SOME SHARP ESTIMATES FOR THE HAAR SYSTEM AND OTHER BASES IN L-1(0,1)

作者：Osekowski Adam^*

来源：Mathematica Scandinavica, 2014, 115(1): 123-142.

DOI：10.7146/math.scand.a-18006

Let h = (h(k))(k >= o) denote the Haar system of functions on [0, 1]. It is well known that h forms an unconditional basis of L-P (0, 1) if and only if 1 < p < infinity, and the purpose of this paper is to study a substitute for this property in the case p = 1. Precisely, for any lambda > 0 we identify the best constant beta = beta(h)(lambda) is an element of [0, 1] such that the following holds. If n is an arbitrary nonnegative integer and alpha(0), alpha(1), alpha(2),...,alpha(n), are real numbers such that parallel to Sigma(n)(k=0) alpha(k)h(k)parallel to(1) <= 1, then
vertical bar{x is an element of [0, 1] : vertical bar Sigma(n)(k=0) epsilon k alpha khk(x) vertical bar >= lambda}vertical bar <= beta,
for any sequence epsilon(0), epsilon(1), epsilon(2),...epsilon(n) of signs. A related bound for an arbitrary basis of L-1(0, 1) is also established. The proof rests on the construction of the Bellman function corresponding to the problem.

• 出版日期2014