On Commutators of Singular Integrals and Bilinear Singular Integrals

Abstract

ABSTRACT. Lp estimates for multilinear singular integrals generalizingCalderón's commutator integral are obtained. The methods introduced involve Fourier and Mellin analysis.

Bilinear Singular Integral Operators, Smooth Atoms and Molecules

“…Let k ∈ Z + . We will say that K(x, y, z) is a Calderón-Zygmund kernel of order k if it is a C ∞ function defined away from the diagonal x = y = z in (R n ) 3 , which satisfies the size and smoothness estimates of order k…”

“…The study of multilinear operators was pioneered by Coifman and Meyer [3,4,5] who proposed to use them as intermediate tools to study specific linear and nonlinear partial differential equations. The recent work of Lacey and Thiele on the bilinear Hilbert transform [16,17] has produced new interest in the subject of multilinear operators.…”

Propagation of oscillations, complete trajectories and attractors for compressible flows

“…where v ∈ W 1,2 , w ∈ L p , 1/p + 1/2 < 1 (see [4]). This approach requires some modifications when γ ≤ 3.…”

Untitled

“…The corresponding operators T σ were studied by R. Coifman and Y. Meyer [1978;, C. Kenig and E. M. Stein [1999] and recently by L. Grafakos and R. Torres [2002]. We know that under (1-3), the operator T σ is bounded from L p ‫)ޒ(‬ × L q ‫)ޒ(‬ into L r ‫)ޒ(‬ for all exponents p, q, r satisfying (1-1) and 1 < p, q < ∞.…”

“…We know that under (1-3), the operator T σ is bounded from L p ‫)ޒ(‬ × L q ‫)ޒ(‬ into L r ‫)ޒ(‬ for all exponents p, q, r satisfying (1-1) and 1 < p, q < ∞. In fact if the symbol is x-independent, one can just assume an homogeneous decay in (1-3) (that is with (|α|+|β|) −b−c ) and then these operators can be decomposed with paraproducts, which were first exploited by J. M. Bony [1981] and R. Coifman and Y. Meyer [1978]. The paraproducts are studied with the linear tools (the Calderón-Zygmund decomposition, the Littlewood-Paley theory and the concept of Carleson measure).…”