> [!theorem] > > Let $p, q \in (1, \infty)$ be [[Hölder's Inequality|Hölder conjugates]], then for any $a, b > 0$, > $ > ab \le \frac{a^p}{p} + \frac{b^q}{q} > $ > *Proof*. Since the mapping $x \mapsto e^x$ is convex, > $ > \begin{align*} > ab &= e^{\ln(a) + \ln(b)} = e^{\frac{1}{p}\ln(a^p) + \frac{1}{q}\ln(b^q)} \\ > &\le \frac{1}{p}e^{\ln(a^p)} + \frac{1}{q}e^{\ln(b^q)} = \frac{a^p}{p} + \frac{b^q}{q} > \end{align*} > $ > [!theorem] > > If $f \in L^1$ and [$g \in L^p$](Lp%20Space), then [$f * g$](Convolution%20of%20Functions.md) exists [[Almost Everywhere|a.e.]], with $f * g \in L^p$, and $\norm{f * g}_p \le \norm{f}_1 \cdot \norm{g}_p$. > > *Proof*. Using [[Minkowski's Inequality|Minkowski's inequality for integrals]], > $ > \begin{align*} > \norm{f * g}_p &= \braks{\int \paren{\int f(y)g(x - y)dy}^{p}dx}^{1/p} \\ > &\le \int \paren{\int [f(y)g(x - y)]^{p}dx}^{1/p}dy \\ > &= \int f(y)\norm{g}_pdy = \norm{f}_1 \cdot \norm{g}_p > \end{align*} > $ > [!theorem] > > Let $p \ge 1$, then there exists $C_p \ge 0$ such that > $ > \abs{a + b}^p \le C_p(\abs{a}^p + \abs{b}^p) > $ > for all $a, b \in \real$. > > *Proof*. By convexity of $x \mapsto \abs{x}^p$, > $ > \begin{align*} > \frac{1}{2^p}\abs{a + b}^p &= \abs{\frac{a + b}{2}}^p \le \frac{\abs{a}^p + \abs{b}^p}{2} \\ > \abs{a + b}^p &\le 2^{p - 1}(\abs{a}^p + \abs{b}^p) > \end{align*} > $