On theT(1) theorem on product domains

Abstract

The direct proof by R. R. Coifman and Y. Meyer of theT(1) Theorem of G. David and J. L. Journé is based on the following result.

LetT be an operator associated to a kernelk(x, y) satisfying

$$|k(y,x) - k(z,x)| \leqslant C\frac{{|y - z|^\delta }}{{|x - z|^{n + \delta } }}, if 2|y - z|< |x - z|$$

for some 0<δ≤1. Suppose thatT has the weak boundedness property and thatT(1)∈BMO (ℝⁿ). Then, the operator ∫ ^∞₀ q _t Q _t TP ² _t dt/t, defined in the weak sense, is continuous onL ² (ℝⁿ).

Here the operatorsq _t andQ _t are convolution operators with functions of integral 0, andP _t is also a convolution operator similar to the Poisson transform.

We prove a product domain version of this result.