On the primarity ofH ∞-spaces

It is proved that the poly-discH ^∞ spacesH ^∞(D ^m ) are primary. We use the representation of these spaces as the direct sum of certain polynomial spaces and some properties of the Fourier transforms ofL ¹(π ^m ) functions.     

A new class ofℒ 1-spaces

It is shown that the class of separableℒ ¹-spaces with the Radon-Nikodym property only admitsL {sp1} as universal element.     

Distribution of points on spheres and approximation by zonotopes

It is proved that if we approximate the Euclidean ballB ⁿ in the Hausdorff distance up toɛ by a Minkowski sum ofN segments, then the smallest possibleN is equal (up to a possible logarithmic factor) toc(n)ε ^{−2(n−1)/(n+2)}. A similar result is proved ifB ⁿ is replaced by a general zonoid inR ⁿ .     

Dunford-pettis operators onL 1 and the Radon-Nikodym property

Using the duality between Dunford-Pettis operators onL ¹ and Pettis-Cauchy martingales, we prove that the Dunford-Pettis operators fromL ¹ intoL ¹ form a lattice. We show also that a Banach spaceX has the Radon-Nikodym property provided the Dunford-Pettis members of ℒ(L ¹,X) are representable. The lifting of dual valued Dunford-Pettis operators is investigated. Some remarks are included.     

An averaging result forl 1-sequences and applications to weakly conditionally compact sets inL x 1

We establish a theorem onl ¹-sequences obtained by averaging of semi-norms. The result is applied in the study of weakly conditionally compact subsets ofL _x ¹ , whereX is a Banach space. Aspirant N.F.W.O. Belgium     

Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order

Our first result is a ‘sum-product’ theorem forsubsets A of the finite field F_p, p prime, providing a lowerbound on max (|A + A|, |A · A|). The second and mainresult provides new bounds on exponential sums