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.

中文翻译：

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加权空间中指数球型整函数的Nikol'skii–Bernstein常数

我们在的整个函数\（f \）的子空间上 的Nikol'skii–Bernstein不等式\（\ | Df \ | __qq \ leq C \ | f \ | _ {p} \）中研究精确常数空间\（L ^ {p}（\ mathbb {R} ^ {d}）\）中具有幂类型权重 \（v _ {\ kappa} \）的指数球形类型。对于微分运算符 \（D \），我们采用与权重\（v _ {\ kappa} \）相关联 的Dunkl Laplacian \（\ Delta _ {\ kappa} \）的非负整数幂 。这种情况包括具有功率权重\（t ^ {2 \ alpha + 1} \）的空间\（L ^ {p}（\ mathbb {R} _ {+}）\）的一维情况。和贝塞尔微分算子。我们的主要结果在于证明\（1 \ leq p \ leq q = \ infty \）的多维加权常量与一维加权常量相等。为此，我们表明范数\（\ | Df \ | __ {\ infty} \）可以替换为值\（Df（0）\），该值仅在一维情况下才知道。函数子空间所需的映射实际上将问题减少到径向，因此是一维的情况，这是通过Dunkl广义平移\（T _ {\ kappa} ^ {t} \ ）。我们在变量\（t \）中证明了解析连续性的新性质 。结果，我们计算了\（p = q = \ infty \），仅在特殊情况下才知道。我们还找到了一些常数的估计值，并给出了未解决问题的简短列表。