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 $(\phi_j)_{j=0}^\infty$, $\|f\|_{\mathscr B_{pq}^{s\tau}(\Omega)}\approx\sup_{P\text{ dyadic cubes}}|P|^{-\tau}\|(2^{js}\phi_j\ast f)_{j=\log_2\ell(P)}^\infty\|_{\ell^q(L^p(\Omega\cap P))};$ $\|f\|_{\mathscr F_{pq}^{s\tau}(\Omega)}\approx\sup_{P\text{ dyadic cubes}}|P|^{-\tau}\|(2^{js}\phi_j\ast f)_{j=\log_2\ell(P)}^\infty\|_{L^p(\Omega\cap P;\ell^q)};$ $\|f\|_{\mathcal N_{uqp}^{s}(\Omega)}\approx\big\|\big(\sup_{P\text{ dyadic cubes}}|P|^{\frac1u-\frac1p}\cdot 2^{js}\|\phi_j\ast f\|_{L^p(\Omega\cap P)}\big)_{j=0}^\infty\big\|_{\ell^q}.$ We also show that $\|f\|_{\mathscr B_{pq}^{s\tau}(\Omega)}$, $\|f\|_{\mathscr F_{pq}^{s\tau}(\Omega)}$ and $\|f\|_{\mathcal N_{uqp}^{s}(\Omega)}$ 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 $\|f\|_{\mathscr X^s(\Omega)}\approx\sum_{|\alpha|\le m}\|\partial^\alpha f\|_{\mathscr X^{s-m}(\Omega)}$. In particular $\|f\|_{\mathscr F_{\infty q}^s(\Omega)}\approx\sum_{|\alpha|\le m}\|\partial^\alpha f\|_{\mathscr F_{\infty q}^{s-m}(\Omega)}\approx\sup_{P}|P|^{-n/q}\|(2^{js}\phi_j\ast f)_{j=\log_2\ell(P)}^\infty\|_{\ell^q(L^q(\Omega\cap P))}$ for all $0<q<\infty$.