Luis Rodríguez-Piazza

Every separable Banach space is isometric to a space of continuous nowhere differentiable functions

We prove the result stated in the title; that is, every separable Banach space is linearly isometric to a closed subspace E of the space of continuous functions on [0, 1], such that every nonzero function in E is nowhere differentiable.

Composition operators on Hardy-Orlicz spaces

We investigate composition operators on Hardy-Orlicz spaces when the Orlicz function Ψ grows rapidly: compactness, weak compactness, to be psumming, order bounded, . . . , and show how these notions behave according to the growth of Ψ. We introduce an adapted version of Carleson measure. We construct various examples showing that our results are essentially sharp. In the last part, we study the case of Bergman-Orlicz spaces. Mathematics Subject Classification. Primary: 47 B 33 – 46 E 30; Secondary: Key-words. Bergman-Orlicz space – Carleson measure – composition operator – Hardy-Orlicz space

On the Muskat problem: global in time results in 2D and 3D

Abstract:This paper considers the three-dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an

SOME OBSERVATIONS ABOUT THE SPACE OF WEAKLY CONTINUOUS FUNCTIONS FROM A COMPACT SPACE INTO A BANACH SPACE

L^2

Banach spaces in which every compact lies inside the range of a vector measure

maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the available estimates to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in spaces with critical regularity, giving solutions with bounded slope and time integrable bounded curvature.

The range of a vector measure determines its total variation

For emergency cooling of a nuclear reactor plant, there is provided a primary flow circuit for cooling gas through the reactor core, and then through the primary sides of a reheater and a main steam generator, to a main circulator pump which returns the gas to the core. Another flow circuit is provided which conducts condensed water from a turbine condenser to the secondary side of a main steam generator, and conducts the steam produced therein through a high pressure part of a main turbine and then to the secondary side of the reheater and to the low pressure part of the main turbine and to the condenser. Condensed water from the condenser is also pumped into an auxiliary generator parallel with the main steam generator where it is vaporized and the steam produced is conducted to the secondary part of the reheater 9, and then to a turbine which drives the main circulator. A shunt conduit in connection with a shunt valve is arranged to connect the outlet of the secondary side of the main steam generator to the inlet of the drive turbine of the main circulator.

Some new thin sets of integers in harmonic analysis

We prove that the compact subsets of a Banach space X lie inside ranges of X-valued measures if and only if X* can be embedded in an LI space. In these spaces we prove that every compact is, in fact, a subset of a compact range. We also prove that if every compact of X is a subset of the range of an X-valued measure of bounded variation, then X is finite dimensional. Thus we answer a question by R. Anantharaman and J. Diestel.

Finite metric spaces needing high dimension for lipschitz embeddings in banach spaces

We prove that if the ranges of two finitely additive measures with values in a normed space have the same closed convex hull, then the measures have the same total variation. We also study the monotonicity of this variation with respect to the range, proving that a normed space X is C-isomorphic to a subspace of an L space if and only if, for every pair ,u, v of X-valued measures such that the range of u lies inside the closed convex hull of the range of v , the total variation of ,u is less than or equal to C times the total variation of v . This allows us to answer two questions raised by R. Anantharaman and J. Diestel. INTRODUCTION AND NOTATION In [AD] the authors asked whether there can exist two vector measures with the same range, exactly one of them having bounded variation. In Theorem 3 we prove that this is impossible; moreover, we prove that the range of a vector measure determines its total variation: measures with the same range have the same total variation. Actually the question had a finite-dimensional nature, as the proof of Theorem 3 reveals; it suffices to prove it for finite-dimensional spaces to extend it to all normed spaces. Anantharaman and Diestel also proved, via a result of Grothendieck, that a subspace X of L1 enjoys the property that if the range of an X-valued measure ,u lies inside the range of another measure of bounded variation, then ,u has bounded variation too. We again prove this result in Theorem 5 and show that this property characterizes the subspaces of L1, answering another question in [AD]. Here we use the local structure of L1 and a finite-dimensional characterization of its subspaces due to Lindenstrauss and Pelczynski [LP]. Let us introduce some notation. If X is a real, normed space, X* will denote its dual space. If K is a subset of X, co(K) will be the closed convex hull of K; we will use the fact that, for every f in X*, the supremum over Received by the editors November 16, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 28C05, 46G10; Secondary 46B20.

Strongly compact normal operators

We randomly construct various subsets A of the integers which have both smallness and largeness properties. They are small since they are very close, in various senses, to Sidon sets: the continuous functions with spectrum in Λ have uniformly convergent series, and their Fourier coefficients are in ℓp for all p > 1; moreover, all the Lebesgue spaces LΛq are equal forq < +∞. On the other hand, they are large in the sense that they are dense in the Bohr group and that the space of the bounded functions with spectrum in Λ is nonseparable. So these sets are very different from the thin sets of integers previously known.

Operators into Hardy spaces and analytic Pettis integrable functions

We construct a sequence of metric spaces (Mn) with cardMn=3n satisfying that for everyc<2, there exists a real numbera(c)>0 such that, if the Lipschitz distance fromMn to a subset of a Banach spaceE is less thanc, then dim(E) ≥a(c)n. We also prove several results about embeddings of metric spaces whose non-zero distance values are in the interval [1,2].