On functions having coincident p-norms

Abstract

In a measure space (X , A, μ), we consider two measurable functions f , g : E → R, for some E ∈ A. We prove that the property of having equal p-norms when p varies in some infinite set P ⊆ [1, +∞) is equivalent to the following condition: μ({x ∈ E : | f (x)| > α}) = μ({x ∈ E : |g(x)| > α}) for all α ≥ 0.

Recovering a variable exponent

“…If γ is not constant, then the previous result is still true, but the distribution function will be with respect to a measure that depends on the unknown power p. This still gives a restatement of the original problem, but not a satisfactory characterization of the exponents p which give the same DN map. Recently Klun [27] and Erdélyi [20] proved that the equality of L n norms implies the equimeasurability of the functions. However, if γ is not identically one, then we only know the weighted L n norms (8), where the weight depends on the unknown power p. For this reason we need the slightly more general statement of lemma 11 with the two different weights.…”

Recovering a variable exponent

“…Recently Klun [33] and Erdélyi [24] proved that the equality of L n norms implies the equimeasurability of the functions. However, if γ is not identically one, then we only know the weighted L n norms (8), where the weight depends on the unknown power p. For this reason we need the slightly more general statement of lemma 11 with the two different weights.…”