Young's convolution inequality - Imported Knowlpedia Content

Statement

Let $G$ be a unimodular locally compact group with a fixed Haar measure. Suppose $1\leq p,q,r\leq\infty$ satisfy

\frac1p+\frac1q=1+\frac1r.

For $f\in L^p(G)$ and $g\in L^q(G)$ , their convolution, initially defined where the integral exists, has an $L^r(G)$ representative and obeys Young's convolution inequality

\lVert f*g\rVert_r\leq \lVert f\rVert_p\lVert g\rVert_q.

Consequently convolution extends uniquely to a bounded bilinear map $L^p(G)\times L^q(G)\to L^r(G)$ .

Endpoint estimates

The case $p=1$ and $q=r$ says that averaging left translates of $g$ against $f$ costs at most $\lVert f\rVert_1$ . The case $p=q=2$ , $r=\infty$ , follows directly from the integral Hölder inequality. Taking $p=q=r=1$ proves that $L^1(G)$ is closed under convolution and supplies its Banach-algebra norm estimate.

Proof mechanism and extension

For compactly supported continuous functions, translation invariance of Haar measure, Hölder's inequality, and Tonelli's theorem give the endpoint bounds. Interpolation and density yield the remaining exponent range. The extension is independent of the approximating functions because the displayed estimate makes convolution jointly continuous in the indicated norms Folland, Chapter 2.

Scope and nonunimodular groups

Euclidean Young inequality is the case $G=\mathbb R^n$ . The same constant-one estimate holds on discrete groups with counting measure and on compact groups with normalized Haar measure.

References

G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 2, convolution estimates on locally compact groups.
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Volume I, Springer, 1963. DOI record. Relevant: convolution and $L^p$ -space inequalities.