Bilinear operators with non-smooth symbol, I

Abstract

×ØÖغ This paper proves the L p -boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding L p -boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part II, our subsequent paper [11], using phase-plane analysis.

Bilinear Pseudodifferential Operators on Modulation Spaces

“…A natural question then is to find sufficient (nontrivial) conditions on the symbol that ensure the boundedness of the operator on products of certain Banach spaces such as Lebesgue, Sobolev, or Besov spaces; see the works of Coifman and Meyer [4], [5], [6], Gilbert and Nahmod [14], [15], Muscalu, Tao and Thiele [25], Grafakos and Torres [16], [17], Bényi and Torres [2], [3], and Bényi [1] and the references therein for more details. For instance, it is known that the condition (2) |∂ for (x, ξ, η) ∈ R 3d and all multi-indices α, β, γ is enough to prove the boundedness of the operator defined by (1) …”

“…In particular, Coifman-Meyer-type conditions (2) are not satisfied by the symbols we consider. Note also that while the symbols considered in [15] or [25] are very singular along the anti-diagonal in the frequency plane, they are independent of the space variable x. Furthermore, the techniques used there to prove the boundedness of the corresponding operators fit the one dimensional situation, but they are yet to be developed in a multidimensional setting.…”

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

“…The first significant boundedness results concerning non-smooth symbols were proved by Lacey and Thiele [18], [19] who established that W σ , with σ(ξ, η) = sign(ξ + αη), α ∈ R \ {0, 1} has a bounded extension from L p (R n ) × L q (R n ) to L r (R n )) when r > 2/3. Extensions of this result were subsequently obtained by Gilbert and Nahmod [7]. Bilinear operators can also be defined on quasi-Banach spaces, such as the Hardy spaces H p ; see for instance [10] for the action of bilinear Calderón-Zygmund operators on real Hardy spaces.…”

On multilinear singular integrals of Calderón-Zygmund type

“…The type of singularity that the symbol of the bilinear Hilbert transform presents requires a delicate time-frequency analysis. Such techniques have been recently extended to other bilinear singular multiplier operators by Gilbert and Nahmod [30], [31] and also to m-linear operators by Muscalu, Tao, and Thiele [48]. For example, in the bilinear case the symbols satisfy the estimates…”