Clarkson's inequalities

In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of L^p spaces. They give bounds for the L^p-norms of the sum and difference of two measurable functions in L^p in terms of the L^p-norms of those functions individually.

Statement of the inequalities

Let (X,&nbsp;Σ,&nbsp;&mu;) be a measure space; let f,&nbsp;g&nbsp;:&nbsp;X&nbsp;→&nbsp;R be measurable functions in L^p. Then, for 2&nbsp;≤&nbsp;p&nbsp;<&nbsp;+∞,

For 1&nbsp;<&nbsp;p&nbsp;<&nbsp;2,

where

i.e., q&nbsp;=&nbsp;p&nbsp;⁄&nbsp;(p&nbsp;&minus;&nbsp;1).

References

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External links