Amalgams of Lᴾ and ℓ^q

Abstract

<p>An amalgam of Lᴾ and ℓ^q is a Banach space (Lᴾ, ℓ^q)(G) (1 ≤ p, q ≤ ∞) of (classes of) functions on a locally compact abelian group G which belong locally to Lᴾ and globally to ℓ^q. Similarly, the space of unbounded measures of type q is a Banach space Mq(G) (1 ≤ q ≤ ∞) of unbounded measures which belong locally to the space of bounded, regular, Borel measures on G and globally to ℓ^q.</p> <p>The Fourier transform of funcions in (Lᴾ, ℓ^q) and measures in Mq is defined to be a linear functional on the subspace Ac(G) of the Fourier algebra A(G), and its relation with other known definitions of Fourier transforms is established.</p> <p>We introduce the space of strong resonance class of functions relative to the test space Φq and find its relation with respect to the linear space generated by the positive definite funcions for (L^q, ℓ¹).</p> <p>We generalize known results for amalgam spaces on the real line spaces to locally compact abelian groups, extend some results in the theory of Lᴾ spaces to amalgams and develop a theory of multipliers for amalgam spaces and spaces of unbounded measures of type q.</p>