\(L^p\)-multipliers sensitive to the group structure on nilpotent Lie groups
For \(x, y \in \mathbb{R}^N\) a group operation can be defined on \(\mathbb{R}^N\) by \[ (x, y) \mapsto xy = x + y + P(x,y) \] where \(P : \mathbb{R}^N \times \mathbb{R}^N \rightarrow \mathbb{R}^N\) is a polynomial mapping \[ P(x, y) = (P_1(x,y), P_2(x,y), \dots, P_N(x,y)) \] with terms of order at least two. For every \(j\), the polynomial \(P_j\) depends only on the variables \(x_k, y_k\), where \(1 \leq k < j\). Furthermore, \[ P(x, 0) = P(0, x) = P(x, -x) = 0, \] for all \(x \in \mathbb{R}^N\). Additionally assume the group \((\mathbb{R}^N, P)\) is homogeneous. Now, \((\mathbb{R}^N, P)\) is a connected, simply connected nilpotent group. Denote by \(\mathcal{S}(\mathbb{R}^N)\) the Schwartz space on \(\mathbb{R}^N\). The multiplier operators on \((\mathbb{R}^N, P)\) are the convolution operators \[ f \star K = \int_{\mathbb{R}^N} f(xy^{-1})K(y)\, dy, \] where \(f \in \mathcal{S}(\mathbb{R}^N)\) and \(K\) is a tempered distribution on \(\mathbb{R}^N\). In this paper, the author gives sufficient conditions on the Fourier transform \(\widehat{K}\) that ensures the operator \(f \mapsto f \star K\) is bounded on \(L^p(\mathbb{R}^N)\), which improves a previous result of \textit{A. Nagel} et al. [Rev. Mat. Iberoam. 28, 631--722 (2012; Zbl 1253.42009)]. The paper is self-contained and well-written.
scientific article; zbMATH DE number 7077717
scientific article