SET-CONTRACTIONS AND BALL-CONTRACTIONS IN LP-SPACES

Consider the space Lp(Ω), 1 ≤ p < + ∞, where Ω is a σ-finite measure space. Defined δ and γ by: δ = 2max{ 1 p, (p - 1) p} and γ = 2|p - 2| p. The following relationship between set-contractions and ball-contractions in separable Lp(Ω) spaces is proved: If T: Lp(Ω) → Lp(Ω) is a k-set-contraction (respectively set-condensing mapping) then T γ is a k-ball-contraction (respectively ball-condensing mapping). If T: Lp(Ω) → Lp(Ω) is a k-ball-contraction, then T δ is a k-set-contraction. Furthermore these constants γ and δ are the best possible. © 1991.