Spectral multipliers on quotients¶by amenable subgroups

We show that spectral multipliers for operators on quotients by amenable subgroups have holomorphic extension to a strip which is contained in the corresponding one for the group. Received: 16 September 1999     

L-Fourier multipliers for the Dunkl operator on the real line

We consider Fourier multipliers for L^p associated with the Dunkl operator on and establish a version of Hörmander's multiplier theorem. In applying this version, we come up with some results regarding the oscillating multipliers, partial sum operators and generalized Bessel potentials.     

Maximal functions for multipliers on stratified groups

In this paper we obtain a refined $L^p$ bound for maximal functions of the multiplier operators on stratified groups and maximal functions of the multi‐parameter multipliers on product spaces of stratified groups. As an application we find a refined $L^p$ bound for maximal functions of joint spectral multipliers on Heisenberg group.     

Eigenfunctions of the Laplace operators for a building of type[(A)\tilde]2\tilde A_2

Let Δ be a thick affine building of type and of order q. We prove that each eigenfunction of the Laplace operators of Δ is the Poisson transform of a suitable finitely additive measure on the maximal boundary Ω.     

On existence of finite universal Korovkin sets in Segal algebras

We prove that there exists a finite universal Korovkin set w.r.t positive operators for the centre of a Segal algebra on a compact groupG if and only ifG is metrizable. As a consequence it follows that a Segal algebra on a compact abelian group admits a finite universal Korovkin set w.r.t. positive operators iff the group is metrizable.     

Wavelet representations and Fock space on positive matrices

We show that every biorthogonal wavelet determines a representation by operators on Hilbert space satisfying simple identities, which captures the established relationship between orthogonal wavelets and Cuntz-algebra representations in that special case. Each of these representations is shown to have tractable finite-dimensional co-invariant doubly cyclic subspaces. Further, motivated by these representations, we introduce a general Fock-space Hilbert space construction which yields creation operators containing the Cuntz-Toeplitz isometries as a special case.     

Support theorem for the fundamental solution to the Schrödinger equation on certain compact symmetric spaces

In this paper we construct the fundamental solution to the Schödinger equation on a compact symmetric space with even root multiplicities using shift operators of Heckman and Opdam. Next, we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time. Moreover, we also show that generalized Gauss sums appear in the expression of the fundamental solution.     

Pseudodifferential operators on<Emphasis Type="Italic">SU</Emphasis>(2)

We construct an algebra of left-invariant pseudodifferential operators on SU(2). We require only that the symbols be homogeneous and C². For Fourier-bandlimited symbols, we derive the expected formulae for composition and commutators and construct an orthonormal basis of common approximate eigenvectors that could be used to study spectral theory. Some remarks on applications to matrices of operators are made.     

Wiener-Hopf operators on a subsemigroup of a discrete torsion free abelian group

In the paper Wiener-Hopf operators on a semigroup of nonnegative elements of a linearly quasi-ordered torsion free Abelian group are considered. Wiener-Hopf factorization of an invertible element of the group algebra is constructed, notions of a topological index and a factor index are introduced. It turns out that the set of factor indices for invertible elements of the group algebra is a linearly ordered group. It is shown that Wiener-Hopf operator with an invertible symbol is an one-side invertible operator and its invertibility properties are defined by the sign of the factor index of its symbol. Groups on which there exist nontrivial Fredholm Wiener-Hopf operators are described. As an example, all linear quasi-orders on the group ⁿ are found and corresponding Wiener-Hopf operators are considered.     

