Bilinear mappings interpolation (Q2719480)

The authors generalize the result by \textit{V. I. Ovchinnikov} [Math. USSR, Sb. 64, No. 1, 229-242 (1989; Zbl 0668.46042); Math. Rep. 1, 349-515 (1984; Zbl 0875.46007)] for the class of bilinear mappings \(X_{\alpha_1^ip_1^i}\) and \(Z_{\beta^iq^i}\), \( i=0,1\), denote the interpolation spaces, generated by the consistent Banach's pairs \((X_0,X_1)\) and \((Z_0,Z_1)\) respectively. The main result of the paper is as follows. Let \( 0<\alpha_i, \beta_i<1\), \( 1\leq p_i, q_i\leq\infty\), \( i=0,1\), \( \alpha_1\neq\alpha_0\), \( \beta_1\neq\beta_0 \) and \(T\) be a bilinear operator. If NEWLINE\[NEWLINE T: X_{\alpha_ip_i}\times Y_i\to Z_{\beta^iq^i},\quad i=0,1. NEWLINE\]NEWLINE Then NEWLINE\[NEWLINE T: X_{\alpha p}\times Y_{\theta r}\to Z_{\beta q}, NEWLINE\]NEWLINE where \( \alpha = \alpha_0(1-\theta)+\alpha_1\theta\), \( \beta = (1-\theta)\beta_0+\theta\beta_1\), NEWLINE\[NEWLINE 1+\frac 1q=\frac 1p + \frac 1r + (1-\theta)\bigg(\frac 1{q_0} - \frac 1{p_0}\bigg)_{+} + \theta\bigg(\frac 1{q_1} - \frac 1{p_1}\bigg)_{+}, NEWLINE\]NEWLINE here \( x_+ = \max(x,0)\).