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 2 d [ ∫ 0 1 m (ξ + t η) d t ] f ˆ (ξ) g ˆ (η) e 2 π i x ⋅ (ξ + η) d ξ d η. Our results are obtained under two different conditions of the multiplier m. The first result is that when K ∈ S ′ ∩ L l o c 1 (R d ∖ { 0 }) is a regular Calderón-Zygmund convolution kernel of regularity 0 < δ ≤ 1 , T K ˆ maps L p (R d) × L q (R d) into L r (R d) for all 1 < p , q ≤ ∞ , 1 r = 1 p + 1 q as long as r > d d + 1. The second result is that when the multiplier m ∈ C d + 1 (R d ∖ { 0 }) satisfies the Hörmander derivative conditions | ∂ ξ α m (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 ≤ ∞ , 1 r = 1 p + 1 q as long as r > d d + 1. These two results are sharp except for the endpoint case r = d d + 1. In case d = 1 and K (x) = 1 / x , it is well-known that T K ˆ maps L p (R) × L q (R) into L r (R) for 1 < p , q ≤ ∞ , 1 r = 1 p + 1 q as long as r > 1 / 2. In higher dimensional case d ≥ 2 , in 2016, when K ˆ (ξ) = ξ j / | ξ | d + 1 is the Riesz multiplier on R d , P. W. Fong, in his Ph.D. Thesis [9] , obtained ‖ T K ˆ (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. [ABSTRACT FROM AUTHOR]