Modulus of Continuity - The Translation Group of Lp Functions, and Moduli of Continuity Lp.

The Translation Group of Lp Functions, and Moduli of Continuity Lp.

Let let a function of class and let The h-translation of, that is the function

belongs to the class; moreover, if, there holds

as

Therefore, since translations are in fact linear isometries, also

as

uniformly on .

In other words, the map defines a strongly continuous group of linear isometries of . In the case 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 for a measurable function is a modulus of continuity such that

This way, moduli of continuity also give a quantitative account of the continuity property shared by all functions.

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