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SUBATOMIC REPRESENTATION OF BESSEL POTENTIAL SPACES MODELLED ON LORENTZ SPACES
ANTO´NIO M. CAETANO
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
Abstract. Subatomic representation of Bessel potential spaces modelled on Lorentz spaces are obtained via interpolation techniques.
1. Introduction
Subatomic representation for function spaces of Besov and Triebel-Lizorkin type have been considered in recent years (see [8], [9], [5], [2], for example). The proofs are quite involved, so that one can expect a lot of work if one sets as our task the use of the same type of approach in order to get subatomic representations for other function spaces. On the other hand, subatomic representations can be quite useful as a means of attack of some questions. For example, Triebel [8, 20.6, 28.6] has successfully used it in order to determine sharp upper estimates for the asymptotic behaviour of entropy numbers of embeddings between fractal-based function spaces of Besov type and, as a consequence, was able to determine the right asymptotic behaviour for the eigenvalues of some fractal pseudo-differential operators.
Our objective here is to show that with the help of interpolation techniques it is possible in some cases to take advantage of the already known subatomic representations in order to get corresponding ones for other spaces, instead of trying to overcome the many difficulties one can expect to meet if a more traditional approach is used. So, rather than trying to be as general as possible, we exemplify what we mean by setting as our task here to get subatomic representations for
2000 Mathematics Subject Classification. 46E35, 46E30, 46E40, 46B70.
Key words and phrases. Bessel potential spaces, Lorentz spaces, interpolation, subatomic, quarkonial, representation, decomposition.
This research was partially supported by Unidade de Investiga¸ca˜o Matema´tica e Aplica¸co˜es of Universidade de Aveiro through Programa Operacional “Ciˆencia, Tecnologia, Inovac¸˜ao” (POCTI) of the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT), co-financed by the European Community fund FEDER.
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