In mathematical analysis, the Young's inequality for integral operators, is a bound on the Lp→Lq operator norm of an integral operator in terms of Lr norms of the kernel itself.
Contents
Statement
Assume that X and Y are measurable spaces, K:X×Y→ℝ is measurable and q,p,r≥1 are such that 1q=1p+1r−1. If
∫Y|K(x,y)|rdy≤Cr for all x∈X
and
∫X|K(x,y)|rdx≤Cr for all y∈Y
then
∫X|∫YK(x,y)f(y)dy|qdx≤Cq(∫Y|f(y)|pdy)qp.
Particular cases
Convolution kernel
If X=Y=ℝd and K(x,y)=h(x−y), then the inequality becomes Young's convolution inequality.