Theorems on the representation of spaces and Schur's lemma (Q1363292)

The set of two-variable continuous concave functions of the form \(\varphi: R^2_+\to R^1_+\) is denoted by \(U(2)\). If \((V,\Sigma,\mu)\) is a measure space and \(S(\mu)= S(V,\Sigma,\mu)\) is a space of real-valued measurable functions on \(V\), then the Banach space \((X,\|\cdot\|)\) is said to be an ideal if \(x\in S(\mu)\), \(y\in X\), \(| x(t)|\leq| y(t)|\) for almost all \(t\), imply that \(x\in X\), and \(\| x|_X\|\leq\| y|_X\|\). If \(X_0\), \(X_1\) are ideal subsets of \(S(\mu)\), and \(\varphi\in U(2)\), then the ideal space \(\varphi(X_0, X_1)\) is derived by the Calderon-Lozanovskii construction with norm \(\| x|_{\varphi(X_0, X_1)}\|= \inf\{V>0:| x(t)|\leq \gamma\varphi(x_0(t), x_1(t))\) a.e. for some \(x_j\) in \(B(1,X_j)\) the unit ball of \(X_j\), \(j= 0,1\}\). If \(0<\theta< 1\), then the case \(\varphi(t, s)= t^\theta s^{1-\theta}\), \(X_0= L^\infty\), \(X_1= X\), is denoted by \(X^\theta\). The main form of general factorization theorem of this paper indicates that if \(0<\theta_j< 1\), \(0<\gamma_j< 1\), \(X^{\theta_0\gamma_1}\subseteq X^{\theta_0\gamma_0}\), \(X^{\theta_1\gamma_0}\subseteq X^{\theta_0\gamma_0}\), if \(T_j\), \(j= 0,1\), are subadditive operators, and if \[ \| w_1T_0x|_{X^{\theta_1}}\|\leq c_0\| w_0 x|_{X^{\theta_0}}\|,\quad \| v_1T_1x|_{X^{\gamma_1}}\|\leq c_1\| v_0 x|_{X^{\gamma_0}}\|, \] then there is a function \(u> 0\), a.e., \(u\in X^{\theta_0\gamma_0}\), such that \[ w_1T_0(u^{1/\gamma_0} w^{-1}_0)\leq 2((c_0+ c_1)u)^{1/\gamma_0};\;v_1 T_1(u^{1/\theta_0}v^{-1}_0)\leq 2((c_0+ c_1)u)^{1/\theta_0}. \] Other results of the paper include constructions of weight functions for which the types of conditions relating to Muckenhoupt \(A_p\)-conditions are not valid.