Weighted bounds for a class of singular integral operators in variable exponent Herz-Morrey spaces
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909311415877632 |
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| author | Yang, Yanqi Wu, Qi |
| author_facet | Yang, Yanqi Wu, Qi |
| contents | Let T be the singular integral operator with variable kernel defined by $Tf(x)= p.v. \int_{\mathbb{R}^{n}}K(x,x-y)f(y)\mathrm{d}y$ and $D^γ(0\leqγ\leq1)$ be the fractional differentiation operator, where $K(x,z)=\frac{Ω(x,z')}{|z|^{n}}$, $z'=\frac{z}{|z|},~~z\neq0$. Let $~T^{\ast}~$and $~T^\sharp~$ be the adjoint of $T$ and the pseudo-adjoint of $T$, respectively. In this paper, via the expansion of spherical harmonics and the estimates of the convolution operators $T_{m,j}$, we shall prove some boundedness results for $TD^γ-D^γT$ and $(T^{\ast}-T^{\sharp})D^γ$ under natural regularity assumptions on the exponent function on a class of generalized Herz-Morrey spaces with weight and variable exponent, which extend some known results. Moreover, various norm characterizations for the product $T_{1}T_{2}$ and the pseudo-product $T_{1}\circ T_{2}$ are also established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07152 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted bounds for a class of singular integral operators in variable exponent Herz-Morrey spaces Yang, Yanqi Wu, Qi Classical Analysis and ODEs Let T be the singular integral operator with variable kernel defined by $Tf(x)= p.v. \int_{\mathbb{R}^{n}}K(x,x-y)f(y)\mathrm{d}y$ and $D^γ(0\leqγ\leq1)$ be the fractional differentiation operator, where $K(x,z)=\frac{Ω(x,z')}{|z|^{n}}$, $z'=\frac{z}{|z|},~~z\neq0$. Let $~T^{\ast}~$and $~T^\sharp~$ be the adjoint of $T$ and the pseudo-adjoint of $T$, respectively. In this paper, via the expansion of spherical harmonics and the estimates of the convolution operators $T_{m,j}$, we shall prove some boundedness results for $TD^γ-D^γT$ and $(T^{\ast}-T^{\sharp})D^γ$ under natural regularity assumptions on the exponent function on a class of generalized Herz-Morrey spaces with weight and variable exponent, which extend some known results. Moreover, various norm characterizations for the product $T_{1}T_{2}$ and the pseudo-product $T_{1}\circ T_{2}$ are also established. |
| title | Weighted bounds for a class of singular integral operators in variable exponent Herz-Morrey spaces |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2409.07152 |