On 2-microlocal spaces with all exponents variable

Abstract

Abstract. In this paper we study various key properties for 2-microlocal Besov and Triebel-Lizorkin spaces with all exponents variable, including the lifting property, embeddings and Fourier multipliers. We also clarify and improve some statements recently published.

Variable exponent Triebel-Lizorkin-Morrey spaces

“…We outline the necessary changes to the proof in [Ke09] (cf. also [AC16], where the version with the k was introduced), which are small, since the proof technique works mainly with pointwise estimates.…”

Decompositions with Atoms and Molecules for Variable Exponent Triebel–Lizorkin–Morrey Spaces

“…with a > 0 at our disposal and c a > 0 independent of x ∈ R n , ν ∈ N 0 and f ∈ S ′ (R n ). Here θ ν = 2 νn θ(2 ν ·), ν ∈ N, where θ 0 , θ fit, for some k ∈ (1, 2] and ε > 0, in the requirements of Theorem 3.1 of [1] for the ψ 0 , ψ in that theorem and D βθ 1 (0) = 0 for any β ∈ N n 0 . Defining now ψ 0 :=θ 0 (−·) and ψ 1 :=θ(− · 2 ), it is easy to see that these ψ 0 , ψ 1 satisfy the conditions of our Theorem 4.5 for the same k and ε as above.…”

Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions, local means and non‐smooth atomic decompositions

“…The case is covered by [, Theorem 3.1(ii)], where we can find also a discussion on the importance of having a k in the conditions and in contrast with the case usually found in the literature in such type of result, cf. e.g.…”