Editing Modulus of continuity (section)

==The translation group of ''L<sup>p</sup>'' functions, and moduli of continuity ''L<sup>p</sup>''.==
Let 1 ≤ ''p''; let ''f'' : '''R'''<sup>''n''</sup> → '''R''' a function of class ''L<sup>p</sup>'', and let ''h'' ∈ '''R'''<sup>''n''</sup>. The ''h''-[[Translation (geometry)|translation]] of ''f'', the function defined by (τ<sub>''h''</sub>''f'')(''x'') := ''f''(''x''−''h''), belongs to the ''L<sup>p</sup>'' class; moreover, if 1 ≤ ''p'' < ∞, then as ǁ''h''ǁ → 0 we have:

:<math>\|\tau_h f - f\|_p=o(1).</math>

Therefore, since translations are in fact  linear isometries, also

:<math>\|\tau_{v+h} f - \tau_v f\|_p=o(1),</math>

as ǁ''h''ǁ → 0, uniformly on ''v'' ∈ '''R'''<sup>''n''</sup>.

In other words, the map ''h'' → τ<sub>''h''</sub> defines a strongly continuous group of linear isometries of ''L<sup>p</sup>''. In the case ''p'' = ∞ the above property does not hold in general: actually, it exactly reduces to the uniform continuity, and defines the uniform continuous functions. This leads to the following definition, that generalizes the notion of a modulus of continuity of the uniformly continuous functions: a modulus of continuity ''L<sup>p</sup>'' for a measurable function ''f'' : ''X'' → '''R''' is a modulus of continuity ω : [0, ∞] → [0, ∞] such that

:<math>\|\tau_h f - f\|_p\leq \omega(h).</math>

This way, moduli of continuity also give a quantitative account of the continuity property shared by all ''L<sup>p</sup>'' functions.