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Dini continuity $\Rightarrow$ Hörmander condition

I'm trying to show that a Dini-continuous function $\Omega\in L^1(\mathbb{S}^{n-1})$ satisfies the Hörmander condition, i.e. $$\sup_{y\ne0}\int_{|x|\ge2|y|}|K_\Omega(x-y)-K_\Omega(x)|dx:=A_2<\infty,$$ …
Lille Nordmann's user avatar