In this paper we study the behavior of dilation operators Dλ:f↦f(λ⋅) with λ>1
Nearby in the stack
in the context of Triebel-Lizorkin-Morrey spaces
Eu,p,qs(Rd)
. For that purpose we prove upper and lower bounds for the operator (quasi-)norm
∥Dλ∣L(Eu,p,qs(Rd))∥
. We show that for
s>σp
the operator (quasi-)norm
∥Dλ∣L(Eu,p,qs(Rd))∥
up to constants behaves as
λs−ud
. For the borderline case
s=σp
we observe a behavior of the form
λσp−ud
, multiplied with logarithmic terms of
λ
that also depend on the fine index
q
. For
s<σp
and
p≥1
we find the relation
∥Dλ∣L(Eu,p,qs(Rd))∥∼λ−ud
. The case
s<σp
and
p<1
is investigated as well. Our proofs are mainly based on the Fourier analytic approach to Triebel-Lizorkin-Morrey spaces. As byproducts we show an advanced Fourier multiplier theorem for band-limited functions in the context of Morrey spaces and derive some new equivalent (quasi-)norms and characterizations of