Sharp estimates for maximal operators associated to the wave equation

Abstract

We give almost sharp conditions under which the maximal operator associated with the wave equation with initial data in Sobolev space H^s(R^n) is bounded from H^s(R^n) to L^q(R^n).Comment: 7 pages, 1 figur

A global space-time estimate for dispersive operators through its local estimate

“…In higher dimensions n ≥ 2 it was proven by Vega [21] that (1.2) holds for q ≥ 2(n+2) n and n+1 q + 1 r ≤ n 2 . When n = 2 Rogers [15] showed it for 2 ≤ r < ∞, q > 16 5 and 3 q + 1 r < 1, and later the excluded endline 3 q + 1 r = 1 was obtained by Lee-Rogers-Vargas [11]. When n ≥ 3, Lee-Rogers-Vargas [11] improved the previous known result to r ≥ 2, q > 2(n+3) n+1 and n+1 q + 1 r = n 2 .…”

“…For a case of the wave operator it is known that (1.2) holds for (r, q) pairs such 2 ≤ r ≤ q, q = ∞ and 1 r + n−1 2q ≤ n−1 4 (see [8,9,13,18]). Particularly, when r = ∞, Rogers-Villarroya [16] showed that (1.2) with regularity s > n( 12 − 1 q ) − 1 r is valid for q ≥ 2(n+1) n−1 . For the fractional Schrödinger operator the known range of (r, q) for which the estimates hold is that 2 ≤ r ≤ q, q = ∞ and n 2q + 1 r ≤ n 4 (see [1,2,4,12,20]).…”

Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds

“…Another application is a sharp maximal function estimate. In [9], Rogers and Villarroya proved the following sharp maximal estimate for the wave operator in R d :…”

Pointwise convergence along a tangential curve for the fractional Schrödinger equation with 0 < m < 1

“…In dimension one, Carleson 1 proved that the pointwise convergence () holds if , which is proved to be sharp by Dahlberg and Kenig 2 through constructing the counterexample. In higher dimensions, one can see previous studies 3–14 for details.…”