In this note, we study the connection between the fractional Laplacian operator that appeared in the recent work of Caffarelli-Silvestre and a class of conformally covariant operators in conformal geometry.

We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.

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Introduction. In this article we wish to discuss a theory which is still developing very rapidly. It is only quite recently that many of the aspects of Fourier analysis of several parameters have been discovered, even though much of the corresponding one-parameter theory has been well known for some time. The topics to be covered include differentiation theory, singular integrals, Littlewood-Paley theory, weighted norm inequalities, Hardy spaces, and functions of bounded mean oscillation, as well as many other related topics. We shall begin in Part I by attempting to give a broad overview of some of the one-parameter results about these topics. The discussion here is, however, anything but encyclopedic. (For more detailed treatments of these matters in the one-parameter setting, the reader can consult such excellent treatments as E. [46].) In Part II we take up these same areas in the two-parameter setting. Since this theory is less well known than the material of Part I, we go into greater detail and devote separate sections to each of several of the above topics.

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