Some Intrinsic Characterizations of Besov–Triebel–Lizorkin–Morrey–Type Spaces on Lipschitz Domains

Abstract

We give Littlewood–Paley type characterizations for Besov–Triebel–Lizorkin–type spaces $\mathscr {B}_{pq}^{s\tau },\mathscr {F}_{pq}^{s\tau }$ and Besov-Morrey spaces $\mathcal N_{uqp}^s$ on a special Lipschitz domain $\Omega \subset \mathbb R^n$: for a suitable sequence of Schwartz functions $(\phi _j)_{j=0}^\infty$,

$$\begin{aligned} \Vert f\Vert _{\mathscr {B}_{pq}^{s\tau }(\Omega )}&\textstyle \approx \sup _{P\text { dyadic cubes}}|P|^{-\tau }\Vert (2^{js}\phi _j*f)_{j=\log _2\ell (P)}^\infty \Vert _{\ell ^q(L^p(\Omega \cap P))};\\ \Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}&\textstyle \approx \sup _{P\text { dyadic cubes}}|P|^{-\tau }\Vert (2^{js}\phi _j*f)_{j=\log _2\ell (P)}^\infty \Vert _{L^p(\Omega \cap P;\ell ^q)};\\ \Vert f\Vert _{\mathcal N_{uqp}^{s}(\Omega )}&\textstyle \approx \big \Vert \big (\sup _{P\text { dyadic cubes}}|P|^{\frac{1}{u}-\frac{1}{p}}\cdot 2^{js}\Vert \phi _j*f\Vert _{L^p(\Omega \cap P)}\big )_{j=0}^\infty \big \Vert _{\ell ^q}. \end{aligned}$$

We also show that $\Vert f\Vert _{\mathscr {B}_{pq}^{s\tau }(\Omega )}$, $\Vert f\Vert _{\mathscr {F}_{pq}^{s\tau }(\Omega )}$ and $\Vert f\Vert _{\mathcal N_{uqp}^{s}(\Omega )}$ have equivalent (quasi-)norms via derivatives: for $\mathscr {X}^\bullet \in \{\mathscr {B}_{pq}^{\bullet ,\tau },\mathscr {F}_{pq}^{\bullet ,\tau },\mathcal N_{uqp}^\bullet \}$, we have $\Vert f\Vert _{\mathscr {X}^s(\Omega )}\approx \sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathscr {X}^{s-m}(\Omega )}$. In particular $\Vert f\Vert _{\mathscr {F}_{\infty q}^s(\Omega )}\approx \sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathscr {F}_{\infty q}^{s-m}(\Omega )}\approx \sup _{P}|P|^{-n/q}\Vert (2^{js}\phi _j*f)_{j=\log _2\ell (P)}^\infty \Vert _{\ell ^q(L^q(\Omega \cap P))}$.