Throughout my work in harmonic analysis and its applications
families of function spaces have played a big role, sometimes as an
technical tool, sometimes as an independent subject of interest.
For me the design of families of function spaces (for me this is
Banach spaces of distributions over locally compact Abelian
groups, like the Euclidean space) is one of the key challenges. It is
natural, that often progress is made by going from a few concrete
cases to more general situations, always asking to which extent the
existing function spaces are appropriate for a discussion of the
continuity properties of linear operators arising in a certain context.
Of course the classical Lebesgue spaces are the prototypical
examples of such a “scale of spaces”, and the masters of
interpolation theory (Jaak Peetre and Hans Triebel) have
shown us that the well-known Besov-Triebel-Lizorkin spaces
form families with similar properties.

For me their work was the motivation to introduce Wiener
amalgam spaces, modulation spaces and decomposition spaces.
The short message of this short talk will be: If you extend or
generalize a given result to a larger family, you have to do it in the
right way. Just to give an example: Having a multi-parameter
family of spaces (like weighed Lp-spaces with a family of weights)
it is a big difference whether one claims that a given operator is
bounded in any of these spaces of this family, or whether
(hopefully constructively) a uniform bound can be established. I
have seen various cases where the second claim can be obtained
from the proofs, while the first one was stated in the results of a
paper. Further examples are related to the irregular sampling
problem or the theory of Banach frames.