Examples of solvable and nonsolvable convolution equations in 𝒦_p′, p ≥ 1

Olaf von Grudzinski

Vol. 80 (1979), No. 2, 561–574

DOI: 10.2140/pjm.1979.80.561

Abstract

For p ∈ [1,+∞) let 𝒦′_p be the space of distributions on Rⁿ not growing faster than some power of exp(|⋅|^p), and let 𝒦_∞′ be the space of distributions on Rⁿ of finite order. For every p ∈ (1,+∞] the existence of convolutors f is proved such that f ∗𝒦_p′ = 𝒦_p′ but f ∗𝒦_s′≠𝒦_s′ for every s < p. The main step in the proof is a construction of slowly decreasing entire functions which satisfy suitable estimates of Paley-Wiener type and which have countably many zeros of orders as high as possible.

Mathematical Subject Classification 2000

Primary: 45E10

Secondary: 46F05

Milestones

Received: 20 March 1978

Published: 1 February 1979