Local solvability of partial differential operators on the Heisenberg group

In this article the following class of partial differential operators is examined for local solvability: Let P(X, Y) be a homogeneous polynomial of degree n ≥ 2 in the non-commuting variables X and Y. Suppose that the complex polynomial P(iz, 1) has distinct roots and that P(z, 0) = zⁿ. The operators which we investigate are of the form P(X, Y) where X = δ_x and Y = δ_y + xδ_w for variables (x, y, w) ∈ ?³. We find that the operators P (X, Y) are locally solvable if and only if the kernels of the ordinary differential operators P(iδ_x, ± x)* contain no Schwartz-class functions other than the zero function. The proof of this theorem involves the construction of a parametrix along with invariance properties of Heisenberg group operators and the application of Sobolev-space inequalities by Hörmander as necessary conditions for local solvability.     

D'Alembert's equation and spherical functions

Summary The observation that the solutions to d'Alembert's functional equation are Z₂-spherical functions onR ² gives us a natural way of extending d'Alembert's functional equation to groups. We deduce in this setting that the general solutions are joint eigenfunctions for a system of partial differential operators, and we find a formula for the bounded solutions.     

Some remarks on a paper by Liu and van Rooij

Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability.     

Maximal Abelian von Neumann Algebras and Toeplitz Operators with Separately Radial Symbols

This paper mainly concerns abelian von Neumann algebras generated by Toeplitz operators on weighted Bergman spaces. Recently a family of abelian w^*-closed Toeplitz algebras has been obtained (see [5,6,7,8]). We show that this algebra is maximal abelian and is singly generated by a Toeplitz operator with a “common” symbol. A characterization for Toeplitz operators with radial symbols is obtained and generalized to the high dimensional case. We give several examples for abelian von Neumann algebras in the case of high dimensional weighted Bergman spaces, which are different from the one dimensional case.     

Transplanting maximal inequalities between Laguerre and Hankel multipliers

We supplement our previous paper [9] by adding a theorem that transplantsL ^p -norm maximal inequalities for Laguerre multipliers. As an immediate consequence we obtain negative results concerningL ^p -estimates of partial sum maximal operators for Laguerre expansions.Research supported in part by KBN grant No. 2 PO3A 030 09.     

The lower estimate for linear positive operators,I

A method to prove lower estimates for linear operators is introduced. As a result the best lower estimate for certain convolution operators, for the multivariate Bernstein-Durrmeyer operators in part I and the Bernstein polynomial operators in part II (see [10]), are obtained.Communicated by Hubert Berens     

Singular integral and maximal operators associated to hypersurfaces:L p theory

We consider singular integral and maximal operators associated to hypersurfaces given by the graph of a function whose level sets are defined by a convex function of finite type. We investigate the L^p theory for these operators which depend on geometric properties of the hypersurface.     

Positive curvature property for some hypoelliptic heat kernels

In this note, we look at some hypoelliptic operators arising from nilpotent rank 2 Lie algebras. In particular, we concentrate on the diffusion generated by three Brownian motions and their three Lévy areas, which is the simplest extension of the Laplacian on the Heisenberg group H. In order to study contraction properties of the heat kernel, we show that, as in the case of the Heisenberg group, the restriction of the sub-Laplace operator acting on radial functions (which are defined in some precise way in the core of the paper) satisfies a non-negative Ricci curvature condition (more precisely a CD(0,∞) inequality), whereas the operator itself does not satisfy any CD(r,∞) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel.     

Paley-Wiener type theorems by transmutations

The classical Paley-Wiener theorem for functions in L _dx ² relates the growth of the Fourier transform over the complex plane to the support of the function. In this work we obtain Paley-Wiener type theorems where the Fourier transform is replaced by transforms associated with self-adjoint operators on L _dμ ² , with simple spectrum, where dμ is a Lebesgue-Stieltjes measure. This is achieved via the use of support preserving transmutations. Communicated by Paul L. Butzer     

Cesaro Summability with Respect to Two-Parameter Walsh Systems

 The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H ^p to L ^p for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T _α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T _α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the usual two-parameter maximal operators of the (C, 1)-means are bounded from L ^p spaces to L ^p if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from the two-dimensional case can be extended to higher dimension in a simple way. (Received 20 April 2000; in revised form 25 September 2000)