Inhomogeneous Strichartz estimatesin some critical cases

Abstract

Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a weak-type norm in the temporal variable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstract setting. The application to the wave equation rests on a slightly stronger form of the standard dispersive estimate in terms of certain Besov spaces.

Strichartz and uniform Sobolev inequalities for the elastic wave equation

“…For the classical wave equation (1.6), this phenomenon has been observed by Harmse [12] and Oberlin [27] for the diagonal case q = r and q = r. Later, Foschi [10] followed the scheme of Keel-Tao [15] and obtained the inhomogeneous estimates for the currently known widest range with q = r and q = r. 4 Furthermore, Taggart [31] obtained more estimates involving the Besov spaces. More recently, Bez, Cunanan and the third author [3] obtained certain weak type (in temporal variable) estimates in borderline cases. All of these results are essentially based on the dispersive estimate…”