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# A Simple Example of Transference

DOI link for A Simple Example of Transference

A Simple Example of Transference book

# A Simple Example of Transference

DOI link for A Simple Example of Transference

A Simple Example of Transference book

## ABSTRACT

A Simple Example of Transference Fernando Soria, Guido Weiss, and Ales Zaloznik

Many operators in Analysis can be expressed in terms of a convolution kernel acting

on a group G and a representation of this group acting on a space of functions, say LP(X),

defined on a general measure space X. Sharp estimates for the norm of such operators on

LP(X) can be expressed in terms of the norm of the convolution operator on LP(G). Let us be more precise. Suppose G is an amenable group, X is a u-finite measure space and R is a

uniformly bounded representation of G acting on LP(x). That is, for each u E G, Ru is a

linear transformation on LP(X) satisfying

(1)

continuous. Let k E L 1(G) have compact support and Np(k) denote the operator norm of the

convolution operator cp(u) __. l;?*k(u) = f Gcp(uv -l)k(v)dv; that is, NP(k) is the infimum of all a > 0 such that

(2)

for all tp E LP(x). We then introduce the operator Hk acting on functions F on X by letting

The second author supported, in part, by NSF Grant MCS 8200884

348 Soria et al.