Ali Ebadian[a] and Ali Jabbari[b]
An amalgamation of Orlicz spaces
Pages: 433-445
Received: 17 July 2023
Accepted in revised form: 12 December 2023
Mathematics Subject Classification: 46B25, 43A15.
Keywords: Amalgams of Banach space, locally compact group, Orlicz space, Orlicz sequence space.
Authors address:
[a],[b]: Department of Mathematics, Urmia University, Urmia, Iran
Abstract: Let \((\Phi,\Psi)\) be a complementary pair of \(N\)-functions and \((\Theta,\Upsilon)\) be a complementary pair of \(M\)-functions. Let \(L^\Phi\) and \(\ell^\Theta\) the be Orlicz and Orlicz sequence spaces on a measure space. We define an Orlicz space on \(L^\Phi\) such that it exhibits \(\ell^\Theta\) behavior at infinity, and we refer to it as amalgams of \(L^\Phi\) and \(\ell^\Theta\), denoted by \((L^\Phi,\ell^\Theta)\). We establish that \((L^\Phi,\ell^\Theta)\) is a Banach space, and under certain conditions, we demonstrate that its dual space is \((L^\Psi,\ell^\Upsilon)\). For a locally compact group \(G\) with the Haar measure \(\mu\), we define \((L^\Phi(G),\ell^\Theta(G))\), and we prove that if \(G\) is abelian, \(\Phi\in\Delta_2\), and \((L^\Phi(G),\ell^\Theta(G))\) forms a Banach algebra with the convolution product, then \(G\) must be compact.