A Banach algebra of functions of several variables of finite total variation and Lipschitzian superposition operators. I

Abstract

It is shown that the space of functions of n real variables with finite total variation in the sense of Vitali, Hardy and Krause, defined on a rectangle I a b ⊂ R n , is a Banach algebra under the pointwise operations and Hildebrandt–Leonov's norm. This result generalizes the classical case of functions of bounded Jordan variation on an interval I a b = [ a , b ] for n = 1 and a previous result of the author in [Monatsh. Math. 137(2) (2002) 99–114] for n = 2 .