Köthe Amalgams: The Ideal Type of Infinite Direct Sums

Abstract We study a special type of infinite direct sums $$E({\mathcal {X}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> which can be seen as the amalgam spaces characterized by a local component given by a countable family $${\mathcal {X}}=\left( X_{\alpha }\right) _{\alpha \in I}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mfenced> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mfenced> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> of quasi-normed function spaces and by a global component E , which is a quasi-normed sequence space. We characterize some fundamental properties of $$E({\mathcal {X}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> such as completeness, Köthe-duality, order continuity and the Fatou property. We also provide its Banach function space characterization. Then, we apply our general results to the appropriate amalgamations of Lorentz (Orlicz) function spaces and Lebesgue sequence spaces. Moreover, for the Lorentz-type amalgams, we derive interpolation results and prove the boundedness of a class of sublinear integral operators whose kernels satisfy a size condition.