On the Classification of Homogeneous Multipliers Bounded on H 1 ( R 2 )

Convolution Calderón-Zygmund singular integral operators with rough kernels

“…Since Ω is even, the left hand side of (13) is equal to K Ω L ∞ in view of (8). We conclude that T Ω is L 2 bounded, and hence condition (10) implies L 2 boundedness.…”

“…For x / ∈ 2Q subtract K(x)a Q (x) from T (a Q )(x) and then use condition (2). [8]. Below, we give the an example communicated to us by M. Christ.…”

“…Remark. A fundamental result of J. Daly and K. Phillips [8] says that if If Ω is allowed to be a distribution on the sphere, such an Ω is shown to exist by abstract methods. To be more precise, let us introduce the following Banach spaces of distributions on the sphere.…”

A necessary condition for Calder�n-Zygmund singular integral operators

“…Grafakos and Stefanov [9,13] have been continuing the study of these sufficient conditions which implies the LP-boundedness or weak boundedness of T~ and have found a new condition on the kernel involving variation. In 1989 Daly and Phillips [6] gave an example of a Calder6n-Zygmund singular integral operator that was bounded on L2(R n) and whose kernel is not a function. In this paper necessary conditions rather than sufficient conditions were considered for the case n = 2, and it is shown that if T~ is bounded on H 1 (R2), then f2 is in H 1 ($1).…”

“…The first is an n-dimensional generalization of the 2-dimensional result that says that homogeneous of degree zero multipliers that give rise to bounded operators on H 1 (R 2) have absolutely convergent Fourier series on the circle. See Daly-Phillips [6]. …”