Spaces of Lorentz multipliers (Q2715684)

Let \(G\) be an infinite compact abelian group and \(m\) be the normalized Haar measure on \(G\). Given a measurable function \(f\) on \(G\), we put \(m_f(y)=m \{x\in G:|f(x)|>y\}\) for \(y\geq 0\) and \(f^*(t)= \inf\{y >0:m_f(y)\leq t\}\) for \(t\geq 0\). Then, the Lorentz space \(L^{p,q}(G)\) is the space of functions \(f\) for which \(\|f\|^*_{p,q} <\infty\), where NEWLINE\[NEWLINE\|f\|^*_{p,q} =\begin{cases} \left( {q\over p}\int^1_0 \bigl(x^{1/p} f^*(x)\bigr)^q {dx\over x}\right)^{1/q} \quad & \text{if }1\leq p,\;q<\infty\\ \sup_x x^{1/p} f^*(x)\quad & \text{if }1\leq p\leq\infty,\;q=\infty. \end{cases}NEWLINE\]NEWLINE A Lorentz multiplier is defined as a bounded linear map from \(L^{p,q}(G)\) to \(L^{r,s}(G)\), for some \(p,q,r,s\), which commutes with translation. The index \(p'\), conjugate to \(p\), is defined by \(1/p+1/p'=1\).NEWLINENEWLINENEWLINEThe authors study when the spaces \(M(p, q;r,s)\) of Lorentz multipliers from \(L^{p,q}(G)\) to \(L^{r,s}(G)\) are distinct. One of their results is as follows. If \(1<p<\infty\) and \(0<1/s- 1/t\neq 1/r-1/q\), then \(M(p,t; p,s)\neq M(p_1,q; p_1,r)\) if \(p_1\) is either \(p\) or \(p'\). In particular, \(M(p,q;p,r)\) is strictly contained in \(M(p,q;p,t)\) whenever \(1\leq r< \min(t,q)\). They also prove that if \(1<p\), \(r<\infty\) and \(r\neq p\), \(p'\), then \(M(p,t; p,s)\neq M(r,v,r,u)\) when \(s\leq t\), \(u\leq v\).NEWLINENEWLINENEWLINETheir method is based on careful constructions of integral functions belonging to the one space of multipliers but not to the other. These functions are formed of linear combinations of Fejer or Dirichlet kernels [cf. \textit{M. Cowling} and \textit{J. Fournier}, Trans. Am. Math. Soc. 221, 59-95 (1976; Zbl 0331.43007); \textit{R. Hunt}, Enseign. Math. (2) 12, 249-276 (1966; Zbl 0181.40301)].