Extremals for $\alpha$-Strichartz inequalities

A necessary and sufficient condition on the precompactness of extremal sequences for one dimensional $\alpha$-Strichartz inequalities, equivalently $\alpha$-Fourier extension estimates, is established based on the profile decomposition arguments. One of our main tools is an operator-convergence dislocation property consequence which comes from the van der Corput Lemma. Our result is valid in asymmetric cases as well. In addition, we obtain the existence of extremals for non-endpoint $\alpha$-Strichartz inequalities.