Equivalent Conditions for Integrability of Distributions

Abstract

We consider several general conditions for integrability of two Roumieu ultradistributions on R d in the space D ′{Mp} and prove their equivalence. The discussed sequential conditions are based on two classes U {Mp} and U {Mp} of approximate units and allow one to introduce sequential definitions of the convolution in D ′{Mp} , analogous to the known definitions in the space D ′ of distributions and in the space D ′(Mp) of ultradistributions of Beurling type.

A sequential approach to the convolution of Roumieu ultradistributions

“…It follows, by induction, from (M.1) that M p Á M q M 0 M pþq for p; q 2 N 0 (see [16]). Under the assumption that M 0 ¼ 1, which we adopt hereinafter for simplicity, the last inequality admits the form:…”

“…The conditions are based on two types of R-approximate units (Definitions 4.2 and 4.3), being the counterparts of the approximate units in the sense of Dierolf and Voigt (see [5]). The respective classes U fM p g and U fM p g of R-approximate units are used in a sequential characterization of integrable Roumieu ultradistributions (see [16]), analogous to that proved by Pilipović in [18] in case of integrable ultradistributions of Beurling type. As a consequence, we give in this paper several sequential definitions of the convolution of Roumieu ultradistributions (Definition 7.2).…”

Convolution of distributions in sequential approach

“…We formulate below a characterization of integrable Roumieu ultradistributions, being an analog of the theorem of Dierolf and Voigt concerning integrable distributions (see [3]) and of the theorem of Pilipović concerning ultradistributions of Beurling type (see [20]). The proof of the theorem is given in [18].…”

“…The conditions are based on two types of R-approximate units (Definition 2 and Definition 3), being the counterparts of the approximate units in the sense of Dierolf and Voigt (see [3]). The respective classes U {Mp} and U {Mp} of R-approximate units are used in sequential characterization of integrable Roumieu ultradistributions ( [18]), analogous to those proved by Pilipović in [20] in case of integrable ultradistributions of Beurling type. As a consequence, we give several sequential definitions of the convolution of Roumieu ultradistributions (Definition 6).…”

Existence theorems on convolution of functions, distributions and ultradistributions

“…There are several general definitions of the convolution of distributions in D given consecutively by C. Chevalley [3], L. Schwartz [26], R. Shiraishi [27], V. S. Vladimirov [28], [29], P. Dierolf -J. Voigt [4], A. Kamiński [9] and S. Mincheva-Kamińska [19], [20] (see also [32], [5], [30], [31], [21], [18] and [15]). These general definitions allow one to define the convolution f * g in D for arbitrary distributions f , g ∈ D (R d ) and to determine, for each pair ( f , g) of distributions, whether the convolution f * g exists in D (then f * g ∈ D (R d )) or not.…”