D.V. Gorbachev, V.I. Ivanov. Nikol’skii–Bernstein constants for entire functions of exponential spherical type in weighted spaces ... P. 75-87

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 weight 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 generalized Dunkl 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 constant and give a short list of open problems.

Keywords: Nikol'skii-Bernstein inequality, exact constant, entire function of exponential spherical type, power-type weight, Dunkl Laplacian

Received April 8, 2019

Revised May 6, 2019

Accepted May 13, 2019

Funding Agency: This work was supported by the Russian Science Foundation (project no. 18-11-00199).

Dmitry Viktorovich Gorbachev, Dr. Phys.-Math. Sci., Prof., Tula State University, Tula, 300012 Russia, e-mail: dvgmail@mail.ru

Valerii Ivanovich Ivanov, Dr. Phys.-Math. Sci., Prof., Tula State University, Tula, 300012 Russia, e-mail: ivaleryi@mail.ru

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Cite this article as: D.V.Gorbachev, V.I.Ivanov. Nikol’skii–Bernstein constants for entire functions of exponential spherical type in weighted spaces. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 75–87.