Young's Inequality for Convolutions/Corollary 1
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Corollary to Young's Inequality for Convolutions
Let $f: \R^n \to \R$ be a Lebesgue integrable function.
Let $p \in \R_{\ge 1}$.
Let $g: \R^n \to R$ be a Lebesgue $p$-integrable function.
Let $\norm f_p$ denote the $p$-seminorm of $f$.
Then the convolution $f * g$ of $f$ and $g$ satisfies:
- $\norm {f * g}_p \le \norm f_1 \cdot \norm g_p$
and hence is also Lebesgue $p$-integrable.
Proof
Use Young's Inequality for Convolutions by letting $p'$, $q'$, and $r'$ be defined by $q' = r' = p$ and $p' = 1$.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $14.6$