Remarks on some interpolation spaces

One purpose of this paper is to introduce a new idea of associating microlocal measures to bounded functions of bounded variation, and even to functions in the interpolation space (H(Q)yL (Q))1.7oo, in order to replace and improve some methods from geometric measure theory. Some properties of similar interpolation spaces are reviewed first. I do not remember when I met Enrico MAGENES for the first time. My first visit to Italy was for a CIME course in Varenna in 1970, and he may have come there, or I may have met him a few months after at ICM70 in Nice, but I definitely met him three years after on my second visit to Italy, when I stopped for a few days in Pavia on my way to Trieste. He kindly guided my visit to the famous Certosa di Pavia, and insisted that I should come back again as my visit could not be complete without seeing the facade; it was not that some work was being done with scaffoldings all around as it unfortunately happens sometimes when one visits a famous monument, but it was in November and the fog limited our visibility to a few meters. I did visit Pavia a few other times, not quite as often as I would have liked, and in that warm atmosphere that emanated from E. MAGENES I always felt as if there was now an Italian branch on my family tree. I cannot remember if my thesis advisor, Jacques-Louis LIONS, had told me to read any of the three volumes that he had written with Enrico MAGENES, but I did read the first one entirely and some parts of the others two and it was then that I first learned about the theory of interpolation in the case of HlLBERT spaces [LlM the second part also answered another question of J.-L. LIONS but never appeared as the corrections proposed by the editor never reached me due to some defect in the transatlantic mail system, and as my personal difficulties about writing (and also reading) were genuine, I could not put my mind on questions of publication for quite a long time after this incident. Many of my results remained then unpublished, sometimes mentioned by J.-L. LIONS as "to appear" instead of "personal communication" which would have been more appropriate, as I could not have initiated the too arduous process for me of preparing a manuscript for publication without someone insisting that a particular result of mine was worth publishing. Progressively, I moved away from functional analysis and set my new goal of understanding more about Continuum Mechanics and Physics, so I could not find enough motivation to write down some of these technical results which I had obtained before. Today, however, I realize that even for questions of Continuum Mechanics, some technical results from interpolation theory might be more important than I had thought previously. I did mention orally some remarks in this sense in two meetings during the Summer of 1992, the first one in Oberwolfach and the second one in TYento, where I had the pleasure to meet again Enrico MAGENES, and it is therefore with great pleasure that I dedicate to him some of these old remarks, mixed with new ideas. I. A problem in singular perturbations and boundary layers. In the early 70s, J.-L. LIONS was working on questions of singular perturbations, some of them involving questions of boundary layers; as he often did, he was teaching the subject and writing a book on it at the same time [L4]. One of the many questions that he was considering was to describe the boundary layer correction in a variational elliptic problem of the type -eAue + ut = / in fi, w c =Oon0f2 , (I.I)