Young's Inequality For Integral Operators

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, $K : X \times Y \to \mathbb$ is measurable and $q,p,r\geq 1$ are such that $\frac = \frac + \frac -1$. If
:$\int_ , K (x, y), ^r \,\mathrm y \le C^r$ for all $x\in X$
and
:$\int_ , K (x, y), ^r \,\mathrm x \le C^r$ for all $y\in Y$
then Theorem 0.3.1 in: C. D. Sogge, ''Fourier integral in classical analysis'', Cambridge University Press, 1993.
:$\int_ \left, \int_ K (x, y) f(y) \,\mathrm y\^q \, \mathrm x \le C^q \left( \int_ , f(y), ^p \,\mathrm y\right)^\frac.$

Particular cases

Convolution kernel

If $X = Y = \mathbb^d$ and $K (x, y) = h (x - y)$, then the inequality becomes Young's convolution inequality.

See also

Young's inequality for products

Notes

Inequalities

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