Ultrafast imaging based on plane-wave (PW) insonification is an active area of research due to its capability of reaching high frame rates. Among PW imaging methods, Fourier-based approaches have demonstrated to be competitive compared to traditional delay and sum methods. Motivated by the success of compressed sensing techniques in other Fourier imaging modalities, like magnetic resonance imaging, we propose a new sparse regularization framework to reconstruct high quality ultrasound images. The framework takes advantage of both the ability to formulate the imaging inverse problem in the Fourier domain and the sparsity of ultrasound images in a sparsifying domain. We show, by means of simulations, in vitro and in vivo data, that the proposed framework significantly reduces image artifacts, i.e. measurement noise and side lobes, compared to classical methods, leading to an increase of the image quality.