On the Inequality of Different Metrics for Multiple Fourier-Haar Series

Abstract

Let 1 < p < q < ∞, f ∈ 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 {‖S_2^k(f)‖Lq}∞_k=0, the following relation is true ‖S_{2^k(f)‖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 _{‖S2^k12^k2(f)‖Lq̄ = o(2^{k1(1/p1 - 1/q1) + k2(1/p2 - 1/q2)}) for f ∈ Lp̄,τ̄[0, 1]².}