Young's inequality for integral operators explained

In mathematical analysis, the Young's inequality for integral operators, is a bound on the

L ^{p\to} L ^{q} operator norm of an integral operator in terms of L ^{r} norms of the kernel itself.

Statement

Assume that

X and Y are measurable spaces, K : X x Y \to R is measurable and q,p,r\geq 1 are such that (1)/(q) = (1)/(p) + (1)/(r) -1 . If \int _{Y} |K (x, y)| ^{r} d y \le C ^{r} for all x\in X and \int _{X} |K (x, y)| ^{r} d x \le C ^{r} for all y\in Y then ^[1] \int _{X} \left|\int _{Y} K (x, y) f(y) d y\right| ^{q} d x \le C ^{q} \left( \int _{Y} |f(y)| ^{p} d y\right) ^{(q)/(p)} .

Particular cases

Convolution kernel

If

X = Y = R ^{d} and K (x, y) = h (x - y) , then the inequality becomes Young's convolution inequality.

See also

Young's inequality for products

Notes and References

1. Theorem 0.3.1 in: C. D. Sogge, Fourier integral in classical analysis, Cambridge University Press, 1993.