Translation Invariant Operators on Lp-Type Spaces

摘要

The continuous, translation invariant, linear operators from L(sub loc, sup p)(R)into L(sub loc, sup p)(R) and from L(sub comp, sup p)(R) into L(sub comp, sup p)(R), 1 = or < p = or < infinity are characterized. This characterization is in terms of the convolution ring ba(sub c)(R) consisting of all compactly varying, right continuous functions of bounded variation. It turns out that for p = 1 and p = infinity, each translation invariant operator on L(sub loc, sup p)(R) leaves invariant the space C(R) of continuous functions on R.