Some Intrinsic Characterizations of Besov-Triebel-Lizorkin-Morrey-type Spaces on Lipschitz Domains
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866910439003127808 |
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| author | Yao, Liding |
| author_facet | Yao, Liding |
| contents | We give Littlewood-Paley type characterizations for Besov-Triebel-Lizorkin-type spaces $\mathscr B_{pq}^{sτ},\mathscr F_{pq}^{sτ}$ and Besov-Morrey spaces $\mathcal N_{uqp}^s$ on a special Lipschitz domain $Ω\subset\mathbb R^n$: for a suitable sequence of Schwartz functions $(ϕ_j)_{j=0}^\infty$,
$\|f\|_{\mathscr B_{pq}^{sτ}(Ω)}\approx\sup_{P\text{ dyadic cubes}}|P|^{-τ}\|(2^{js}ϕ_j\ast f)_{j=\log_2\ell(P)}^\infty\|_{\ell^q(L^p(Ω\cap P))};$
$\|f\|_{\mathscr F_{pq}^{sτ}(Ω)}\approx\sup_{P\text{ dyadic cubes}}|P|^{-τ}\|(2^{js}ϕ_j\ast f)_{j=\log_2\ell(P)}^\infty\|_{L^p(Ω\cap P;\ell^q)};$
$\|f\|_{\mathcal N_{uqp}^{s}(Ω)}\approx\big\|\big(\sup_{P\text{ dyadic cubes}}|P|^{\frac1u-\frac1p}\cdot 2^{js}\|ϕ_j\ast f\|_{L^p(Ω\cap P)}\big)_{j=0}^\infty\big\|_{\ell^q}.$
We also show that $\|f\|_{\mathscr B_{pq}^{sτ}(Ω)}$, $\|f\|_{\mathscr F_{pq}^{sτ}(Ω)}$ and $\|f\|_{\mathcal N_{uqp}^{s}(Ω)}$ have equivalent (quasi-)norms via derivatives: for $\mathscr X^\bullet\in\{\mathscr B_{pq}^{\bullet,τ},\mathscr F_{pq}^{\bullet,τ},\mathcal N_{uqp}^\bullet\}$, we have $\|f\|_{\mathscr X^s(Ω)}\approx\sum_{|α|\le m}\|\partial^αf\|_{\mathscr X^{s-m}(Ω)}$.
In particular $\|f\|_{\mathscr F_{\infty q}^s(Ω)}\approx\sum_{|α|\le m}\|\partial^αf\|_{\mathscr F_{\infty q}^{s-m}(Ω)}\approx\sup_{P}|P|^{-n/q}\|(2^{js}ϕ_j\ast f)_{j=\log_2\ell(P)}^\infty\|_{\ell^q(L^q(Ω\cap P))}$ for all $0<q<\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_03996 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Some Intrinsic Characterizations of Besov-Triebel-Lizorkin-Morrey-type Spaces on Lipschitz Domains Yao, Liding Functional Analysis Classical Analysis and ODEs 46E35 (primary), 42B35 and 42B25 (secondary) We give Littlewood-Paley type characterizations for Besov-Triebel-Lizorkin-type spaces $\mathscr B_{pq}^{sτ},\mathscr F_{pq}^{sτ}$ and Besov-Morrey spaces $\mathcal N_{uqp}^s$ on a special Lipschitz domain $Ω\subset\mathbb R^n$: for a suitable sequence of Schwartz functions $(ϕ_j)_{j=0}^\infty$, $\|f\|_{\mathscr B_{pq}^{sτ}(Ω)}\approx\sup_{P\text{ dyadic cubes}}|P|^{-τ}\|(2^{js}ϕ_j\ast f)_{j=\log_2\ell(P)}^\infty\|_{\ell^q(L^p(Ω\cap P))};$ $\|f\|_{\mathscr F_{pq}^{sτ}(Ω)}\approx\sup_{P\text{ dyadic cubes}}|P|^{-τ}\|(2^{js}ϕ_j\ast f)_{j=\log_2\ell(P)}^\infty\|_{L^p(Ω\cap P;\ell^q)};$ $\|f\|_{\mathcal N_{uqp}^{s}(Ω)}\approx\big\|\big(\sup_{P\text{ dyadic cubes}}|P|^{\frac1u-\frac1p}\cdot 2^{js}\|ϕ_j\ast f\|_{L^p(Ω\cap P)}\big)_{j=0}^\infty\big\|_{\ell^q}.$ We also show that $\|f\|_{\mathscr B_{pq}^{sτ}(Ω)}$, $\|f\|_{\mathscr F_{pq}^{sτ}(Ω)}$ and $\|f\|_{\mathcal N_{uqp}^{s}(Ω)}$ have equivalent (quasi-)norms via derivatives: for $\mathscr X^\bullet\in\{\mathscr B_{pq}^{\bullet,τ},\mathscr F_{pq}^{\bullet,τ},\mathcal N_{uqp}^\bullet\}$, we have $\|f\|_{\mathscr X^s(Ω)}\approx\sum_{|α|\le m}\|\partial^αf\|_{\mathscr X^{s-m}(Ω)}$. In particular $\|f\|_{\mathscr F_{\infty q}^s(Ω)}\approx\sum_{|α|\le m}\|\partial^αf\|_{\mathscr F_{\infty q}^{s-m}(Ω)}\approx\sup_{P}|P|^{-n/q}\|(2^{js}ϕ_j\ast f)_{j=\log_2\ell(P)}^\infty\|_{\ell^q(L^q(Ω\cap P))}$ for all $0<q<\infty$. |
| title | Some Intrinsic Characterizations of Besov-Triebel-Lizorkin-Morrey-type Spaces on Lipschitz Domains |
| topic | Functional Analysis Classical Analysis and ODEs 46E35 (primary), 42B35 and 42B25 (secondary) |
| url | https://arxiv.org/abs/2112.03996 |