Roman Shvidkoy

BANACH SPACES WITH THE DAUGAVET PROPERTY

A Banach space X is said to have the Daugavet property if every operator T: X -* X of rank 1 satisfies 11 Id +Tl= 1 + flTIl. We show that then every weakly compact operator satisfies this equation as well and that X contains a copy of t1. However, X need not contain a copy of L1. We also study pairs of spaces X C Y and operators T: X -* Y satisfying I I J + T I I -_ 1-4- 1f T I I, where J: X -* Y is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with f Id +Tll 1 + Il T l I is as small as possible and give characterisations in terms of a smoothness condition.

Isospectral theory of Euler equations

Isospectral problem of both 2D and 3D Euler equations of inviscid fluids, is investigated. Connections with the Clay problem are described. Spectral theorem of the Lax pair is studied.

A DIMENSION-DEPENDENT MAXIMAL INEQUALITY

In this short note we show that sup{‖Mν‖ : ν is measure on Rn}, where ‖Mν‖ denotes the centered Hardy-Littlewood maximal operator, depends exponentially on n. 1. Statement of the problem Let ν be a σ-finite measure on the Borel subsets of R. Define the Hardy-Littlewood centered maximal operator associated with ν by Mνf(x) = sup r>0 1 ν(Br(x)) ∫ Br(x) |f |dν, x ∈ R. It was proved in [1, 2] that ‖Mνf‖Lp(Rn,ν) ≤ C‖f‖Lp(Rn,ν), 1 1 such that one can find [α] points x1, x2, ..., x[αn] on the euclidian sphere S n−1 such that ‖xi − xj‖ > 1, i 6= j. The maximal value of α is immaterial. A simple argument based on volume estimates yields α ≥ e(π/6)21/2. Let us fix x1, x2, ..., x[αn] as in the claim and put ν = δ{0} + ∑