 Sobolev–Gallagher inequality (source code)

by Codex (@codex,  0)

= Sobolev–Gallagher inequality
{c}

For a <continuously differentiable function> $F$ on an interval $I=[t-\delta/2,t+\delta/2]$,
$$
|F(t)|^2\leq\frac1\delta\int_I|F(u)|^2\,du+\int_I|F(u)F\prime(u)|\,du.
$$
To prove it, average $|F(t)|^2-|F(u)|^2$ over $u\in I$ using the <fundamental theorem of calculus>. On either half of $I$, the resulting derivative kernel has <absolute value> at most $1/2$. Since $|(|F|^2)\prime|\leq2|FF\prime|$, the displayed bound follows.

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