On the multiplier of a rearrangement invariant space with respect to the tensor product
Let \(E\) be an r.i. space (= rearrangement invariant real function space) on \(I=[0,1]\) and \({\mathbf M}(E)\) denote the set of all measurable functions \(x= x(s)\) on \(I\) such that \(x(s)y(t)\in E(I\times I)\) for each \(y= y(t)\in E\). Then \({\mathbf M}(E)\) equipped with the norm \(|x|_{{\mathbf M}(E)}= \sup\{|x(s)y(t)|_{E(I\times I)}:|y|_E\leq 1\}\) is a r.i. and \({\mathbf M}(E)\subset E\). Denote \(\chi_e\) the indicator function of a measurable set \(e\), \(\varphi\) the fundamental function of an r.i. space \(E:\varphi(t)= |\chi_{(0,t)}|_E\) \((t>0)\) and let \(\sigma_t\) be the (bounded in any r.i. space) dilation operator \(\sigma_t(y(u))= y(u/t)\chi_{(0,1)}(u/t)\) \((t>0)\). Theorem 1. For any r.i. space \(E\) on \(I\) the following relations are valid: \[ \varphi_{{\mathbf M}(E)}(t)= |\sigma_t|_{E\to E},\quad 0<t\leq 1, \] \[ |\sigma_t|_{{\mathbf M}(E)\to{\mathbf M}(E)}= |\sigma_t|_{E\to E},\quad 0<t\leq 1, \] \[ |\sigma_t|^{-1}_{E\to E}\leq |\sigma_t|_{{\mathbf M}(E)\to{\mathbf M}(E)}\leq|\sigma_t|_{E\to E}. \] Let \(\alpha_E\) and \(\beta_E\) denote the lower and upper Boyd indices of an r.i. space \(E\), respectively, \(0\leq\alpha_E\leq\beta_E\leq 1\). Corollary 1. The inequalities \(\alpha_E\leq\alpha_{{\mathbf M}(E)}\leq \beta_{{\mathbf M}(E)}\leq \beta_E\) hold. Theorem 2. Put \(p= 1/\alpha_E\). Then \({\mathbf M}(E)\subset I_p\) and the embedding constant does not depend on \(E\). Corollary 2. If \(\alpha_E=0\), then \({\mathbf M}(E)= L_\infty\). Theorem 3. Denote \(\psi(t)= |\sigma_t|_{E\to E}\) \((0\leq t\leq 1)\), and let \(\Lambda(\psi)\) denote the Lorentz space given by the concave function \(\psi\). Then \({\mathbf M}(E)\supset\Lambda(\psi)\), where the embedding constant does not depend on \(E\). For \(1<p<\infty\) denote \({\mathfrak M}_p\) the Marcinkiewicz space given by the concave function \(t\), where \(q= 1-1/p\). Theorem 4. If \(E\) is an interpolation space between \(L_p\) and \({\mathfrak M}_p\), then \({\mathbf M}(E)= L_p\). Corollary 4. If \(1<p<\infty\), \(p\leq q\leq\infty\), then \({\mathbf M}(L_{pq})= L_p\). Using the above approach, the author refines an O'Neil's result about the tensor product of \(L_{pq}\) spaces.
scientific article; zbMATH DE number 1053620