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Properties and examples of \((L^ p,L^ q)\) multipliers. (English) Zbl 0655.43003

We discuss conditions pertaining to the “rate of decay” of bounded sequences which almost characterize the space of multipliers from \(L^ 2(G)\) to \(L^ q(G)\), denoted by M(2,q), for G a compact abelian group and \(q>2\). Specific examples of multipliers are constructed to illustrate that M(2,q)\(\subseteq M(2,p)\) if \(q>p\). The spectrum of a multiplier \(\phi\in M(2,p)\), for \(p>2\), as an operator from \(L^ p\) to \(L^ p\) is seen to be \(\{\) \(\phi\) (Ĝ)\(\}^{ci}\). It is shown that for some values of p and q there are examples of multipliers from \(L^ p\) to certain quotient spaces of \(L^ q\) which are not the restrictions of multipliers from \(L^ p\) to \(L^ q\). Quasi-idempotent multipliers from \(H^ 1(T)\) to \(H^ p(T)\) for \(p>1\) are characterized.
Reviewer: K.E.Hare

MSC:

43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
42A45 Multipliers in one variable harmonic analysis
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