The Multipliers for the Pair (L _p (G), L _q (G)), 1 ≦ p, q ≦ ∞

Abstract

In the previous chapter we discussed multipliers for the pair (L _P (G), L _q (G)) when p = q. Our attention in this chapter will be focused on the case where p ≠ q. The problem of describing the multipliers in this situation is equally if not more difficult than in the case p = q. In order to obtain a description of the multipliers as convolution operators we shall have to introduce a class of mathematical objects which properly contains the space of pseudomeasures employed previously, namely, the space of quasimeasures. The description of multipliers as multiplication by bounded functions is no longer possible, but an analogous result will be obtained using the Fourier transform of certain quasimeasures. Unfortunately these transforms are again quasimeasures and not in general functions. We shall define a Fourier transform for L _P (G), p>2, in terms of quasimeasures and show, in particular, that there exist f∈L _p (G), p>2,whose Fourier transforms are not measures. We shall also examine various inclusion relationships for the spaces M(L _p (G), L _q (G)) and obtain a characterization of these spaces as dual spaces of certain Banach spaces.