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 ${\mathcal{B}}_{pq}^{s\tau },{\mathcal{F}}_{pq}^{s\tau }$ 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 $({\varphi }_j{)}_{j=0}^{\infty }$, $\Vert f{\Vert }_{{\mathcal{B}}_{pq}^{s\tau }(\Omega )}\approx {\sup }_{P\text{ dyadic cubes}}|P{|}^{-\tau }\Vert (2^{js}{\varphi }_j\ast f{)}_{j={\log }_2\ell (P)}^{\infty }{\Vert }_{{\ell }^q(L^p(\Omega \cap P))};$ $\Vert f{\Vert }_{{\mathcal{F}}_{pq}^{s\tau }(\Omega )}\approx {\sup }_{P\text{ dyadic cubes}}|P{|}^{-\tau }\Vert (2^{js}{\varphi }_j\ast f{)}_{j={\log }_2\ell (P)}^{\infty }{\Vert }_{L^p(\Omega \cap P;{\ell }^q)};$ $\Vert f{\Vert }_{{\mathcal{N}}_{uqp}^s(\Omega )}\approx \Vert ({\sup }_{P\text{ dyadic cubes}}|P{|}^{\frac{1}{u}-\frac{1}{p}}\cdot 2^{js}\Vert {\varphi }_j\ast f{\Vert }_{L^p(\Omega \cap P)}{)}_{j=0}^{\infty }{\Vert }_{{\ell }^q}.$ We also show that $\Vert f{\Vert }_{{\mathcal{B}}_{pq}^{s\tau }(\Omega )}$, $\Vert f{\Vert }_{{\mathcal{F}}_{pq}^{s\tau }(\Omega )}$ and $\Vert f{\Vert }_{{\mathcal{N}}_{uqp}^s(\Omega )}$ have equivalent (quasi-)norms via derivatives: for ${\mathcal{X}}^{\bullet }\in \{{\mathcal{B}}_{pq}^{\bullet ,\tau },{\mathcal{F}}_{pq}^{\bullet ,\tau },{\mathcal{N}}_{uqp}^{\bullet }\}$, we have $\Vert f{\Vert }_{{\mathcal{X}}^s(\Omega )}\approx {\sum }_{|\alpha |\le m}\Vert {\partial }^{\alpha }f{\Vert }_{{\mathcal{X}}^{s-m}(\Omega )}$. In particular $\Vert f{\Vert }_{{\mathcal{F}}_{\infty q}^s(\Omega )}\approx {\sum }_{|\alpha |\le m}\Vert {\partial }^{\alpha }f{\Vert }_{{\mathcal{F}}_{\infty q}^{s-m}(\Omega )}\approx {\sup }_P|P{|}^{-n/q}\Vert (2^{js}{\varphi }_j\ast f{)}_{j={\log }_2\ell (P)}^{\infty }{\Vert }_{{\ell }^q(L^q(\Omega \cap P))}$ for all $0<q<\infty$.

Final version; 13 pages. Appeared in Journal of Fourier Analysis and Applications. Fixed a few typoes of citation due to change of version

References in corpus (1)

• Interpolation of Morrey-Campanato and Related Smoothness Spaces

Cited by in corpus (2)

• New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications
• Sobolev and Hölder estimates for homotopy operators of the $\overline{\partial }$-equation on convex domains of finite multitype