Reciprocal of Hölder's inequality

Asked Apr 22 '20 at 22:51

Active Apr 24 '20 at 22:25

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Let $(X,S,\mu)$ be a $\sigma$-finite measure space, and $g \in M(X,S)$ such that $gs \in L_1(\mu)$ for any simple function $s \in L_p(\mu)$, with $p \in (1, \infty)$. Suppose $A \geq 0$ exists such that $|\int gs \mu| \leq A \|s\|_p$ for any function $s$. Prove:

$g \in L_q$ with $\dfrac{1}{p} + \dfrac{1}{q}=1$ and $\|g\|_q\leq A$

edited Apr 24 '20 at 22:25

asked Apr 22 '20 at 22:51

Davshock

There is an answer here: https://math.stackexchange.com/questions/3619580/reciprocal-of-holders-inequality?rq=1 – Kavi Rama Murthy Apr 22 '20 at 23:17
You can also use : https://math.stackexchange.com/questions/3434325/if-fg-in-l-1-for-every-f-in-l-p-show-that-g-in-l-q?rq=1 – Kavi Rama Murthy Apr 22 '20 at 23:19
Thank you, it helped me – Davshock Apr 23 '20 at 14:17

Reciprocal of Hölder's inequality

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