Singular integrals and estimates for the Cauchy-Riemann equations

Abstract

1. Introduction. Our purpose here is to give new estimates for the inhomogeneous Cauchy-Riemann equations du = ƒ, in a smooth strictly pseudo-convex domain in C". The estimates we shall present should not, however, be regarded as isolated calculations ; they are part of a larger pattern of results which arise from adopting the following general point of view 1 : There is an intimate connection between some new singular integrals that arise (and the estimates to be made for them) in the following areas :(1) In the context of nilpotent groups, motivated partly by the study of intertwining operators. See From the qualitative point of view what is common to these areas (and distinguishes them from the more classical integrals and estimates) is the splitting of directions at each point, together with the nonisotropic way that singularities behave and estimates are made. The nonisotropy is not always apparent for LP estimates, but its role becomes clear when calculating with the appropriate Lipschitz (or Holder) inequalities. It is with the latter type of estimates that we are principally concerned here. Detailed proofs and further results will be given at another occasion.

Harmonic forms and Riesz transforms for rank one symmetric spaces

“…General theorems about these type of operators on nilpotent groups have also been obtained by and Stein [6].…”

Optimal Hölder and $L\sp p$ estimates for $\overline\partial\sb b$ on the boundaries of real ellipsoids in ${\bf C}\sp n$

“…The The Holder space of exponent Q on aD, 0 < Q < 1, is denoted by Aa. (aD) and is defined by the following norm (see Stein [25]): Note that 8 b f does not depend on the extension f. We shall say 8 b u = f in the weak sense for u E L;,q(aD) and f E L;,q+l(aD) if for any smooth (n-p,n-q-l) form'll on aD, ( 1.2) It is easy to see that when u and f are smooth, (1.2) agrees with (1.1).…”

Tangential Lipschitz gain for holomorphic functions on domains of finite type

“…i.e., Lip α (D) = {f : D → C : ∃ C f > 0, |f (x) − f (y)| ≤ C f · |x − y| α for x, y ∈ D} . Theorem 1.1 (Stein [17]). Suppose f ∈ O(Ω)∩C(Ω).…”