Atomic representations in function spaces and applications to pointwise multipliers and diffeomorphisms, a new approach

In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents Fp(·),q(·)sfalse(·false),ϕfalse(Rnfalse). These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents Fp(·),q(·)s(·)false(Rnfalse) as well as the classical Triebel–Lizorkin spaces Fp,qsfalse(Rnfalse), even the case when p=∞. Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces Fp,qs,τfalse(Rnfalse) with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.

“…Remark In the particular case of the classical Triebel–Lizorkin spaces

F_{p, q}^{s} false(R^{n} false)

this result is well‐known and coincides with [, Corollary 2.8.2]; see also to .…”

Section: Pointwise Multiplierssupporting

confidence: 76%

“…Remarks (i)In the case of classical Triebel–Lizorkin spaces

F_{p, q}^{s} false(R^{n} false)

the above result has been proved in . Regarding the Triebel–Lizorkin spaces with variable exponents

F_{p (\cdot), q (\cdot)}^{s (\cdot)} false(R^{n} false)

, it is covered by [, Theorem 3.14]. (ii)Our proof relies mainly on the proof of the smooth atomic decomposition from , although we were able to avoid the use of the maximal operator. …”

Section: Non‐smooth Atomic Decompositionmentioning

confidence: 95%

“…The next result states that the product of a

[K, L]

‐non‐smooth atom with a function

φ \in {scriptC}^{ρ} false(R^{n} false)

is still a

[K, L]

‐non‐smooth atom, and represents a slight normalization of Lemma 4.3 in . Lemma There exists a constant c with the following property: for all

ν \in {double-struckN}_{0}

m \in {double-struckZ}^{n}

, all

[K, L]

‐non‐smooth atoms

a_{ν, m}

with support in

3 Q_{ν, m}

and all

φ \in {scriptC}^{ρ} false(R^{n} false)

with

ρ \geq max (K, L)

, the product

c false∥ φ ∣ {scriptC}^{ρ} false(R^{n} false) ∥^{- 1} \cdot φ \cdot a_{ν, m}

is a

[K, L]

‐non‐smooth atom with support in

3 Q_{ν, m}

.…”

Section: Pointwise Multipliersmentioning

confidence: 98%

“…Remark The conditions and were firstly used together in , but they were adapted from similar conditions previously suggested in and , respectively. Since

C^{k} false(R^{n} false) ↪ {scriptC}^{k} false(R^{n} false)

for

k \in {double-struckN}_{0}

, it is clear that condition follows from .…”

Section: Non‐smooth Atomic Decompositionmentioning

confidence: 99%

{scriptC}^{s} false(R^{n} false)

is still a function in this space, as in Lemma 4.2 in .…”

Section: Pointwise Multipliersmentioning

confidence: 99%

See 3 more Smart Citations

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

Gonçalves

Moura

2018

Local means, wavelet bases and wavelet isomorphisms in Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces

Rosenthal

2012

We consider local means with bounded smoothness for Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r-regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov-Triebel-Lizorkin spaces.

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

Izuki

Noi

2019

In this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.