Local space–time estimate with Besov data (Conjecture 1.9) and its higher-dimensional status
Establish, in dimensions n ≥ 2, the local space–time inequality ∥e^{itΔ}f∥_{L_x^p L_t^r(R^n×[0,1])} ≤ C ∥f∥_{B_{α,p}(R^n)} with α = n(1 − 2/p) − 2/r, for exponents 2 ≤ p < ∞ and 2 ≤ r ≤ ∞ satisfying the constraints stated in Conjecture 1.9, thereby extending the one-dimensional case where the conjecture is known to hold.
References
Conjecture 1.9. Let 2 ≤ p < ∞, 2 ≤ r ≤ ∞ satisfy p + r< 2 and p + r < n. Then, it∆ (1.9) e f LxL tR ×[0,1]) f B α,pR ) p hold for α = n(1 − p) − .rHere B α,p denotes the non-homogeneous Besov space with the norm X kαp p 1/p f Bα,p = ( 2 P k p) . For n = 1, the conjecture was proved by Lee, Rogers, and Seeger [24]. The higher dimensional case is still open.