Nikol'skii–Bernstein Constants for Entire Functions of Exponential Spherical Type in Weighted Spaces

We study the exact constant in the Nikol'skii–Bernstein inequality $\|Df\|_{q}\le C\|f\|_{p}$ on the subspace of entire functions $f$ of exponential spherical type in the space $L^{p}(\mathbb{R}^{d})$ with a power-type weight $v_{\kappa}$. For the differential operator $D$, we take a nonnegative integer power of the Dunkl Laplacian $\Delta_{\kappa}$ associated with the weight $v_{\kappa}$. This situation encompasses the one-dimensional case of the space $L^{p}(\mathbb{R}_{+})$ with the power weight $t^{2\alpha+1}$ and Bessel differential operator. Our main result consists in the proof of an equality between the multidimensional and one-dimensional weighted constants for $1\le p\le q=\infty$. For this, we show that the norm $\|Df\|_{\infty}$ can be replaced by the value $Df(0)$, which was known only in the one-dimensional case. The required mapping of the subspace of functions, which actually reduces the problem to the radial and, hence, one-dimensional case, is implemented by means of the positive operator of Dunkl generalized translation $T_{\kappa}^{t}$. We prove its new property of analytic continuation in the variable $t$. As a consequence, we calculate the weighted Bernstein constant for $p=q=\infty$, which was known in exceptional cases only. We also find some estimates of the constants and give a short list of open problems.