부산대학교 도서관

Wave front sets are designed to detect singularities in a phasespace/time-frequency plane. The main classical applications of suchsets are connected to questions in the theory of distributions andpropagation of singularities in partial differential equations. Morerecently, wave front sets have been successfully applied in renormalizedquantum field theory in curved spacetime. Some results related toprogress in that direction were mentioned in [C. Brouder, N.~V. Dang andF. Hélein, J. Phys. A {\bf 47} (2014), no.~44, 443001; MR3270560].\parThe properties of wave front sets were used in [N.~V. Dang, Ann. HenriPoincaré {\bf 17} (2016), no.~4, 819--859; MR3472625] toprove thatperturbative quantum field theories are renormalizable on curvedspacetimes. In fact, if $ \Cal{D}' _{\Gamma} $ denotes the set ofdistributions whose wave front sets are contained in $ \Gamma $, thenone should observe continuity properties on $ \Cal{D}' _{\Gamma} $ forthe most common operations on distributions, such as pull-back, push-forward,tensor and convolution products. Traditionally, it is done bythe use of sequential convergence or, equivalently, by sequentialconvergence in the topology of $ \Cal{D}' _{\Gamma} $ introduced byJ.~J. Duistermaat in [{\it Fourier integral operators}, Progr. Math., 130,Birkhäuser Boston, Boston, MA, 1996; MR1362544].\parIt turns out that the above-mentioned topology is too weak for thecontinuity of the pull-back and for the separate continuity of thetensor product [S. Alesker, Geom. Funct. Anal. {\bf 20} (2010), no.~5,1073--1143; MR2746948]. The main motivation of the authors,which comesfrom counterexamples in [S. Alesker, op. cit.], is to define atopology forwhich the fundamental operations have optimal continuity properties.More precisely, it is shown that in the so-called {\it normal} topologyintroduced by the authors the following is true: the tensor product,the convolution product and multiplication of distributions arehypocontinuous, the pull-back and the push-forward by a smooth map arecontinuous, and the pull-back and the push-forward by a family ofsmooth maps depending smoothly on parameters are uniformly continuous.\parThe paper is well motivated and carefully written and may be a valuable sourceof information for researchers in the field.