Partial Converse of Holder's Theorem

Question

Holder's Theorem is the following: Let $E\subset \mathbb{R}$ be a measurable set. Suppose $p\ge 1$ and let $q$ be the Holder conjugate of $p$ - that is, $q=\frac{p}{p-1}.$ If $f\in L^p(E)$ and $g\in L^q(E),$ then $$\int_E \vert fg\vert\le \Vert f\Vert _p\cdot\Vert g \Vert_q$$

I am trying to prove the following partial converse is true. Suppose $g$ is integrable, $p>1$ and $$\int_E \vert fg\vert\le M\Vert f\Vert_p$$ when $f\in L^p(E)$ is bounded, for some $M\ge0$. I am trying to prove that this implies that $g\in L^q(E)$, where $q$ is the conjugate of $p$. I am interested in knowing if my proof is correct or if it has any potential.

Attempt: Assuming this is true, I want to show $\int_E \vert g\vert^q<\infty$. Note that $g^{q-1}\in L^p(E)$ since $\int_E \vert g\vert^{pq-p}=\int_E \vert g \vert ^q<\infty$ (I think this is true since $g$ is integrable - I know it holds when $q$ is a natural number). Take $f=g^{q-1}$ and the hypothesis tells us that $$\int_E \vert g \vert^q\le M\Vert g^{q-1}\Vert_P\\<\infty \,\,(\text{since }g^{q-1}\in L^p)$$so $g$ is integrable.

Answer 1

This is not a correct proof. You want to show that $\int_E |g|^q < \infty$, but then you are using this in the proof. So nothing has been shown.

To write a correct proof, you will have to approximate $g$, as you wrote in your last comment. E.g.set $E_k = E \cap B_k(0)$ (so these sets now have finite measure) and $$ g_k(x) = \begin{cases} g(x) \quad (x \in E_k, |g(x)| \le k) \\ 0 \quad \text{otherwise} \end{cases} $$ Clearly $\int_E |g_k|^q < \infty$ for all $k$. Now use the assumption and a suitable limit, e.g. with Lebesgue's Theorem.

Answer 2

An alternative proof.

Let $T_g(f):=\int_E f g\,,\, \forall f\in L^p.$

It's clear that $T_g$ is linear.

By the condition $\int_E \vert fg\vert\le M\Vert f\Vert_p$, we see $T_g$ is bounded.

So, $T_g\in (L^p)^*.$

By the Riesz Representation Theorem for the Dual of $L^p$(c.f. p. 160, Real Analysis, 4th Edition, Royden), we know there exists an unique $L^q$ function $\tilde{g}$ s.t. $T_g(f)=\int_E f \tilde{g}\,,\, \forall f\in L^p.$ By the uniqueness, we conclude that $g=\tilde{g}\in L^q.$

$\tag*{$\blacksquare$}$

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The following is the theorem I quoted in my answer.

$\tag*{$\blacksquare$}$