In this paper, we establish Lp norm bounds for spectral clusters on compact manifolds, under the assumption that the metric is C1,1. Precisely, we show that the Lp estimates proven by Sogge in the case of smooth metrics hold under this limited regularity assumption. It is known by examples of Smith-Sogge that such estimates fail for C1,α metrics if α < 1.

Intrinsic Lp metrics are defined and shown to satisfy a dimension–free bound with respect to the Hausdorff metric. MSC 2000: 52A20, 52A27, 52A40, 60G15.

In this paper, we establish Lp norm bounds for spectral clusters on compact manifolds, under the assumption that the metric is C1,1. Precisely, we show that the Lp estimates proven by Sogge [12] in the case of smooth metrics hold under this limited regularity assumption. It is known by examples of Smith-Sogge [11] that such estimates fail for C1,α metrics if α < 1.

Codes and associated lattices are studied in the lp metric, particularly in the l1 (Lee) and the l∞ (maximum) distances. Discussions and results on decoding processes, classification and analysis of perfect or dense codes in these metrics are presented. Keywords—Codes and lattices, lp metric, Lee metric, perfect codes.

In our note, we prove the result that the Hilbert’s cube equipped with lp−metrics, p ≥ 1, cannot be isometrically embedded into c.