# Tagged Questions

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### L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
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### How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p}$ from Jensen's inequality?

Let $f$ be a positive function with $\lVert f \rVert_{L^2}=1$. Let $p= 2n/(n-2)$. How to obtain $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p}$ from Jensen's inequality? Here all ...
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### Estimation of a scalar product

I encountered the following, which shouldn't be that hard, but I can't get my head around it. The problem is the following estimate (part of a bigger equation, but here's just the difficult part): ...
The following version of Jensens inequality is used in lecture notes, But i don't seem to get it . if $\phi : \mathbb R^n \to R$ is a convex function and $f_i \in L^1 (\Omega)$ for all $i$ and ...
Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that \left|\left|f+g\right|\right|_p^p + ...