Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Betancor, Jorge J. Dalmasso, Estefanía Quijano, Pablo |
| author_facet | Betancor, Jorge J. Dalmasso, Estefanía Quijano, Pablo |
| contents | In this paper we consider fractional Kolmogorov operators defined, in $\mathbb{R}^d$, by
\[Λ_κ=(-Δ)^{α/2}+\fracκ{|x|^α} x\cdot \nabla,\]
with $α\in (1,2)$, $α<(d+2)/2$ and $κ\in \mathbb{R}$. The operator $Λ_α$ generates a holomorphic semigroup $\{T_t^α\}_{t>0}$ in $L^2(\mathbb{R}^d)$ provided that $κ<κ_c$ where $κ_c$ is a critical coupling constant. We establish $L^p$-boundedness properties for the variation operators $V_ρ\left(\{t^\ell\partial_t^\ell T_t^α\}_{t>0}\right)$ with $ρ> 2$, $\ell\in \mathbb{N}$ and $1\vee \frac{d}β<p<\infty$, where $β$ depends on $κ$. We also study the behavior of these variation operators in the endpoint $L^{1\vee \frac{d}β}(\mathbb{R}^d)$ and we prove that $V_2(\{T_t^α\}_{t>0})$ is not bounded from $L^p(\mathbb{R}^d)$ to $L^{p,\infty}(\mathbb{R}^d)$ for any $1< p<\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14631 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators Betancor, Jorge J. Dalmasso, Estefanía Quijano, Pablo Analysis of PDEs 42B20, 42B25, 42B35, 47D03 In this paper we consider fractional Kolmogorov operators defined, in $\mathbb{R}^d$, by \[Λ_κ=(-Δ)^{α/2}+\fracκ{|x|^α} x\cdot \nabla,\] with $α\in (1,2)$, $α<(d+2)/2$ and $κ\in \mathbb{R}$. The operator $Λ_α$ generates a holomorphic semigroup $\{T_t^α\}_{t>0}$ in $L^2(\mathbb{R}^d)$ provided that $κ<κ_c$ where $κ_c$ is a critical coupling constant. We establish $L^p$-boundedness properties for the variation operators $V_ρ\left(\{t^\ell\partial_t^\ell T_t^α\}_{t>0}\right)$ with $ρ> 2$, $\ell\in \mathbb{N}$ and $1\vee \frac{d}β<p<\infty$, where $β$ depends on $κ$. We also study the behavior of these variation operators in the endpoint $L^{1\vee \frac{d}β}(\mathbb{R}^d)$ and we prove that $V_2(\{T_t^α\}_{t>0})$ is not bounded from $L^p(\mathbb{R}^d)$ to $L^{p,\infty}(\mathbb{R}^d)$ for any $1< p<\infty$. |
| title | Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators |
| topic | Analysis of PDEs 42B20, 42B25, 42B35, 47D03 |
| url | https://arxiv.org/abs/2506.14631 |