Some Intrinsic Characterizations of Besov-Triebel-Lizorkin-Morrey-type Spaces on Lipschitz Domains (2112.03996v5)

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 $0<q<\infty$.