A weak type $(p,a)$ criterion for operators, and applications
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| Format: | Preprint |
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2025
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| author | Haak, Bernhard H. Ouhabaz, El-Maati |
| author_facet | Haak, Bernhard H. Ouhabaz, El-Maati |
| contents | Let $(X, d, μ)$ be a space of homogeneous type and $Ω$ an open subset of $X$. Given a bounded operator $T: L^p(Ω) \to L^q(Ω)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for $p_0$ and $a$ such that $\frac{1}{p_0} - \frac{1}{a} = \frac{1}{p}-\frac{1}{q}$. These results are illustrated by several applications including estimates of weak type $(p_0, a)$ for Riesz potentials $L^{-\fracα{2}}$ or for Riesz transform type operators $\nabla Δ^{-\fracα{2}}$ as well as $L^p-L^q$ boundedness of spectral multipliers $F(L)$ when the heat kernel of $L$ satisfies a Gaussian upper bound or an off-diagonal bound. We also prove boundedness of these operators from the Hardy space $H^1_L$ associated with $L$ into $L^a(X)$. By duality this gives boundedness from $L^{a'}(X)$ into $\text{BMO}_L$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08334 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A weak type $(p,a)$ criterion for operators, and applications Haak, Bernhard H. Ouhabaz, El-Maati Functional Analysis Analysis of PDEs Classical Analysis and ODEs 42B20, 47G10, 42B30, 47A60 Let $(X, d, μ)$ be a space of homogeneous type and $Ω$ an open subset of $X$. Given a bounded operator $T: L^p(Ω) \to L^q(Ω)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for $p_0$ and $a$ such that $\frac{1}{p_0} - \frac{1}{a} = \frac{1}{p}-\frac{1}{q}$. These results are illustrated by several applications including estimates of weak type $(p_0, a)$ for Riesz potentials $L^{-\fracα{2}}$ or for Riesz transform type operators $\nabla Δ^{-\fracα{2}}$ as well as $L^p-L^q$ boundedness of spectral multipliers $F(L)$ when the heat kernel of $L$ satisfies a Gaussian upper bound or an off-diagonal bound. We also prove boundedness of these operators from the Hardy space $H^1_L$ associated with $L$ into $L^a(X)$. By duality this gives boundedness from $L^{a'}(X)$ into $\text{BMO}_L$. |
| title | A weak type $(p,a)$ criterion for operators, and applications |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs 42B20, 47G10, 42B30, 47A60 |
| url | https://arxiv.org/abs/2509.08334 |