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

Abstract

We study the exact constant in the Nikol’skii–Bernstein inequality $$\|Df\|_{q}\leq C\|f\|_{p}$$ 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 $$1\leq p\leq q=\infty$$ . 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.