The Bernstein-Nikolskii limit conjecture for finite q

Let $a>0$, $N\in{\mathbb Z}^1_+$, $V\subset{\mathbb R}^m$, and $0<p\le q<\infty$. The quantities $P_{p,q,D_N,a,V}$ and $E_{p,q,D_N,V}$ are the Bernstein-Nikolskii constants defined for the corresponding function classes and differential operator $D_N$.

Bernstein-Nikolskii limit conjecture. The limit

$$\lim_{a\to\infty}P_{p,q,D_N,a,V}$$

exists and

$$E_{p,q,D_N,V}=\lim_{a\to\infty}P_{p,q,D_N,a,V}.$$

The conjecture extends the equality proved in the paper for $q=\infty$ to all finite $q$ with $0<p\le q$. Its status is not resolved in the supplied text.