Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929471329665024 |
|---|---|
| author | Cruz-Uribe, David Roberts, Troy |
| author_facet | Cruz-Uribe, David Roberts, Troy |
| contents | In this paper we prove necessary conditions for the boundedness of fractional operators on the variable Lebesgue spaces. More precisely, we find necessary conditions on an exponent function $\pp$ for a fractional maximal operator $M_α$ or a non-degenerate fractional singular integral operator $T_α$, $0 \leq α< n$, to satisfy weak $(\pp,\qq)$ inequalities or strong $(\pp,\qq)$ inequalities, with $\qq$ being defined pointwise almost everywhere by
%
\[ \frac{1}{p(x)} - \frac{1}{q(x)} = \fracα{n}. \]
%
We first prove preliminary results linking fractional averaging operators and the $K_0^α$ condition, a qualitative condition on $\pp$ related to the norms of characteristic functions of cubes, and show some useful implications of the $K_0^α$ condition. We then show that if $M_α$ satisfies weak $(\pp,\qq)$ inequalities, then $\pp \in K_0^α(\R^n)$. We use this to prove that if $M_α$ satisfies strong $(\pp,\qq)$ inequalities, then $p_->1$. Finally, we prove a powerful pointwise estimate for $T_α$ that relates $T_α$ to $M_α$ along a carefully chosen family of cubes. This allows us to prove necessary conditions for fractional singular integral operators similar to those for fractional maximal operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces Cruz-Uribe, David Roberts, Troy Classical Analysis and ODEs Primary: 42B20, 42B25, 42B35, Secondary: 46A80, 46E30} In this paper we prove necessary conditions for the boundedness of fractional operators on the variable Lebesgue spaces. More precisely, we find necessary conditions on an exponent function $\pp$ for a fractional maximal operator $M_α$ or a non-degenerate fractional singular integral operator $T_α$, $0 \leq α< n$, to satisfy weak $(\pp,\qq)$ inequalities or strong $(\pp,\qq)$ inequalities, with $\qq$ being defined pointwise almost everywhere by % \[ \frac{1}{p(x)} - \frac{1}{q(x)} = \fracα{n}. \] % We first prove preliminary results linking fractional averaging operators and the $K_0^α$ condition, a qualitative condition on $\pp$ related to the norms of characteristic functions of cubes, and show some useful implications of the $K_0^α$ condition. We then show that if $M_α$ satisfies weak $(\pp,\qq)$ inequalities, then $\pp \in K_0^α(\R^n)$. We use this to prove that if $M_α$ satisfies strong $(\pp,\qq)$ inequalities, then $p_->1$. Finally, we prove a powerful pointwise estimate for $T_α$ that relates $T_α$ to $M_α$ along a carefully chosen family of cubes. This allows us to prove necessary conditions for fractional singular integral operators similar to those for fractional maximal operators. |
| title | Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces |
| topic | Classical Analysis and ODEs Primary: 42B20, 42B25, 42B35, Secondary: 46A80, 46E30} |
| url | https://arxiv.org/abs/2408.12745 |