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I am working on this homework problem, and I am totally stuck:

Let $(X,\mu)$ be a measure space, and let $1 \leq r < s < t < \infty$. Prove that there exist constants $\alpha,\beta>0$ so that $$ \|f\|_s \;\leq\; \|f\|_r^\alpha\,\|f\|_t^\beta $$ for any measurable function $f\colon X\to\mathbb{R}$. Use this to show that $$ L^r(X) \cap L^t(X) \,\subset\, L^s(X). $$

I know this is supposed not to be difficult. But I cannot solve it.

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1 Answer

up vote 5 down vote accepted

Hint: Write $\frac{1}{s} = \frac{\alpha}{r} + \frac{\beta}{t}$ and apply Hölder's inequality.

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I am trying to work out the detail. Thanks –  Kerry Apr 8 '11 at 6:01
    
@user7887: I should have added that you want $\alpha + \beta = 1$, so $0 \lt \alpha, \beta \lt 1$. –  t.b. Apr 8 '11 at 6:02
    
@Theo Buehler: I want to ask how to apply it, the left hand side only has $f$ available, I am thinking I need to make $f$ into $f^{\frac{1}{r}}$ or something, but that looks very ugly. –  Kerry Apr 8 '11 at 6:05
    
Thanks. Now got it. –  Kerry Apr 8 '11 at 6:18
    
@user7887. Very good. Playing around with conjugate exponents can be very tricky sometimes :) –  t.b. Apr 8 '11 at 6:23
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