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

A. Wi´snicki / Journal of Functional Analysis 236 (2006) 447–456 455
Theorem 4.3. (See [27, Theorem 1]) Let X be a Banach space which is uniformly convex in
every direction. If Y is a Banach space such that R ⊕ Y , endowed with a strictly monotone norm,
has WFPP, then X ⊕ Y also has WFPP.
Theorems 3.3 and 4.3 give
Proposition 4.4. Let X be a Banach space which is uniformly convex in every direction. If Y has
SFPP, then X ⊕ Y , endowed with a strictly monotone norm, has WFPP.
We conclude with the observation concerning spaces with the Schur property. The proof is
similar to that in Section 3.
Proposition 4.5. Let X be a Banach space with the Schur property and let Y has SFPP. Then
X ⊕ Y , endowed with a norm of type (UL) or a strictly monotone norm, has WFPP.
Problem. It is natural to ask whether the results of this paper are valid for the product space
endowed with the “maximum” norm or, more generally, with a monotone norm.
Acknowledgment
The author thanks the referee for valuable comments on the manuscript.
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