Off-diagonal estimates for the first order commutators in higher dimensions

In this paper we study natural generalizations of the first order Calderón commutator in higher dimensions d≥2. We study the bilinear operator T_m which is given by T_m(f,g)(x):=∬R^2d[∫01m(ξ+tη)dt]fˆ(ξ)gˆ(η)e^{2πix⋅(ξ+η)}dξdη. Our results are obtained under two different conditions of the multiplier m. The first result is that when K∈S^′∩L_loc ¹(R^d∖{0}) is a regular Calderón-Zygmund convolution kernel of regularity 0<δ≤1, T_Kˆ maps L^p(R^d)×L^q(R^d) into L^r(R^d) for all 1<p,q≤∞, [Formula presented] as long as [Formula presented]. The second result is that when the multiplier m∈C^d+1(R^d∖{0}) satisfies the Hörmander derivative conditions |∂_ξ ^αm(ξ)|≤D_α|ξ|^−|α| for all ξ≠0, and for all multi-indices α with |α|≤d+1, T_m maps L^p(R^d)×L^q(R^d) into L^r(R^d) for all 1<p,q≤∞, [Formula presented] as long as [Formula presented]. These two results are sharp except for the endpoint case [Formula presented]. In case d=1 and K(x)=1/x, it is well-known that T_Kˆ maps L^p(R)×L^q(R) into L^r(R) for 1<p,q≤∞, [Formula presented] as long as r>1/2. In higher dimensional case d≥2, in 2016, when Kˆ(ξ)=ξ_j/|ξ|^d+1 is the Riesz multiplier on R^d, P. W. Fong, in his Ph.D. Thesis [9], obtained ‖T_Kˆ(f,g)‖_r≤C‖f‖_p‖g‖_q for 1<p,q≤∞ as long as r>d/(d+1). As far as we know, except for this special case, there has been no general results for the off-diagonal case r<1 in higher dimensions d≥2. To establish our results we develop ideas of C. Muscalu and W. Schlag [18,19] with new methods.