View source for Young's inequality for integral operators

{{technical|date=July 2017}}

In [[mathematical analysis]], the '''Young's inequality for integral operators''', is a bound on the <math>L^p\to L^q</math> [[operator norm]] of an [[integral operator]] in terms of <math>L^r</math> norms of the kernel itself.

==Statement==

Assume that <math>X</math> and <math>Y</math> are measurable spaces, <math>K : X \times Y \to \mathbb{R}</math> is measurable and <math> q,p,r\geq 1 </math> are such that <math>\frac{1}{q} = \frac{1}{p} + \frac{1}{r} -1</math>. If 
:<math>
\int_{Y} |K (x, y)|^r \,\mathrm{d} y \le C^r 
</math>  for all <math> x\in X </math>
and 
:<math>
\int_{X} |K (x, y)|^r \,\mathrm{d} x \le C^r
</math>   for all <math> y\in Y </math>
then <ref>Theorem 0.3.1 in: [[Christopher D. Sogge|C. D. Sogge]], ''Fourier integral in classical analysis'', Cambridge University Press, 1993. {{ISBN|0-521-43464-5}}</ref>
:<math>
  \int_{X} \left|\int_{Y} K (x, y) f(y) \,\mathrm{d} y\right|^q \, \mathrm{d} x
\le C^q \left( \int_{Y} |f(y)|^p \,\mathrm{d} y\right)^\frac{q}{p}.
</math>

==Particular cases==

=== Convolution kernel ===

If <math>X = Y = \mathbb{R}^d</math> and <math>K (x, y) = h (x - y) </math>, then the inequality becomes [[Young's convolution inequality]].

==See also==
[[Young's inequality for products]]

==Notes==
{{reflist}}

[[Category:Inequalities]]


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