📂Lebesgue Spaces

Fundamental Theorem of Algebra

Theorem¹

Suppose that $p, q, r \ge 1$ satisfies $\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r} = 2$. Then, for all ${u \in L^{p}(\mathbb{R}^{n})}$, ${v \in L^{q}(\mathbb{R}^{n})}$, ${w \in L^{r}(\mathbb{R}^{n})}$, the following equation holds.

$$\begin{equation} \left| \int_{\mathbb{R}^{n}} (u \ast v)(x)w(x)dx \right| \le \left\| u \right\|_{p} \left\| v \right\|_{q} \left\| w \right\|_{r} \end{equation}$$

Here, $u \ast v$ is the convolution of $u$ and $v$.

Description

This is called Young’s theorem.

The inequality $(1)$ holds when there is a constant $K=K(p, q, r, n)<1$ on the right side. The best (smallest) constant is as follows.

$$K(p, q, r, n)=\left( \dfrac{ p^{1/p} q^{1/q} r^{1/r} }{ (p^{\prime})^{1/p^{\prime}}(q^{\prime})^{1/q^{\prime}}(r^{\prime})^{1/r^{\prime}} } \right)^{n/2}$$

Proof

Let’s call $p, q, r$’s Hölder conjugates as $p^{\prime}, q^{\prime}, r^{\prime}$ each.

$$\frac{1}{p} + \frac{1}{p^{\prime}} = 1 \quad \text{and} \quad \frac{1}{q} + \frac{1}{q^{\prime}} = 1 \quad \text{and} \quad \frac{1}{r} + \frac{1}{r^{\prime}} = 1$$

Then, the following equation holds.

$$\dfrac{1}{p^{\prime}} + \dfrac{1}{q^{\prime}} + \dfrac{1}{r^{\prime}} = 3 - \dfrac{1}{p} - \dfrac{1}{q} - \dfrac{1}{r} = 1$$

$$\frac{p}{q^{\prime}}+\dfrac{p}{r^{\prime}}=p\left( \frac{1}{q^{\prime}}+\dfrac{1}{r^{\prime}} \right) =p\left(1-\dfrac{1}{p^{\prime}}\right)=p\left(1-\frac{p-1}{p} \right)=p-p+1=1$$

Similarly,

$$\dfrac{r}{p^{\prime}} + \dfrac{r}{q^{\prime}} = 1 \quad \text{and} \quad \dfrac{q}{p^{\prime}} + \dfrac{q}{r^{\prime}} = 1$$

Therefore, for three functions

$$U(x, y)=|v(y)|^{q/p^{\prime}}|w(x)|^{r/p^{\prime}}$$

$$V(x, y)=|u(x-y)|^{p/q^{\prime}}|w(x)|^{r/q^{\prime}}$$

$$W(x, y)=|u(x-y)|^{p/r^{\prime}}|v(y)|^{q/r^{\prime}}$$

the next equation is satisfied.

$$(UVW)(x, y)=u(x-y)v(y)w(x)$$

Let’s calculate $\left\| V \right\|_{q^{\prime}}$.

$$\begin{align*} |V |_{q^{\prime}} =&\ \left( \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |u(x-y)|^p |w(x)|^r dxdy\right)^{1/q^{\prime}} \\ =&\ \left( \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} |u(x-y)|^p dy \right) |w(x)|^rdx \right)^{1/q^{\prime}} \end{align*}$$

The second equality is due to Fubini’s theorem. When you look at the inner bracket on the second line, you can see that it is the norm of $u$ irrespective of the value of $x$. Replacing it with $x-y=z$,

$$\int_{\mathbb{R}^n} |u(x-y)|^p dy = \int_{\mathbb{R}^n} |u(z)|^pdz = \left\| u \right\|_{p}^p$$

Therefore, the above equation is

$$\begin{align*} \left\| V \right\|_{q^{\prime}} =&\ \left\| u \right\|_{p}^{p/q^{\prime}}\left( \int_{\mathbb{R}^n}|w(x)|^rdx \right)^{1/q^{\prime}} \\ =&\ \left\| u \right\|_{p}^{p/q^{\prime}} \left\| w \right\|_{r}^{r/q^{\prime}} \end{align*}$$

Since $\left\| u \right\|_{p}, \left\| w \right\|_{r}$ exists, $\left\| v \right\|_{q^{\prime}}$ also exists, and its value is as above. Similarly,

$$\left\| U \right\|_{p^{\prime}}=\left\| v \right\|_{q}^{q/p^{\prime}}\left\| w \right\|_{r}^{r/q^{\prime}}$$

is valid, and

$$\left\| W \right\|_{r^{\prime}} = \left\| u \right\|_{p}^{p/r^{\prime}} \left\| v \right\|_{q}^{q/r^{\prime}}$$

Using these results, when applying the Hölder’s inequality for three functions,

$$\begin{align*} \left| \int_{\mathbb{R}^n} (u \ast v)(x)w(x)dx \right| \le& \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |u(x-y)|\ |v(y)|\ |w(x)| dy dx \\ =&\ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n}U(x, y) V(x, y) W(x, y) dy dx \\ \le & \left\| U \right\|_{p^{\prime}} \left\| V \right\|_{q^{\prime}} \left\| W \right\|_{r^{\prime}} \\ =&\ \left\| u \right\|_{p} \left\| v \right\|_{q} \left\| w \right\|_{r} \end{align*}$$

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