부산대학교 도서관

Wave front sets are designed to detect singularities in a phase space/time-frequency plane. The main classical applications of such sets are connected to questions in the theory of distributions and propagation of singularities in partial differential equations. More recently, wave front sets have been successfully applied in renormalized quantum field theory in curved spacetime. Some results related to progress in that direction were mentioned in [C. Brouder, N.~V. Dang and F. Hélein, J. Phys. A {\bf 47} (2014), no.~44, 443001; MR3270560]. \par The properties of wave front sets were used in [N.~V. Dang, Ann. Henri Poincaré {\bf 17} (2016), no.~4, 819--859; MR3472625] to prove that perturbative quantum field theories are renormalizable on curved spacetimes. In fact, if $ \Cal{D}' _{\Gamma} $ denotes the set of distributions whose wave front sets are contained in $ \Gamma $, then one should observe continuity properties on $ \Cal{D}' _{\Gamma} $ for the most common operations on distributions, such as pull-back, push-forward, tensor and convolution products. Traditionally, it is done by the use of sequential convergence or, equivalently, by sequential convergence in the topology of $ \Cal{D}' _{\Gamma} $ introduced by J.~J. Duistermaat in [{\it Fourier integral operators}, Progr. Math., 130, Birkhäuser Boston, Boston, MA, 1996; MR1362544]. \par It turns out that the above-mentioned topology is too weak for the continuity of the pull-back and for the separate continuity of the tensor product [S. Alesker, Geom. Funct. Anal. {\bf 20} (2010), no.~5, 1073--1143; MR2746948]. The main motivation of the authors, which comes from counterexamples in [S. Alesker, op. cit.], is to define a topology for which the fundamental operations have optimal continuity properties. More precisely, it is shown that in the so-called {\it normal} topology introduced by the authors the following is true: the tensor product, the convolution product and multiplication of distributions are hypocontinuous, the pull-back and the push-forward by a smooth map are continuous, and the pull-back and the push-forward by a family of smooth maps depending smoothly on parameters are uniformly continuous. \par The paper is well motivated and carefully written and may be a valuable source of information for researchers in the field.