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.
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This work was supported by the Russian Science Foundation (project no. 18-11-00199).
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Translated by M. Deikalova
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Gorbachev, D.V., Ivanov, V.I. Nikol’skii–Bernstein Constants for Entire Functions of Exponential Spherical Type in Weighted Spaces. Proc. Steklov Inst. Math. 309 (Suppl 1), S24–S35 (2020). https://doi.org/10.1134/S0081543820040045
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DOI: https://doi.org/10.1134/S0081543820040045