Comparison theorems for a generalized modulus of continuity

TLDR: In this paper, the authors define a vector notation for Borel measures on real Euclidean w-space, where the product of the measures is the convolution a * r and the norm of a measure is its total variation.

Abstract: 1. Notation and definitions. Let R denote real Euclidean w-space. We shall employ standard vector notations, whereby / = (tu • • •, tm), u = («i, • • • , um) denote points of R , tu = ^ ï* /*w» and 11\ = (tt). In connection with Fourier transforms x = (xi, • • •, xm) denotes a point of a "dual" copy of R. M=M(R) shall denote the totality of bounded complex-valued Borel measures on R, made into a Banach algebra in the usual way, i.e. the "product" of the measures <r, r is the convolution a * r and the norm of a measure is its total variation. & shall denote the Fourier transform of <r, and W=W(R) the Banach algebra of Fourier transforms of elements of M. In W the "multiplication" is ordinary point-wise multiplication of functions. W is isometrically isomorphic to M under the map a-*fr. For f£;L(R) with 1 ^p^ <*> and <r(E.M we write ƒ * a to denote the function g such that g(t) —ff(t—u)d(r(u). (This is defined for almost all /; moreover g £ I X ) For a>0 let <T(fl) denote the measure defined by <T(a)(E) =cr(a" £) for all Borel sets £ . We have \\<T(a)\\ = |k | l For ƒ £Z>, a G I f and a > 0 let us define