关于\((L_p([0,1]),L_q([0,1]))\)的弱最大化性质
The weak maximizing property for $(L_p([0,1]),L_q([0,1]))$
AI总结：
研究\((L_p([0,1]),L_q([0,1]))\)的弱最大化性质，通过高斯构造等方法证明，对于\(1 < p < \infty\)且\(1 \leq q < \infty\)，该对具有弱最大化性质当且仅当\(p = q = 2\)。
AI中文摘要：
我们解决了丹塔斯、荣格和马丁内斯 - 塞尔万特斯留下的[4，问题4.2]，给出了具有弱最大化性质的\((L_p([0,1]),L_q([0,1]))\)对的完整刻画。具体而言，证明了对于\(1 < p < \infty\)且\(1 \leq q < \infty\)，\((L_p([0,1]),L_q([0,1]))\)具有弱最大化性质当且仅当\(p = q = 2\)。证明中的关键步骤是一种高斯构造，它表明当\(r > 2\)时，\((\ell_2,L_r([0,1]))\)对不存在紧扰动性质。
英文摘要：
We solve [4, Question 4.2] left open by Dantas, Jung and Martínez-Cervantes by providing a complete characterization for the pairs $(L_p([0,1]),L_q([0,1]))$ having the weak maximizing property. More precisely, we prove that, for $1<p<\infty$ and $1\leq q<\infty$, the pair $(L_p([0,1]),L_q([0,1]))$ has the weak maximizing property if and only if $p=q=2$. A key step in the proof is a Gaussian construction that yields the failure of the compact perturbation property for the pair $(\ell_2,L_r([0,1]))$ whenever $r>2$.
