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Let $(X,\mathcal{A},\mu$) be a finite measure space and $p,q\in(0,\infty)$ such that $1/p+1/q=1$. Let $g\in\mathcal{M}(\mathcal{A})$ measurable function such that

$$\int |fg|d\mu\leq C\|f\|_p$$

for all $f\in\mathcal{L}^p(\mu)$ and some constant $C$.

Show that $g\in\mathcal{L}^q(\mu)$

I understand that Holder's Inequality does the reverse argument with $C=\|g\|_q$, but i don't know how to bend this into a proof.

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  • $\begingroup$ What about choosing $f$ in some careful way so that the integral on the left turns into something relevant? $\endgroup$ – Giuseppe Negro Apr 1 '14 at 15:28
  • $\begingroup$ I will write out my answer (which is different from mookid's tomorrow), since I'm in the same course as you. $\endgroup$ – user45878 Apr 1 '14 at 16:01
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We use the relations $p>1;\ \ pq-p=q; \ \ p/(p-1)=q$, and the fact that $\mu(X)<\infty$, so that $\mu\{|g|>A\}<\infty$ too.

$$\int |g|^q1_{|g|\le N} d\mu = \int |g| \left[|g|^{q-1} 1_{|g|\le N}\right]d\mu \\\le C\left[ \int |g|^{(q-1)p} 1_{|g|\le N} d\mu \right] ^{1/p} = C\left[ \int |g|^{q} 1_{|g|\le N} d\mu \right] ^{1/p} \\ \left[\int |g|^q1_{|g|\le N} d\mu \right] ^{1-1/p} = \left[\int |g|^q1_{|g|\le N} d\mu \right] ^{1/q} \le C $$ As the bound does not depend on $N$, we conclude via the Fatou theorem: $$ ||g||_q \le C<\infty. $$

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