## 1 Introduction
Compared to the fully digital precoding scheme, the cost and complexity are greatly reduced with a slight performance degradation. Ref. [16] proposed a robust energy-efficient hybrid precoding scheme under the constraints of transmit power and quality of service (QoS). In [17], the authors introduced a multi-LEO-satellite communication network, which combines a hybrid precoding architecture and heuristic user scheduling algorithms. The superior performance of the proposed scheme was confirmed. In [18], the performance of hybrid precoding for mMIMO LEO satellites is analyzed by utilizing a fully connected hybrid antenna array structure. The performance is also compared with fully digital precoding and analog precoding. Furthermore, a hybrid precoding design scheme considered resource allocation for multi-user mMIMO LEO satellite communication systems was proposed in [19]. In mMIMO downlink LEO satellite communication systems, nonlinearities of power amplifiers (PAs) cause nonlinear distortions in the transmitted signals, leading to performance degradation. In previous works, the authors in [20] investigated the distortion generated by mMIMO base stations with nonlinear power amplifiers (NPAs) and obtainedthe spatial inter-correlation matrix of nonlinear distortion. It helps to predict the behavior of nonlinear distortion. Ref. [21] modeled the nonlinear amplified signal of a multi-antenna transmitter and analyzed the performance of a multiple-input multiple-output (MIMO) millimeter-wave hybrid precoding scheme in the presence of nonlinear beams. This model is employed in [13; 22; 23]. Ref. [22] builds on [21] and investigates the design of distortion-aware precoding utilizing the classical weighted minimum mean square error (WMMSE) framework and proposes an algorithm to solve the equivalent WMMSE optimization problem in precoding design. In [23], a distortion-aware precoding design was proposed for SATCOM downlink system with NPAs at the satellite transmitters. Based on the aforementioned research, this paper focuses on the hybrid precoding design of the downlink in an integrated network architecture combining CF-mMIMO and LEO satellites with the nonlinear distortion of NPAs. The main contributions of this paper are as follows:  
* We establish a comprehensive downlink transmission model for CF-mMIMO-assisted LEO SATCOM systems under the impact of NPAs. To quantify the distortion effects, the NPAs are modeled as finite-order memoryless polynomials, explicitly characterizing their in-band distortion. The formulation provides a tractable framework for analyzing and mitigating NPA-induced signal impairments. * A user-centric SC strategy is proposed to optimize satellite-ground collaboration in CF-mMIMO LEO systems, where multiple satellites cooperatively serve multiple GUs. The strategy adaptively selects serving satellites for each GU by jointly considering the elevation angle and channel fading. Based on the proposed strategy, the link relationships between APs and GUs can be represented through a set of binary diagonal matrices, enabling low-complexity cluster management and resource allocation. * A hybrid precoding problem for EE maximization is formulated by taking the product of tightly coupled analog and digital matrices as a single fully digital precoder. As for the fully digital precoding problem, we transform it into a linear programming problem by adopting an auxiliary variable and then use the WMMSE framework and a gradient descent method with [PERSON]'s accelerated momentum to solve it. Additionally, we design the hybrid analog/digital precoders based on the principle of alternating minimization. The rest of this paper is organized as follows: Section 2 discusses the system model with NPAs and formulates the EE maximization problem. An algorithmic approach is developed to form the SCs in Section 3. Section 4 investigates the precoding design for the EE maximization and discusses the design of hybrid analog/digital precoders with partially connected architectures.