Multipliers and relative completion in weighted Lorentz spaces
The authors identify the multipliers of a certain subspace \(A\) of a weighted Lorentz space in terms of the relative completion of \(A\). The context is weighted Lorentz spaces with respect to a Beurling weight \(w\) (i.e., \(w(x)\geq 1\) and \(w(x+y)\leq w(x)w(y)\)) on a locally compact Abelian group \(G\) with Haar measure \(\mu\). The weight condition makes \(L^1_w(G)\) a Banach algebra under convolution. The Lorentz space \(L(p,q,w)\) is defined with respect to the quasinorm \(\| f\| _{pq,w}^\ast =(\frac{q}{p}\int_0^\infty t^{\frac{q}{p}-1}[f^\ast(t)]^q \,dt)^{1/q}\) when \(0<p,q<\infty\) and \(\sup_{t>0} t^{1/p}f^\ast(t)\) when \(q=\infty\). The rearrangement \(f^\ast(t)\) is defined by \(f^\ast(t)=\inf\{y>0:w(\{x\in G:| f(x)| >y\}\leq t)\}\) where \(w(E)=\int_E w\, d\mu\). One defines \(A\) by the conditions (i) \((A,\| \cdot\| _A)\) is a Banach convolution module over \(L^1_w(G)\) and \(\| \cdot\| _{pq,w}\leq \| \cdot \| _A\) and (ii) there is an approximate identity \(\{e_\alpha\}\) of \(L^1(G)\) such that \(\| e_\alpha\| _{1,w} \) is uniformly bounded and \(\| f\ast e_\alpha -f\| _A\to 0\) for all \(f\in A\). The relative completion \(\tilde{A}\) of \(A\) is defined as those \(f\in L(p,q,w)\) such that \(\| f\ast e_\alpha\| _A\) is bounded. The authors prove that \(\tilde{A}\) is homeomorphic and algebraically isomorphic to the space of continuous homomorphisms from \(L^1_w(G)\) into \(A\).
scientific article; zbMATH DE number 2029329