Mediterr. J. Math. DOI 10.1007/s00009-016-0801-x c  Springer International Publishing 2016 Nemytskii Operator in Riesz-Bounded Variation Spaces with Variable Exponent Ren´ e Erl´ ın Castillo, Oscar Mauricio Guzm´ an and Humberto Rafeiro Abstract. In this note, we characterize global Lipschitz Nemytskii oper- ators in the Riesz-bounded variation space with variable exponent, and in the course of the proof, we also show that this space is a Banach algebra. Mathematics Subject Classification. 26A45, 47H30. Keywords. Bounded variation, Variable exponent spaces, Nemytskii operator. 1. Introduction Functions of bounded variation of different types were studied by many math- ematicians such as F. Riesz, N. Wiener, R.E. Love, H. Ursell, L.C. Young, W. Orlicz, J. Musielak, L. Tonelli, L. Cesari, R. Caccioppoli, E. de Giorgi, O. Oleinik, E. Conway, J. Smoller, A. Vol’pert, S. Hudjaev, L. Ambrosio, G. Dal Maso, among many others. It is noteworthy to mention that many of these generalizations were motivated by applications, see [19] for examples in mathematical physics. Function spaces with variable exponent is a very active area of research nowadays [15, 16, 21, 22, 28], and one of the reasons is the wide variety of applications of such spaces, e.g., in the modeling of electrorheological flu- ids [3, 4, 31, 32] as well as thermorheological fluids [5], in the study of image processing [1, 2, 6, 7, 13, 14, 33] and in differential equations with non-standard growth [18, 27]. Other variable exponent spaces, e.g. Morrey, Campanato, H¨older(cf.[11, 12, 25, 26, 29, 30]), were widely studied. Bounded variation spaces with variable exponent were introduced quite recently, in the Wiener sense in [9] and in the Riesz sense by the authors in [10]. For a different approach related to bounded variation in the Riesz sense, see [20]. The main purpose of this paper is to characterize global Lipschitz Ne- mytskii operators in the variable exponent bounded variation spaces and to