ON THE INEQUALITY OF DIFFERENT METRICS FOR MULTIPLE FOURIER-HAAR SERIES

Let 1 < p < q < infinity, f is an element of L-p[0, 1]. Then, according to the inequality of different metrics due to S.M. Nikol'skii, for the sequence of norms of partial sums of the Fourier-Haar series {parallel to S-2k (f)parallel to(Lq)}(k=0)(infinity) the following relation is true parallel to S-2k (f)parallel to(Lq) = O (2(k(1/p-1/q))) . In this paper, we study the asymptotic behavior of partial sums in the Lorentz spaces. In particular, it is obtained that parallel to S-2k1 2k2 (f)parallel to(Lq) = o (2(k1(1/p1-1/q1))(+)(k2(1/p2-1/q2))) for f is an element of L (p) over bar (,)(tau) over bar [0, 1](2).