## Weak $$\Phi$$-inequalities for the Haar system and differentially subordinated martingales.(English)Zbl 1266.60080

Summary: For a wide class of Young functions $$\Phi:[0,\infty)\to[0,\infty)$$, we determine the best constant $$C_{\Phi}$$ such that the following holds. If $$(h_{k})_{k\geq 0}$$ is the Haar system on $$[0,1]$$, then, for any vector $$a_{k}$$ from a separable Hilbert space $$\mathcal{H}$$ and $$\varepsilon_{k}\in \{-1,1\}$$, $$k=0, 1, 2,\dots$$, we have $\left|\left\{x\in [0,1]:\left|\sum_{k=0}^{n} \varepsilon_{k}a_{k}h_{k}(x)\right|\geq 1\right\}\right|\leq C_{\Phi } \int_{0}^{1}\Phi \left(\left|\sum_{k=0}^{n} a_{k}h_{k}(x)\right|\right)\,\mathrm{d}x,\quad n=0,1,2,\dots.$ This is generalized to the sharp weak-$$\Phi$$ inequality $\operatorname{P}(\sup_{t\geq 0}|Y_{t}|\geq 1)\leq C_{\Phi }\sup_{t \geq 0}\operatorname{E} \Phi (|X_{t}|),$ where $$X, Y$$ stand for $$\mathcal{H}$$-valued martingales such that $$Y$$ is differentially subordinate to $$X$$. These statements complement and generalize the results of Burkholder, Suh, the author and others.

### MSC:

 60G44 Martingales with continuous parameter 60G42 Martingales with discrete parameter 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

### Keywords:

Haar system; martingale; weak-$$\Phi$$ inequality; best constant
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### References:

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