For \(\mu , \beta \in {\mathbb {R}}\), we introduce and study in detail the generalized Stieltjes operators

on Sobolev spaces \({{\mathcal{T}}_{p}^{{(\alpha )}}} (t^{\alpha })\) (where \(\alpha \ge 0\) is the fractional order of derivation and these spaces are embedded in \(L^p({\mathbb {R}}^+)\) for \(p\ge 1\)). If \(0< \beta - \frac{1}{p} < \mu \), then operators \({\mathcal {S}}_{\beta ,\mu }\) are bounded, commute and factorize with generalized Cesàro operator on \({{\mathcal{T}}_{p}^{{(\alpha )}}} (t^{\alpha })\) . We give their norm, and calculate and represent explicitly their spectrum set \(\sigma ({\mathcal {S}}_{\beta ,\mu })\). The main technique is to subordinate these operators in terms of \(C_0\)-groups which allows to transfer new properties from some exponential functions to these operators. We also prove some similar results for generalized Stieltjes operators \({\mathcal {S}}_{\beta ,\mu }\) in the Sobolev–Lebesgue \({{\mathcal {T}}_{p}^{(\alpha )}}(\vert t\vert ^\alpha )\) defined on the real line \({\mathbb {R}}\). We show connections of this family of operators with the Fourier and the Hilbert transform, and a convolution product defined by the Hilbert transform.

对于\(\mu , \beta \in {\mathbb {R}}\)，我们详细介绍和研究了广义 Stieltjes 算子

在 Sobolev 空间\({{\mathcal{T}}_{p}^{{(\alpha )}}} (t^{\alpha })\)（其中\(\alpha \ge 0\)是派生的分数阶和这些空间嵌入在\(L^p({\mathbb {R}}^+)\) 中，用于\(p\ge 1\) )。如果\(0< \beta - \frac{1}{p} < \mu \)，则运算符\({\mathcal {S}}_{\beta ,\mu }\)是有界的，交换和因式分解\({{\mathcal{T}}_{p}^{{(\alpha )}}} (t^{\alpha })\)上的广义 Cesàro 算子。我们给出他们的范数，并计算和明确表示他们的频谱集\(\sigma ({\mathcal {S}}_{\beta ,\mu })\)。主要技术是根据\(C_0\)-groups 允许将新属性从一些指数函数转移到这些运算符。我们还证明了 Sobolev–Lebesgue \({{\mathcal {T}}_{p}^{} 中广义 Stieltjes 算子\({\mathcal {S}}_{\beta ,\mu }\) 的一些类似结果(\alpha )}}(\vert t\vert ^\alpha )\)定义在实线\({\mathbb {R}}\) 上。我们展示了这一系列算子与傅立叶和希尔伯特变换以及由希尔伯特变换定义的卷积乘积的联系。