The Multipliers for Lp(G)
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In this chapter we shall investigate the multipliers for the pair (L p (G), L p (G)). We have proven some scattered results pertaining to M(L p (G)) in the previous chapters. In particular, we have already discussed to some extent the cases when p =1 and p = ∞. Consequently we shall now restrict our attention primarily to the values of p such that 1< p < ∞. We shall show in the following sections that the multipliers for L p (G) can, in a certain sense, be represented either as multiplication of the Fourier transform by a bounded function or as a convolution operator, in this instance convolution with a pseudomeasure. We shall also investigate the relationships between the spaces M(L p (G)) for various values of p, obtain some results on the existence of bounded functions which do not determine multipliers for L p (G), and examine the notion of the derived space for L p (G). As usual we shall consider only the commutative case.
KeywordsBanach Algebra Proper Subset Measurable Subset Compact Abelian Group Algebra Isomorphism
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