p-1)^{-1/p}\Vert M\Vert_{p}\]](/DPubS?service=Repository&version=1.0&verb=Disseminate&view=body&handle=euclid.bj/1251463285&content-type=gif_1)
and the constant is the best possible.
(iii) If 1≤p≤2 and M and N are orthogonal, then
‖N‖p, ∞≤Kp‖M‖p
where
![\[K_{p}^{p}=\frac{1}{\Gamma(p+1)}\cdot\biggl(\frac{\pi}{2}\biggr)^{p-1}\cdot\frac{1+1/3^{2}+1/5^{2}+1/7^{2}+\cdots}{1-1/3^{p+1}+1/5^{p+1}-1/7^{p+1}+\cdots}.\]](/DPubS?service=Repository&version=1.0&verb=Disseminate&view=body&handle=euclid.bj/1251463285&content-type=gif_2)
The constant is the best possible.
We also provide related estimates for harmonic functions on Euclidean domains.
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