Novi Sad J. Math. Vol. 45, No. 1, 2015, 285-301 PSEUDO-DIFFERENTIAL OPERATORS AND LOCALIZATION OPERATORS ON S μ ν (R) SPACE INVOLVING FRACTIONAL FOURIER TRANSFORM Jitendra Kumar Dubey 1 , Anuj Kumar 2 and S.K. Upadhyay 3 Dedicated to Professor Stankovi´ c on the occasion of his 90 th birthday and to Professor Vickers on the on the occasion of his 60 th birthday. Abstract. In this paper, some properties of pseudo-differential opera- tors, SG-elliptic partial differential equations with polynomial coefficients and localization operators on space S μ ν (R), are studied by using fractional Fourier transform. AMS Mathematics Subject Classification (2010): 35S05, 46F12, 47G30. Key words and phrases: Fractional Fourier transform; Pseudo-differential operator; time-frequency analysis; Ultra-differential functions. 1. Introduction: Almeida [1], Namias [5] and others introduced the fractional Fourier trans- form which is a generalization of the Fourier transform. Fractional Fourier transform is the most important tool, which is frequently used in signal pro- cessing and other branches of mathematical sciences and engineerings. The pseudo-differential operator is a generalization of the partial differ- ential operator, and played an important role in study of the properties of Sobolev spaces, partial differential equations and localization operators. Zaid- mann [11], Wong [10] and Cappiello et al. [2, 3] discussed the properties of pseudo-differential operator on S(R n ) space and certain types of Gelfand-Shilov space by using Fourier transformation. Pathak et al. [6], Prasad and Kumar [7] studied the properties of pseudo- differential operator on the Schwartz space S(R) involving fractional Fourier transformation. Motivated by Cappiello et al. [2], our main aim in this paper is to study the properties of pseudo-differential operators and localization ope- rators on S µ ν (R) space involving fractional Fourier transformation. Now, from [2, 4, 6] we recall definitions and properties which are useful for our further investigations: 1 DST-CIMS, Banaras Hindu University, Varanasi - 221005, India, email: jitendradubey1989@gmail.com 2 DST-CIMS, Banaras Hindu University, Varanasi - 221005, India, email: anujk743@gmail.com 3 DST-CIMS, Department of Mathematical Sciences, Indian Institute of Technology, Ba- naras Hindu University, Varanasi - 221005, India, email: sk upadhyay2001@yahoo.com