For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$. Let $0 < a < b < c < \infty$ and prove the following: $$ \|f\|_b \leqslant \max\{\|f\|_a, \|f\|_c\}. $$ Any help is appreciated because I dont understand the solution underneath
Proof of an inequality of $L^p$ norms
1 Answers
We have that
$$ \frac{1}{c}<\frac{1}{b}<\frac{1}{a}. $$ Therefore, there exists $\alpha\in (0,1)$ such that $$ \frac{1}{b}=\frac{\alpha}{c}+\frac{1-\alpha}{a}. $$ We claim that $$ \Vert f\Vert_b\leqslant \Vert f\Vert_c^\alpha\Vert f\Vert_a^{1-\alpha} (\ast) $$ from where it follows that $$ \Vert f\Vert_b\leqslant\max\{\Vert f\Vert_a, \Vert f\Vert_c\} $$ because $$ \Vert f\Vert_b\leqslant \Vert f\Vert_c^\alpha\Vert f\Vert_a^{1-\alpha}\leqslant \max\{\Vert f\Vert_a,\Vert f\Vert_c\}^{\alpha}\max\{\Vert f\Vert_a,\Vert f\Vert_c\}^{1-\alpha}=\max\{\Vert f\Vert_a,\Vert f\Vert_c\}. $$ Now, (*) can be proved easily using Holder's Inequality as follows $$ \int |f|^b=\int |f|^{\alpha b}|f|^{(1-\alpha)b}\leqslant \left(\int|f|^{\alpha b\frac{c}{\alpha b}}\right)^{\frac{\alpha b}{c}}\left(\int|f|^{(1-\alpha)b\frac{a}{(1-\alpha) b}}\right)^{\frac{(1-\alpha)b}{a}}=\left(\int|f|^{c}\right)^{\frac{\alpha b}{c}}\left(\int|f|^{a}\right)^{\frac{(1-\alpha)b}{a}}=\Vert f\Vert_c^{\alpha b}\Vert f\Vert_a^{(1-\alpha) b}, $$ where we have used the fact that $$ \frac{\alpha b}{c}+\frac{(1-\alpha)b}{a}=1. $$ Simiplifying we have the result.
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0Holder inequality states that for any 1
– 2012-11-19
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0How does the result follow from proving * ? I can't see it – 2012-11-19
– 2012-11-19