Category:Young's Inequality for Convolutions

This category contains pages concerning Young's Inequality for Convolutions:

Let $p, q, r \in \R_{\ge 1}$ satisfy:

$1 + \dfrac 1 r = \dfrac 1 p + \dfrac 1 q$

Let $\map {L^p} {\R^n}$, $\map {L^q} {\R^n}$, and $\map {L^r} {\R^n}$ be Lebesgue spaces with seminorms $\norm {\, \cdot \,}_p$, $\norm {\, \cdot \,}_q$, and $\norm {\, \cdot \,}_r$ respectively.

Let $f \in \map {L^p} {\R^n}$ and $g \in \map {L^q} {\R^n}$.

Then the convolution $f * g$ is in $\map {L^r} {\R^n}$ and the following inequality is satisfied:

$\norm {f * g}_r \le \norm f_p \cdot \norm g_q$

Source of Name

This entry was named for William Henry Young.