be the Haar system on
. We show that for any vectors
from a separable Hilbert space
and any Borel subset
, we have the sharp inequality
The above estimate is generalized to the sharp estimate
-valued continuous-time martingales such that
is differentially subordinate to
. An application to Riesz system of harmonic functions is indicated.
Key Words: Haar system, martingale, differential subordination, best constant.
2000 Mathematics Subject Classification: Primary: 60G44;
Secondary: 60G42, 31B05, 46E30.
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