Reverse Hölder's Inequality for Sums

Theorem
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that $\displaystyle \frac 1 p - \frac 1 q = 1$.

Suppose that the sequences $\mathbf{x} = \langle{x_n}\rangle$ and $\mathbf{y} = \langle{y_n}\rangle$ in $\C$ (or $\R$) are such that the series
 * $\displaystyle \left( {\sum_{n=1}^\infty \left\vert{x_n}\right\vert^p} \right)^{1/p}$

and
 * $\displaystyle \left( {\sum_{n=1}^\infty \left\vert{y_n}\right\vert^{-q} } \right)^{-1/q}$

are convergent.

Denote these values by $\left\Vert{ \mathbf{x} }\right\Vert_p$ and $\left\Vert{ \mathbf{y} }\right\Vert_{-q}$, respectively.

Here, the notation $\left\Vert{ \mathbf{x} }\right\Vert_p$ does not denote a norm, but is instead just a convenient notation similar to that of the $p$-norm, which is only defined when $p \ge 1$.

Let $\left\Vert{ \mathbf{x}\mathbf{y} }\right\Vert_1$ denote the $1$-norm of $\mathbf{x}\mathbf{y}$, if $\mathbf{x}\mathbf{y}$ is in the Lebesgue space $\ell^1$.

Then $\left\Vert{ \mathbf{x}\mathbf{y} }\right\Vert_1 \ge \left\Vert{ \mathbf{x} }\right\Vert_p \left\Vert{ \mathbf{y} }\right\Vert_{-q}$.

Proof
Assume WLOG that $\mathbf x$ and $\mathbf y$ are non-zero.

Let:
 * $\displaystyle \mathbf u = \langle {u_n} \rangle = \frac {\mathbf x} {\left\Vert {\mathbf x} \right\Vert_p}$

and:
 * $\displaystyle \mathbf v = \langle {v_n} \rangle = \frac {\mathbf y} { \left\Vert {\mathbf y} \right\Vert_{-q} }$

Then:
 * $\displaystyle \left\Vert {\mathbf u} \right\Vert_p = \frac {1} {\left\Vert {\mathbf x} \right\Vert_p} \left({ \sum_{n=1}^\infty \left\vert{x_n}\right\vert^p }\right)^{1/p} = 1$

Similarly:
 * $\left\Vert {\mathbf{v}} \right\Vert_{-q} = 1$

It then suffices to prove that:
 * $\displaystyle \left\Vert {\mathbf{u}\mathbf{v}} \right\Vert_1 = \frac {\left\Vert {\mathbf{x}\mathbf{y}} \right\Vert_1} { \left\Vert {\mathbf x} \right\Vert_p \left\Vert {\mathbf y} \right\Vert_{-q} } \ge 1$

By Reverse Young's Inequality for Products:
 * $\displaystyle \left\vert u_n v_n \right\vert \ge \frac 1 p \left\vert u_n \right\vert^p - \frac 1 q \left\vert v_n \right\vert^{-q}$

Summing over all $n \in \N$ gives:
 * $\displaystyle \left\Vert {\mathbf{u}\mathbf{v}} \right\Vert_1 \ge \frac 1 p \left\Vert {\mathbf{u}} \right\Vert_p - \frac 1 q \left\Vert {\mathbf{v}} \right\Vert_{-q} = 1$

as desired.

Also see

 * Hölder's Inequality for Sums