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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 30, Fasc. 1,
pages 61 - 72
 

SHARP INEQUALITIES FOR THE SQUARE FUNCTION OF A NONNEGATIVE MARTINGALE

Adam Osękowski

Abstract: We determine the optimal constants C
 p  and C*
  p  such that the following holds: if f is a nonnegative martingale and S(f) and f* denote its square and maximal functions, respectively, then

||S(f)|| < C || f ||, p < 1,
      p    p   p
and
||S(f )|| p < C*p||f*||p, p < 1.

2000 AMS Mathematics Subject Classification: Primary: 60G42; Secondary: 60G46.

Keywords and phrases: Martingale, square function, maximal function.

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