Monitoring vegetation productivity at extremely fine resolutions is valuable for real-world agricultural applications, such as detecting crop stress and providing early warning of food insecurity. Solar-Induced Chlorophyll Fluorescence (SIF) provides a promising way to directly measure plant productivity from space. However, satellite SIF observations are only available at a coarse spatial resolution, making it impossible to monitor how individual crop types or farms are doing. This poses a challenging coarsely-supervised regression (or downscaling) task; at training time, we only have SIF labels at a coarse resolution (3km), but we want to predict SIF at much finer spatial resolutions (e.g. 30m, a 100x increase). We also have additional fine-resolution input features, but the relationship between these features and SIF is unknown. To address this, we propose Coarsely-Supervised Smooth U-Net (CS-SUNet), a novel method for this coarse supervision setting. CS-SUNet combines the expressive power of deep convolutional networks with novel regularization methods based on prior knowledge (such as a smoothness loss) that are crucial for preventing overfitting. Experiments show that CS-SUNet resolves fine-grained variations in SIF more accurately than existing methods.