For the half-wave propagator, the maximal operator that takes the supremum in time of the solution is bounded from the Sobolev space of order s into L^q exactly when s is at least a critical exponent s_c(q,d), which equals (d+1)/4 minus (d-1)/(2q) for q below the threshold 2(d+1)/(d-1) and equals d/2 minus d/q at and above it. Boundedness above the critical exponent and failure below it were already known; this concept is about the critical line itself. A strong-type bound holds at s equal to s_c for every q strictly between 2 and infinity except the single threshold exponent, where only a weak-type substitute from a Sobolev-Lorentz space into weak L^q is available, and the endpoint genuinely fails at q equal to 2 and at q equal to infinity, the latter because the Sobolev embedding into bounded functions fails. The proof route is to prove sharp L^p-Sobolev to L^q estimates and specialize, drawing on bilinear restriction estimates for the light cone, with a real-interpolation upgrade to Lorentz targets. A companion structural point is that the four variant formulations of the maximal estimate, local in time, local in time and space, global in time with local space, and fully global with homogeneous Sobolev data, are all equivalent for q at least 2, by finite speed of propagation, translation invariance and scaling.
A harmonic-analysis research paper that closes the endpoint case of maximal estimates for the half-wave propagator. Setting: for the wave equation in dimension d at least 2 with initial data f and z…