2 JESUS BASTERO, MARIO MILMAN AND FRANCISCO J. RUIZ
setting some of the characteristic features of real interpolation are blended with the
theory of weights.
In the second part of the paper we study the connection of the theory of
weighted interpolation developed in the first half with the so called commutator
estimates arising from the real method of interpolation. Here the motivation for
our work comes from recent striking work by Muller [Mu] and Coifman, Lions,
Meyer, and Semmes [CLMS] and the rapidly growing body of literature gener-
ated through their influence. Indeed, these papers have generated considerable
new impetus for the applications of real variable techniques to estimate the size
of certain nonlinear expressions that appear in nonlinear PDE's. These operations
(e.g. Jacobians, Null Lagrangians, etc) can be estimated because of subtle cancel-
lations. For a sample of recent results we refer to [EM], [Li], [LZ], [LMZ], [Mil],
[Se] and the references quoted therein. The higher integrability of these nonlin-
ear operators is crucial to establish their compactness in suitable weak topologies.
These developments provide a new framework to study questions of the theory of
compensated compactness developed by Murat and Tartar (cf. [CLMS] and the
references therein). Many of these estimates can be established using commuta-
tor theorems for singular integrals and methods arising from interpolation theory
(cf. [RW], [JRW], [IS], [Mi]). These commutator estimates are also important in
the regularity theory of quasilinear second order elliptic equations under minimal
assumptions on the coefficients (cf. the survey [Ch] and the references therein).
Interpolation theory plays an important role in these studies. On the one hand
it can be used to establish many of these concrete estimates while on the other it
provides tools to study higher integrability of non linear operations, due to cancel-
lations, in a very general setting. In fact in this generalized setting many of the
arguments are simpler and the role of the cancellations apparent. Moreover, inter-
polation methods also point to expressions that exhibit higher order cancellations
and which therefore are bounded in better spaces than one could have predicted
from size considerations only. The theory, originally started in [RW] and [JRW],
has been extended and applied in several directions. We refer to [Mi] and [MR]
for an extended discussion, as well as a detailed list of contributions. Apart from
its applications to Elasticity Theory, Harmonic Analysis, and Partial Differential
Equations the theory has also been applied in other areas of functional analysis. In
particular we refer to [Ka], [Kal], [Ka2] and the review papers quoted above for
more references.
It therefore seemed to us of timely interest to extend the general theory of
commutators associated with the real method of interpolation. In our development
in this paper we focus on two directions. On the one hand we treat commutators
of bounded operators with a general class of nonlinear operators including higher
order commutators of fractional order. On the other we also consider weighted
norm inequalities for these operators in a very general context. Our methods are
based, and generalize, the analysis given in [Mi] and [Mi3].
In order to explain in some more detail what we do let us now recall some basic
facts and definitions.
Previous Page Next Page