Guillermo P. Curbera

Optimal Domains for Kernel Operators via Interpolation

The problem of finding optimal lattice domains for kernel operators with values in rearrangement invariant spaces on the interval (0,1) is considered. The techniques used are based on interpolation theory and integration with respect to C((0,1))-valued measures.

Banach lattices with the Fatou property and optimal domains of kernel operators

Abstract New features of the Banach function space L 1 w ( v ), that is, the space of all v -scalarly integrable functions (with v any vector measure), are exposed. The Fatou property plays an essential role and leads to a new representation theorem for a large class of abstract Banach lattices. Applications are also given to the optimal domain of kernel operators taking their values in a Banach function space.

Compactness properties of Sobolev imbeddings for rearrangement invariant norms

Compactness properties of Sobolev imbeddings are studied within the context of rearrangement invariant norms. Attention is focused on the extremal situation, namely, when the imbedding is considered as defined on its optimal Sobolev domain (with the range space fixed). The techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator (of just one variable).

A note on function spaces generated by Rademacher series

Let X be a rearrangement invariant function space on [0, 1] in which the Rademacher functions (»•„) generate a subspace isomorphic to I. We consider the space K(Tt, X) of measurable functions/ such t h a t / g e X for every function g = J^fc.r, where (£>„) 6 I. We show that if X satisfies certain conditions on the fundamental function and on certain interpolation indices then the space A(TC, X) is not order isomorphic to a rearrangement invariant space. The result includes the spaces Lpq and certain classes of Orlicz and Lorentz spaces. We also study the cases X = Le

Vector Measures, Integration and Related Topics

On Mean Ergodic Operators.- Fourier Series in Banach spaces and Maximal Regularity.- Spectral Measures on Compacts of Characters of a Semigroup.- On Vector Measures, Uniform Integrability and Orlicz Spaces.- The Bohr Radius of a Banach Space.- Spaces of Operator-valued Functions Measurable with Respect to the Strong Operator Topology.- Defining Limits by Means of Integrals.- A First Return Examination of Vector-valued Integrals.- A Note on Bi-orthomorphisms.- Compactness of Multiplication Operators on Spaces of Integrable Functions with Respect to a Vector Measure.- Some Applications of Nonabsolute Integrals in the Theory of Differential Inclusions in Banach Spaces.- Equations Involving the Mean of Almost Periodic Measures.- How Summable are Rademacher Series?.- Rearrangement Invariant Optimal Domain for Monotone Kernel Operators.- The Fubini and Tonelli Theorems for Product Local Systems.- A Decomposition of Henstock-Kurzweil-Pettis Integrable Multifunctions.- Non-commutative Yosida-Hewitt Theorems and Singular Functionals in Symmetric Spaces of ?-measurable Operators.- Ideals of Subseries Convergence and Copies of c 0 in Banach Spaces.- On Operator-valued Measurable Functions.- Logarithms of Invertible Isometries, Spectral Decompositions and Ergodic Multipliers.- Norms Related to Binomial Series.- Vector-valued Extension of Linear Operators, and Tb Theorems.- Some Recent Applications of Bilinear Integration.- A Complete Classification of Short Symmetric-antisymmetric Multiwavelets.- On the Range of a Vector Measure.- Measure and Integration: Characterization of the New Maximal Contents and Measures.- Vector Measures of Bounded ?-variation and Stochastic Integrals.- Does a compact operator admit a maximal domain for its compact linear extension?.- A Note on R-boundedness in Bidual Spaces.- Salem Sets in the p-adics, the Fourier Restriction Phenomenon and Optimal Extension of the Hausdorff-Young Inequality.- L-embedded Banach Spaces and a Weak Version of Phillips Lemma.- When is the Space of Compact Range Measures Complemented in the Space of All Vector-valued Measures?.- When is the Optimal Domain of a Positive Linear Operator a Weighted L 1-space?.- Liapounoff Convexity-type Theorems.

Multiplication operators on the space of Rademacher series in rearrangement invariant spaces

Let E be a rearrangement invariant (r.i.) function space on [0; 1]. We consider the space (R;E) of measurable functions f such that fg2E for every a.e. converging seriesg = P anrn2 E, where (rn) are the Rademacher functions. Curbera [4] showed that, for a broad class of spaces E, the space (R;E) is not order-isomorphic to a r.i. space. We study cases when (R;E) is order-isomorphic to a r.i. space. We give conditions on E so that (R;E) is order-isomorphic to L1. This includes certain classes of Lorentz and Marcinkiewicz spaces. We study further when (R;E) is orderisomorphic to a r.i. space different fromL1. This occurs for the Orlicz spacesE = Lq withq(t) asymptotically equivalent to expjtj q 1 and 0

Rademacher multiplicator spaces equal to

Let X be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space Λ(R, X) of measurable functions x such that x h ∈ X for every a.e. converging series h = Σa n r n ∈ X, where (r n ) are the Rademacher functions. We characterize the situation when Λ(R, X) = L ∞ . We also discuss the behaviour of partial sums and tails of Rademacher series in function spaces.

A weighted Khintchine inequality.

i=1 ai 1/2 , for every (ai) ∈ l, where (ri) are the Rademacher functions, that is, ri(t) := sign sin(2πt), t ∈ [0, 1], i ∈ N. A weighted version of the above inequality was recently proved in [18]. Namely, let w be a weight satisfying the following conditions (a) for some q > p we have w ∈ L([0, 1]); (b) the support of w satisfies m(supp(w)) > 2/3. Then there exist constants C1, C2 > 0, depending on p and w, such that for every a = (ai) ∈ l

How Summable are Rademacher Series

Khintchin inequalities show that a.e. convergent Rademacher series belong to all spaces L p ([0, 1]), for finite p. In 1975 Rodin and Semenov considered the extension of this result to the setting of rearrangement invariant spaces. The space L N of functions having square exponential integrability plays a prominent role in this problem.

Multipliers on the Set of Rademacher Series in Symmetric Spaces

Let E be a symmetric space on [0,1]. Let Λ(ℛ,E) be the space of measurable functions f such that fg∈ E for every almost everywhere convergent series g=∑bnrn∈ E, where (rn) are the Rademacher functions. It was shown that, for a broad class of spaces E, the space Λ(ℛ,E) is not order isomorphic to a symmetric space, and we study the conditions under which such an isomorphism exists. We give conditions on E for Λ(ℛ,E) to be order isomorphic to L∞. This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which Λ(ℛ,E) is order isomorphic to a symmetric space that differs from L∞. The answer is positive for the Orlicz spaces E=LΦq with Φq(t)=exp|t|q-1 and 0