Homogeneous fractional integral operators on weighted Lebesgue, Morrey and Campanato spaces
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| Formato: | Preprint |
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2025
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| _version_ | 1866916982283042816 |
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| author | Du, Jingliang Wang, Hua |
| author_facet | Du, Jingliang Wang, Hua |
| contents | Let $0<α<n$ and $T_{Ω,α}$ be the homogeneous fractional integral operator which is defined by \begin{equation*} T_{Ω,α}f(x):=\int_{\mathbb R^n}\frac{Ω(x-y)}{|x-y|^{n-α}}f(y)\,dy, \end{equation*} where $Ω$ is homogeneous of degree zero in $\mathbb R^n$ for $n\geq2$, and is integrable on the unit sphere $\mathbb{S}^{n-1}$. In this paper we study boundedness properties of the homogeneous fractional integral operator $T_{Ω,α}$ acting on weighted Lebesgue and Morrey spaces. Under certain Dini-type smoothness condition on $Ω$, we prove that $T_{Ω,α}$ is bounded from $L^{p}(ω^p)$ to $\mathcal{C}^{γ,\ell}_ω$(a class of Campanato spaces) for appropriate indices, when $n/α<p<\infty$. Moreover, we prove that if $Ω$ satisfies certain Dini-type smoothness condition on $\mathbb{S}^{n-1}$, then $T_{Ω,α}$ is bounded from $\mathcal{M}^{p,κ}(ω^p,ω^q)$ to $\mathcal{C}^{γ,\ell}(ω^q)$(weighted Campanato spaces) for appropriate indices, when $p/q<κ<1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00426 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homogeneous fractional integral operators on weighted Lebesgue, Morrey and Campanato spaces Du, Jingliang Wang, Hua Classical Analysis and ODEs 42B20, 42B25, 42B35 Let $0<α<n$ and $T_{Ω,α}$ be the homogeneous fractional integral operator which is defined by \begin{equation*} T_{Ω,α}f(x):=\int_{\mathbb R^n}\frac{Ω(x-y)}{|x-y|^{n-α}}f(y)\,dy, \end{equation*} where $Ω$ is homogeneous of degree zero in $\mathbb R^n$ for $n\geq2$, and is integrable on the unit sphere $\mathbb{S}^{n-1}$. In this paper we study boundedness properties of the homogeneous fractional integral operator $T_{Ω,α}$ acting on weighted Lebesgue and Morrey spaces. Under certain Dini-type smoothness condition on $Ω$, we prove that $T_{Ω,α}$ is bounded from $L^{p}(ω^p)$ to $\mathcal{C}^{γ,\ell}_ω$(a class of Campanato spaces) for appropriate indices, when $n/α<p<\infty$. Moreover, we prove that if $Ω$ satisfies certain Dini-type smoothness condition on $\mathbb{S}^{n-1}$, then $T_{Ω,α}$ is bounded from $\mathcal{M}^{p,κ}(ω^p,ω^q)$ to $\mathcal{C}^{γ,\ell}(ω^q)$(weighted Campanato spaces) for appropriate indices, when $p/q<κ<1$. |
| title | Homogeneous fractional integral operators on weighted Lebesgue, Morrey and Campanato spaces |
| topic | Classical Analysis and ODEs 42B20, 42B25, 42B35 |
| url | https://arxiv.org/abs/2510.00426 |