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Weak type \((p,q)\)-inequalities for the Haar system and differentially subordinated martingales. (English) Zbl 1244.60045

Summary: For any \(1\leq p,q<\infty\), we determine the optimal constant \(C_{p,q}\) such that the following holds. If \((h_k)_{k\geq 0}\) is the Haar system on \([0,1]\), then for any vectors \(a_k\) from a separable Hilbert space \({\mathcal H}\) and \(\varepsilon_k\in\{-1,1\}\), \(k=0,1,2,\dots\), we have \[ \left\|\sum^n_{k=0}\varepsilon_ka_kh_k\right\|_{q,\infty}\leq C_{p,q}\left\|\sum^n_{k=0}a_kh_k\right\|_p. \] This is generalized to the sharp weak-type inequality \[ \|Y\|_{q,\infty}\leq C_{p,q}\|X\|_p, \] where \(X,Y\) stand for \({\mathcal H}\)-valued martingales such that \(Y\) is differentially subordinate to \(X\).

MSC:

60G42 Martingales with discrete parameter
60G44 Martingales with continuous parameter
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