Weighted weak-type (1, 1) inequalities for pseudo-differential operators with symbol in $S^{m}_{0,δ}$
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| Format: | Preprint |
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2025
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| _version_ | 1866913717628698624 |
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| author | Wang, Guangqing He, Suixin Zhang, Lihua |
| author_facet | Wang, Guangqing He, Suixin Zhang, Lihua |
| contents | Let $T_a$ be a pseudo-differential operator defined by exotic symbol $a$ in Hörmander class $S^m_{0,δ}$ with $m \in \mathbb{R} $ and $0 \leq δ\leq 1 $. It is well-known that the weak type (1,1) behavior of $T_a $ is not fully understood when the index $m $ is equal to the possibly optimal value $-\frac{n}{2} - \frac{n}{2} δ$ for $0 \leq δ< 1 $, and that $T_a $ is not of weak type (1,1) when $m = -n$ and $δ= 1 $.
In this note, we prove that $T_a $ is of weighted weak type (1,1) if $a \in S^{-n}_{0, δ}$ with $0 \leq δ< 1 $. Additionally, we show that the dual operator $T_a^* $ is of weighted weak type (1,1) if $a \in L^\infty S^{-n}_0 $. We also identify $m = -n$ as a critical index for these weak type estimates. As applications, we derive weighted weak type (1,1) estimates for certain classes of Fourier integral operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_10738 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted weak-type (1, 1) inequalities for pseudo-differential operators with symbol in $S^{m}_{0,δ}$ Wang, Guangqing He, Suixin Zhang, Lihua Analysis of PDEs Let $T_a$ be a pseudo-differential operator defined by exotic symbol $a$ in Hörmander class $S^m_{0,δ}$ with $m \in \mathbb{R} $ and $0 \leq δ\leq 1 $. It is well-known that the weak type (1,1) behavior of $T_a $ is not fully understood when the index $m $ is equal to the possibly optimal value $-\frac{n}{2} - \frac{n}{2} δ$ for $0 \leq δ< 1 $, and that $T_a $ is not of weak type (1,1) when $m = -n$ and $δ= 1 $. In this note, we prove that $T_a $ is of weighted weak type (1,1) if $a \in S^{-n}_{0, δ}$ with $0 \leq δ< 1 $. Additionally, we show that the dual operator $T_a^* $ is of weighted weak type (1,1) if $a \in L^\infty S^{-n}_0 $. We also identify $m = -n$ as a critical index for these weak type estimates. As applications, we derive weighted weak type (1,1) estimates for certain classes of Fourier integral operators. |
| title | Weighted weak-type (1, 1) inequalities for pseudo-differential operators with symbol in $S^{m}_{0,δ}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.10738 |