The Hankel Convolution and the Zemanian Spaces β_μ and β'_μ

Abstract

In this paper we give structure theorems for the elements of the ZEMANIAN spaces SP and Also, bounded sets and convergent sequences in pP,o are characterized through representations as derivatives of measurable functions. Finally, we analyze the HANKEL convolution on the above spaces. 1.Math. Nachr. 160 (1993) 278 vestigate the structure of the members of &,a. Finally we characterize bounded sets and convergent sequences in /3h,o through representations of their elements. J. de SOUSA PINTO 11 11 started to investigate the generalized HANKEL convolution. He defined in spaces of distributions the convolution already introduced by F. M. CHOLEWINSKI [l] and I. 1. HIRSCHMAN [6]. Recently [9], the authors have improved the results of J. de SOUSA PINTO by extending the class of generalized functions on which the HANKEL convolution is defined. In Sections 3 and 4 of this work we study the HANKEL convolution on p, and &, respectively.Throughout this paper C will always denote a suitable positive constant (not necessarily the same in each occurrence). Also, to simplify the writing we introduce the function b,(z) = z-'J,(z) ( z E I). The spaces and fl,, and their dualsThe spaces p,,; and 8, were defined by A. H. ZEMANIAN [ 141. For every p E IR and a > 0, the space /3,,,o is constituted by all those smooth functions 4 = 4 ( x ) on I such that 4(x) = 0, for x 2 a, and When endowed with the topology generated by the family of seminorms { Y ; }~~~, becomes a FRECHET space. It is clear that Brsa c fic,b provided that a 5 b. This fact allows to define p, = u /3r,o as the inductive limit of {/3,,,o)o>o. Then fi, is a dense Now, we introduce on fi,,,a new families of seminorms which generate on pp,o the Let 1 I p < 03 and r E N. For every 4 E pp,a, we define subspace of X,. a > O same topology as {Y;}k.N. Proposition 2.1. Let a > 0 and 1 I p I 03. IJ ~E I R respectively p 2system of seminorms { 11 . l l , , , p, , }, EN (respectively {I . I p, p, r } , EN) generates on fic:a the same topology as {YfilLN.

Hypoellipticity of Hankel convolution equations on Schwartz distribution spaces

“…According to [2,Corollary 3.3] and [25,Corollaire 8.2], for every x ∈ (0, ∞), the Hankel translation τ x defines a continuous linear mapping from D * into itself. Then, we can define the Hankel convolution T #φ of T ∈ D * , the dual space of D * , and φ ∈ D * by…”

“…Indeed, since δ#φ = φ is in E * and the Dirac functional δ is not in E * φ can not be hypoelliptic. Moreover, according to [2,Proposition 4.8], S#φ ∈ E * , provided that S ∈ E * and φ ∈ D * . Then if T ∈ E * and φ ∈ D * , T + φ is hypoelliptic if, and only if, T is hypoelliptic.…”

On Besov spaces in the Hankel setting

“…The distributional Hankel convolution was studied in [5], [9] and [21]. If f ∈ S e and φ ∈ S e , the Hankel convolution is…”

A new characterization of the bounded operators commuting with Hankel translation

“…It is an open problem to remove the restriction m b À 1 2 in [2] and [10] by defining the Hankel convolution and the Hankel translation on h m and b m and their respective duals, when m % À 1 2 . After solving this problem a natural question is to obtain Theorems 1 and 2 (established in this note) when m % À 1 2 .…”