Interpolation of Banach spaces and applications 1000-1M25IPB

We mainly study the method(s) of complex interpolation, and the real inter-
polation method via the K-functional and J-functional. Concerning the general
properties of these methods, we establish the equivalence of K-method and J-
method and we consider natural questions related to reiteration and duality.
Also, the useful “reverse reiteration” theorem of Wolff will be considered and
applied in some situations.
The Davies-Figiel-Johnson-Pelczi´nski factorisation theorem is one of the first
major applications of interpolation in the Banach space theory. Apart from this,
we will consider other results concerning functional analytical properties of the
interpolation spaces and of the operators acting on them. In particular, we will
pay attention to Cwikel’s result related to the preservation of compactness.
One of the main motivations for developing the theory of interpolation was
to apply it to the theory of Partial Differential Equations. For instance, in-
terpolation enters as an essential ingredient in the proof of boundedness of the
Calder´on-Zygmund operators. The Littlewood-Paley theory itself aligns with
these ideas as well as embedding theorems for Sobolev spaces that are om-
nipresent in the study of the regularity of solutions. Interpolation is also useful
for other type of problems, that at first sight are unrelated to the theory of
Partial Differential Equations. One such example is provided by the proof of
the endpoint case of the Tomas-Stein restriction theorem. Here, a version of the
complex method (the Stein method) is used.
In order to apply the interpolation methods in the field of Partial Differential
Equations one needs to know how to interpolate the spaces that often appear
in this field. In many cases, thanks to the Littlewood-Paley theory, one can
explicitly interpolate Lp spaces and Sobolev spaces via the complex and the
real interpolation method. In the case of the real interpolation we are led to the
theory of Besov spaces. The theory of traces is, as well, linked naturally to the
subject.
One major drawback of the interpolation theory is the fact that, in general,
we can not easily describe the interpolation spaces of subcouples. This problem
arises, for instance, when the spaces of the subcouple are not given by a bounded
projection. However, in some situations one can overcome this problem. One key
concept is the K-closedness of subcouples introduced by Pisier. Some interesting
examples of K-closed subcouples are present in the work of Pisier and Bourgain.
In this circle of ideas elegant methods where applied in order to interpolate the
Hardy spaces.