Absolute continuity of degenerate elliptic measure
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909323353915392 |
|---|---|
| author | Cao, Mingming Yabuta, Kôzô |
| author_facet | Cao, Mingming Yabuta, Kôzô |
| contents | Let $Ω\subset \mathbb{R}^{n+1}$ be an open set whose boundary may be composed of pieces of different dimensions. Assume that $Ω$ satisfies the quantitative openness and connectedness, and there exist doubling measures $m$ on $Ω$ and $μ$ on $\partial Ω$ with appropriate size conditions. Let $Lu=-\mathrm{div}(A\nabla u)$ be a real (not necessarily symmetric) degenerate elliptic operator in $Ω$. Write $ω_L$ for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) $ω_L \in A_{\infty}(μ)$, (ii) the Dirichlet problem for $L$ is solvable in $L^p(μ)$ for some $p \in (1, \infty)$, (iii) every bounded null solution of $L$ satisfies Carleson measure estimates with respect to $μ$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(μ)$ for all $q \in (0, \infty)$ for any null solution of $L$, and (v) the Dirichlet problem for $L$ is solvable in $\mathrm{BMO}(μ)$. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of $ω_L$ with respect to $μ$ in terms of local $L^2(μ)$ estimates of the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $μ$-almost everywhere of the truncated conical square function for any bounded null solution of $L$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_04860 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Absolute continuity of degenerate elliptic measure Cao, Mingming Yabuta, Kôzô Analysis of PDEs Classical Analysis and ODEs 35J25, 42B37, 31B05, 35J08 Let $Ω\subset \mathbb{R}^{n+1}$ be an open set whose boundary may be composed of pieces of different dimensions. Assume that $Ω$ satisfies the quantitative openness and connectedness, and there exist doubling measures $m$ on $Ω$ and $μ$ on $\partial Ω$ with appropriate size conditions. Let $Lu=-\mathrm{div}(A\nabla u)$ be a real (not necessarily symmetric) degenerate elliptic operator in $Ω$. Write $ω_L$ for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) $ω_L \in A_{\infty}(μ)$, (ii) the Dirichlet problem for $L$ is solvable in $L^p(μ)$ for some $p \in (1, \infty)$, (iii) every bounded null solution of $L$ satisfies Carleson measure estimates with respect to $μ$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(μ)$ for all $q \in (0, \infty)$ for any null solution of $L$, and (v) the Dirichlet problem for $L$ is solvable in $\mathrm{BMO}(μ)$. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of $ω_L$ with respect to $μ$ in terms of local $L^2(μ)$ estimates of the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $μ$-almost everywhere of the truncated conical square function for any bounded null solution of $L$. |
| title | Absolute continuity of degenerate elliptic measure |
| topic | Analysis of PDEs Classical Analysis and ODEs 35J25, 42B37, 31B05, 35J08 |
| url | https://arxiv.org/abs/2109.04860 |