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Let $f(x)$ be defined on $[-1,1]$ spanned by Legendre polynomials $\{P_n(x)\}$. $f(x)$ has a total mass $M=\int_{-1}^1 fdx$. Can we prove the following inequality

$$\int_{-1}^{1} f^2 dx \le C_1+C_2\sqrt{\int_{-1}^{1}(1-x^2)f'f' dx}$$

where constants $C_1$ and $C_2$ only depends on $M$

Note similar inequality can be achieved using standard Fourier method,

$$|f|^2_2\le M\|f\|_\infty \le M C \|f\|_{H_1}$$ However the same trick doesn't apply because $P_n$ are not orthonormal $\int_{-1} ^1P_n^2dx=2/(2n+1)$.

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    So normalize them. Instead of using the $P_n$ themselves as your basis, use $$\sqrt{\frac{2n+1}2}P_n$$ instead. – 2017-02-16
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    But then the maximum of the new basis is growing. I don't have the same expression for infinity norm any more. – 2017-02-16

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