Probabilistic approach to the Littlewood-Paley-Stein theory

The Littlewood-Paley-Stein inequality is a basic tool in the Lp-theory on the Euclidean space. A typical application would be the equivalence of two (apparently) different Sobolev-type norms; one is the summation of Lp-norms of gradients up to degree k, another is the Lp-norm of the k/2-th power of the Laplacian.
In 1976, P. Meyer (ref.2 below) gave an elegant proof of the Littlewood-Paley-Stein inequality by running a Brownian motion. An advantage of this proof is that it can be applied without much extra difficulty to the setting of Riemannian manifold. In fact, following the idea of Meyer, D. Bakry (ref.1 below) proved the norm equivalence referred to above in the case k=1 on the Riemannian manifold with the Ricci curvature bounded from below.
Here, the work of Bakry was limitted to the case of k=1 for the following reason. If one tries to generalize the result to the case k \ge 2, one then has to work on tensor field of higher degree, for which one needs new idea to overcome the complicated commutation relation of the covariant differentiations (For example, the Ricci identity or the Weizenbeck formula for p-tensors (p \ge 2) are much more complicated than they are for p=1).
In the ref. 5 below, the problem mentioned above was solved and the Littlewood-Paley-Stein inequality was extended to general section to the vector bundle on the manifold (tensor fields, differential forms...). As an application, the result of Bakry was extended to arbitrary k \ge 2.

References

  1. Bakry, D. : Etude des transformations de Riesz dans les vari\'{e}t\'{e}s riemanniennes \`{a} courbure de Ricci minor\'{e}e, S\'{e}minaire de Probabilit\'{e}s XXI, Springer Lecture Notes in Math. {\bf 1247}, 137-172 (1987).
  2. Meyer, P. A. : D\'{e}monstration probaliste de certaines in\'{e}galit\'{e}s Littlewood-Paley, S\'{e}minaire de Probabilit\'{e}s. X, Springer Lecture Notes in Math. {\bf 511}, 125-183 (1976).
  3. Yoshida, N. The Littlewood-Paley-Stein inequality on an infinite dimensional manifold,
    J. Funct. Anal. . 122, 402--427, (1994).
  4. Yoshida, N. and Shigekawa, I.: Littlewood-Paley-Stein inequality for symmetric diffusions,
    J. math. Soc. Japan, 44, pp.251-280 (1992).
  5. Yoshida, N: Sobolev spaces on a Riemannian manifold and their equivalence,
    J. Math. Kyoto Univ. 33 621--654,(1992).
    See the abstract