We consider the limiting property of the distribution function of L~p function at endpoints 0 and ∞ and prove that for λ > 0 the following two equations limλ→+∞λ~pm({x : |f(x)| > λ}) = 0, limλ→0+λ~pm({x : |f(x)| > λ}) = 0hold for f ∈ L~p(Rn) with 1 ≤ p < ∞. This result is naturally applied to many operators of type(p, q) as well.
We consider the limiting property of the distribution function of L^p function at endpoints 0 and ∞ and prove that for λ 〉 0 the following two equations limλ→+∞λ^p({x : |f(x)| 〉 λ}) = 0, limλ→0+λ^p({x : |f(x)| 〉 λ}) = 0hold for f ∈ L^p(Rn) with 1 ≤ p 〈 ∞. This result is naturally applied to many operators of type(p, q) as well.