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

Abstract

In this paper we study natural generalizations of the first order Calderon commutator in higher dimensions d >= 2. We study the bilinear operator T-m which is given by T-m (f, g) (x) := integral integral(R2d )[integral(1)(0) m (xi + t eta) dt] (f) over cap(xi)(g) over cap(eta)e(2 pi ix.(xi+eta)) d xi d eta. Our results are obtained under two different conditions of the multiplier m. The first result is that when K is an element of S' boolean AND L-loc(1), (R-d \ {0}) is a regular CalderOn-Zygmund convolution kernel of regularity 0 < delta < 1, T ((K) over cap) maps L-p (R-d) x (q)(L)(R-d) into L-r (R-d) for all 1 < p, q <= infinity, 1/r = 1/p + 1/q as long as r >d/d+1. The second result is that when the multiplier m is an element of Cd+1 (Rd \ {0}) satisfies the Hormander derivative conditions vertical bar partial derivative(alpha)(xi) m(xi)vertical bar <= D-alpha vertical bar xi vertical bar(-vertical bar)(alpha vertical bar ) for all xi not equal 0, and for all multi-indices alpha with vertical bar alpha vertical bar <= d + 1, T-m maps L-p (R-d) x L-q (R-d ) into L-r (R-d ) for all 1 < p, q <= infinity, 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) over cap maps L-p (R) x L-q (R) into L-r (1[8) for 1 < p, q <= infinity, 1/r = 1/p + 1/q as long as r > 1/2. In higher dimensional case d >= 2, in 2016, when (K) over cap(xi) = xi(j)/vertical bar xi vertical bar(d+1) is the Riesz multiplier on R-d, P. W. Fong, in his Ph.D. Thesis [9], obtained parallel to T-(K) over cap (f, g) parallel to(r) <= C parallel to f parallel to(p)parallel to g parallel to(q) for 1 < p, q <= infinity 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. (C) 2020 Elsevier Inc. All rights reserved.