John E. Gilbert

Bilinear operators with non-smooth symbol, I

This article proves the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies earlier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane, a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding Lp-boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part in Part II, our subsequent article [11], using phase-plane analysis.

Bimeasure algebras on locally compact groups

For locally compact groups G and H, let BM(G, H) denote the Banach space of bounded bilinear forms on C0(G) × C0(H). Using a consequence of the fundamental inequality of A. Grothendieck. a multiplication and an adjoint operation are introduced on BM(G, H) which generalize the convolution structure of M(G × H) and which make BM(G, H) into a KG2-Banach ∗-algebra, where KG is Grothendiecks universal constant. Various topics relating to the ideal structure of BM(G, H) and the lifting of unitary representations of G × H to ∗-representations of BM(G, H) are investigated.

Maximal theorems for some orthogonal series II

Abstract If {un} is the orthonormal sequence of eigenfunctions arising from a nonsingular (or sometimes a singular) Sturm-Liouville system with separated boundary conditions defined on (0, π), it has long been known that the eigenfunction expansion with respect to {un} of a function φ on (0, π) has properties similar to those of the Fourier-cosine expansion of φ. For instance, there is the classical equiconvergence theorem of Haar ([5]; see [3] pp. 1616–1622 for a general survey). In this paper, by restricting attention to two specific classes of singular Sturm-Liouville systems, we shall establish a much more precise relationship between the corresponding eigenfunction expansions and the Fourier-cosine expansions. Many results for Fourier series can then be carried over to these eigenfunction expansions. The method of proof includes as special cases Fourier-Bessel functions, ultraspherical polynomials, Jacobi polynomials and any nonsingular Sturm-Liouville system.

Hardy spaces and a Walsh model for bilinear cone operators

The study of bilinear operators associated to a class of non-smooth symbols can be reduced to ther study of certain special bilinear cone operators to which a time frequency analysis using smooth wave-packets is performed. In this paper we prove that when smooth wave-packets are replaced by Walsh wave-packets the corresponding discrete Walsh model for the cone operators is not only LP-bounded, as Thiele has shown in his thesis for the Walsh model corresponding to the bilinear Hilbert transform, but actually improves regularity as it maps into a Hardy space. The same result is expected to hold for the special bilinear cone operators.

Counter-Examples in Interpolation Space Theory from Harmonic Analysis

That the classical interpolation theorems of Riesz-Thorin and Marcinkiewicz are of great importance in harmonic analysis has long since been realized. The functional analytic generalization of these interpolation theorems — abstract interpolation space theory — also is now proving to be of importance in harmonic analysis (cf., for instance, [1]). In this paper we reverse the flow in the sense that a circle of ideas from Lp-multiplier theory and tensor products of Banach spaces is used to provide answers to a number of questions that arise naturally in the general theory of abstract interpolation spaces. While some of these questions have been answered before (though not always published), the unified approach used in this paper introduces techniques which may well be of wider utility in interpolation space theory. We shall be interested in both the real interpolation spaces (X o, X 1)θq;J, (X o, X 1)θ, q;k of Peetre and the complex interpolation spaces (X o,X 1)θ of Lions and Calderon. For all unexplained notation and terminology see [2], [5], [6] or [11].