Suppose for two functions $f, g\in L^{\infty}([0,1])$ and $$||f-g||_{L_1([0,1])}\leq \epsilon$$ Under what condition can we get $||f-g||_{L^\infty([0,1])}\leq K(\epsilon)$ where $K(\cdot)$ is some function (e.g., logarithm function)? What is the weakest condition to ensure $K(\epsilon)=\tilde{K}\cdot\epsilon$ for some constant $\tilde{K}$?
2026-03-29 12:03:56.1774785836
Sufficient condition for $L^\infty$ distance bound
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