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From Folland's book on PDEs
- Lemma
Let f∈L1(ℝn) be continuous.
Let φ∈L1(ℝn) be bounded and such that φ(x) is positive and decreasing in |x| and
Set
(Note that also .) Then for each x∈ℝn
- Proof
Given η>0 we need to show that there exists α>0 such that, for all 0<ε<α, we have
Since f is continuous there exists α1(η)>0 such that
where |B| is the volume of the unit ball in ℝn; in particular
It thus remains to find an α2(η)>0 (then set α:=min{α1,α2}) such that
But this expression is bounded by the sum of
so such a value for α2(η) must exist.