2025/10/13 by Hovemann, Marc, Weimar, Markus
#46E30 #46E35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2510.11439
In this paper we study the behavior of dilation operators Dλ\colon f ↦ f(λ ⋅) with λ> 1 in the context of Triebel-Lizorkin-Morrey spaces Esu,p,q(ℝd). For that purpose we prove upper and lower bounds for the operator (quasi-)norm ‖ Dλ | L(Esu,p,q(ℝd)) ‖ . We show that for s>σp the operator (quasi-)norm ‖ Dλ | L(Esu,p,q(ℝd)) ‖ up to constants behaves as λs - (d)/(u) . For the borderline case s = σp we observe a behavior of the form λσp- (d)/(u), 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(Esu,p,q(ℝd)) ‖ ∼ λ - (d)/(u). 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 Esu,p,q(ℝd). Keywords: Dilation Operator, Morrey space, Triebel-Lizorkin-Morrey space, Fourier multiplier