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Let ⁿ denote the usual n-torus and let $S{̃}_u^{\delta }\left(f\right)$, u > 0, denote the Bochner-Riesz means of order δ > 0 of the Fourier expansion of f ∈ L¹(ⁿ). The main result of this paper states that for f ∈ H¹(ⁿ) and the critical index α: = (n-1)/2, $li{m}_{R\to \infty }1/logR{\int }_0^R\left(\right||S{̃}_u^{\alpha }\left(f\right)-{f\left|\right|}_{H¹\left(ⁿ\right)})/(u+1)du=0$.