Young's inequality for integral operators

In mathematical analysis, the Young's inequality for integral operators, is a bound on the operator norm of an integral operator in terms of norms of the kernel itself.

Statement

Assume that and are measurable spaces, is measurable and are such that . If for all and for all then Theorem 0.3.1 in: C. D. Sogge, Fourier integral in classical analysis, Cambridge University Press, 1993.

Particular cases

Convolution kernel

If and , then the inequality becomes Young's convolution inequality.

See also Young's inequality for products

Notes