In medical image segmentation, convolutional neural networks (CNNs) and transformers are dominant. For CNNs, given the local receptive fields of convolutional layers, long-range spatial correlations are captured through consecutive convolutions and pooling. However, as the computational cost and memory footprint can be prohibitively large, 3D models can only afford fewer layers than 2D models with reduced receptive fields and abstract levels. For transformers, although long-range correlations can be captured by multi-head attention, its quadratic complexity with respect to input size is computationally demanding. Therefore, either model may require input size reduction to allow more filters and layers for better segmentation. Nevertheless, given their discrete nature, models trained with patch-wise training or image downsampling may produce suboptimal results when applied on higher resolutions. To address this issue, here we propose the resolution-robust HNOSeg-XS architecture. We model image segmentation by learnable partial differential equations through the Fourier neural operator which has the zero-shot super-resolution property. By replacing the Fourier transform by the Hartley transform and reformulating the problem in the frequency domain, we created the HNOSeg-XS model, which is resolution robust, fast, memory efficient, and extremely parameter efficient. When tested on the BraTS'23, KiTS'23, and MVSeg'23 datasets with a Tesla V100 GPU, HNOSeg-XS showed its superior resolution robustness with fewer than 34.7k model parameters. It also achieved the overall best inference time (< 0.24 s) and memory efficiency (< 1.8 GiB) compared to the tested CNN and transformer models.