LiDAR super-resolution addresses the challenge of achieving high-quality 3D perception from cost-effective, low-resolution sensors. While recent transformer-based approaches like TULIP show promise, they remain limited to spatial-domain processing with restricted receptive fields. We introduce FLASH (Frequency-aware LiDAR Adaptive Super-resolution with Hierarchical fusion), a novel framework that overcomes these limitations through dual-domain processing. FLASH integrates two key innovations: (i) Frequency-Aware Window Attention that combines local spatial attention with global frequency-domain analysis via FFT, capturing both fine-grained geometry and periodic scanning patterns at log-linear complexity. (ii) Adaptive Multi-Scale Fusion that replaces conventional skip connections with learned position-specific feature aggregation, enhanced by CBAM attention for dynamic feature selection. Extensive experiments on KITTI demonstrate that FLASH achieves state-of-the-art performance across all evaluation metrics, surpassing even uncertainty-enhanced baselines that require multiple forward passes. Notably, FLASH outperforms TULIP with Monte Carlo Dropout while maintaining single-pass efficiency, which enables real-time deployment. The consistent superiority across all distance ranges validates that our dual-domain approach effectively handles uncertainty through architectural design rather than computationally expensive stochastic inference, making it practical for autonomous systems.