L^p-L^q estimates off the line of duality

SOME SHARP BOUNDS FOR THE CONE MULTIPLIER OF NEGATIVE ORDER IN ${\mathbb R}^3$

“…where φ ∈ C ∞ 0 (1,2) and denotes the Fourier transform; r + = r if r > 0, and r + = 0 if r < 0. By analytic continuation, this definition makes sense even when Re(µ) 6 −1.…”

“…where ψ ∈ C ∞ 0 (−1, 1) and ρ ∈ C ∞ 0 (1,2). Using the L 6/5 → L 2 restriction estimate for the cone in R 3 (see [14, pp.…”

“…We only need to set ψ k = F(r (n−1)/2−k ψ(r)). Since ρ is supported in the interval (1,2), it follows that if |ξ| 6 1/2, then |d/dr(2 j rτ + 2 j rθ · ξ)| > C2 j for all θ ∈ S n−1 . Considering expression (3.9), we easily see that (3.8) follows from integration by parts since supp ρ ⊂ (1, 2).…”

Sharp Lp-Lq estimates for Bochner–Riesz operators of negative index in Cho1, Kim2, Lee3 et al. 2005Journal of Functional Analysis

“…In this note we study the problem of L p -L q boundedness of T − , > 0, which already has been studied by some authors ( [1][2][3]5,8,14]). To describe the results we introduce some notations.…”

“…In R 2 , Bak [1] showed that the converse is true for 0 < 3/2. In higher dimensions, n 3, this was established only for 1/2 < (n + 1)/2 (see [1,3,8]). In addition to strong (p, q) boundedness of T − , for (1/p, 1/q) on the line segments (A , B ], (A , B ] (0 < < 3/2 if n = 2 and 1/2 < < (n + 1)/2 if n 3), some weaker endpoint estimates are known (see [1,8]).…”