If $|\int fg| \le M\|f\|_p$ for all $f\in L^p$, show that $g \in L^{q}$ and $\|g\|_q \le M$, where $1/p +1/q=1$

Question

Let $g$ be an integrable function on $[0,1]$ and let $1 \leq p < \infty$. Suppose there is a constant $M$ such that

$$ \left|\int f \;g \right| \leq M \; \|f\|_p $$ for all bounded measurable functions $f$.

Show that

a) $g \in L^q$ and

b) $||g||_q \leq M$ where $q$ is the conjugate number of $p$ (i.e. $1/p + 1/q =1$ ).

For a), I used Holder's inequality to obtain

\begin{align}\left|\int f \; g \right| &\leq \int |f \; g| \\ &= \|f \; g \|_1 \\ &\leq \|f\|_p \|g\|_q \end{align}

So $g \in L^q$. I am not sure how to bound $\|g\|_q $ by $M$. Is Riesz Representation Theorem used here?

Answer 1

You can indeed use the Riesz representation theorem to solve the problem. The hypothesis $$ \left| \int f g \right | \leq M \| f\|_{p}$$ shows in particular that the element $g$ defines a linear continuous functional $T_g$ on the space $L^{p}$ via the relation $$T_g(f) = \int f g.$$ Now, Riesz representation theorem tells you that there exists a unique $h \in L^{q}$ that represents the functional $T_g$ in the sense that $$T_g(f) = \int hf \hspace{0.9 cm} \text{for any $f \in L^{p}$.}$$ Moreover, the theorem states that $\|T_g\| = \|h\|_{q}$. From this you may deduce that $g=h$ and since you already have seen that $\|T_g\| \leq M$, you inmediately obtain that $\|g \| \leq M.$

Answer 2

I will assume that $g$ is real valued and $p>1$. (The case $p=1$ is similar).

I will prove b) directly and a) follows from b).

Let $N$ be a positive integer and $f=(sgn \,g)|g|^{q/p}I_{|g| \leq N}$ where $q$ is the conjugate index. Then $g$ is bounded and the given inequality becomes $\int_{|g| \leq N} |g|^{q} \leq M (\int_{|g| \leq N} |g|^{q})^{1/p}$. This implies that $(\int_{|g| \leq N} |g|^{q})^{1/q} \leq M$. now let $N \to \infty$.