Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

454 A. Wi´snicki / Journal of Functional Analysis 236 (2006) 447–456
4. Consequences
It is a long-standing open problem whether all superreflexive spaces have FPP (and hence
SFPP). However, there are two important classes of spaces which have the super fixed point
property. Recall first that
ɛ0(X) = sup ɛ 0: δX(ɛ) = 0 ,
where δX denotes the modulus of convexity of a Banach space X. It is well known that ɛ0(X) < 2
implies superreflexivity of X. A recent result of García Falset et al. [12] (see also [32]), states that
if ɛ0(X) < 2, then X has FPP. In consequence, X has SFPP since ɛ0(X) = ɛ0((X)U ). Combining
it with Theorems 3.2 and 3.3 we obtain
Proposition 4.1. Let X be a Banach space with ɛ0(X) < 2 and let F be a finite-dimensional
space. Then F ⊕ X, endowed with a norm of type (UL) or a strictly monotone norm, has SFPP.
In 1997 Prus [33] introduced the notion of uniformly noncreasy spaces. A real Banach space
X is uniformly noncreasy if for every ε>0 there is δ>0 such that if f,g ∈ SX∗ and f −g ε,
then diam S(f,g,δ) ε, where
S(f,g,δ) = x ∈ BX: f(x) 1 − δ ∧ g(x) 1 − δ .
To be precise, we put diam ∅=0. It is well known that uniformly convex as well as uniformly
smooth spaces are uniformly noncreasy. The Bynum space l2,∞ , which is l2 space endowed with
the norm
x2,∞ = max x + 2, x −
2 ,
(see [4]), and the space X √ 2 , which is l2 space endowed with the norm
x√ 2 = maxx2, √
2x∞ ,
(see [3]), are examples of uniformly noncreasy spaces without normal structure. It was proved
in [33] that all uniformly noncreasy spaces are superreflexive and have SFPP. This yields
Proposition 4.2. Let X be uniformly noncreasy and let F be a finite-dimensional space. Then
F ⊕ X, endowed with a norm of type (UL) or a strictly monotone norm, has SFPP.
Remark. The above result is also valid for some generalizations of uniformly noncreasy spaces
given in [8,9,11].
Other examples of spaces with the super fixed point property are given by the author’s result
[39, Theorem 2.3]. In particular, the l p -products of uniformly noncreasy spaces have SFPP.
Banach spaces X with the property that R ⊕ X has WFPP were studied in [27]. The following
theorem was established for the l 1 -norm but the proof works for all strictly monotone norms.