Characterization of bi-parametric potentials and rate of convergence of truncated hypersingular integrals in the Dunkl setting
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915319610605568 |
|---|---|
| author | Verma, Sandeep Kumar P, Athulya |
| author_facet | Verma, Sandeep Kumar P, Athulya |
| contents | In this work, we introduce the $β$-semigroup for $β> 0$, which unifies and extends the classical Poisson (for $β=1$) and heat (for $β=2$) semigroups within the Dunkl analysis framework. Leveraging this semigroup, we derive an explicit representation for the inverse of the Dunkl-Riesz potential and characterize the image of the function space $L_k^p(\mathbb{R}^n)$ for $1 \leq p < \frac{n + 2γ}α$. We further define the bi-parametric potential of order $α$ by $$\mathfrak{S}_k^{(α,β)} = \left(I + (-Δ_k)^{β/2}\right)^{-α/β}$$ and establish its inverse along with a detailed description of the associated range space. Our approach employs a wavelet-based method that represents the inverse as the limit of truncated hypersingular integrals parameterized by $ε> 0$. To analyze the convergence of these approximations, we introduce the concept of $η$-smoothness at a point $x_0$ in the Dunkl setting. We show that if a function $f \in L_k^p(\mathbb{R}^n) \cap L_k^2(\mathbb{R}^n)$, for $1 \leq p \leq \infty$, possesses $η$-smoothness at $x_0$, then the truncated hypersingular approximations converge to $f(x_0)$ as $ε\to 0^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15748 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of bi-parametric potentials and rate of convergence of truncated hypersingular integrals in the Dunkl setting Verma, Sandeep Kumar P, Athulya Functional Analysis 41A35, 42A38, 44A35, 47G20 G.0 In this work, we introduce the $β$-semigroup for $β> 0$, which unifies and extends the classical Poisson (for $β=1$) and heat (for $β=2$) semigroups within the Dunkl analysis framework. Leveraging this semigroup, we derive an explicit representation for the inverse of the Dunkl-Riesz potential and characterize the image of the function space $L_k^p(\mathbb{R}^n)$ for $1 \leq p < \frac{n + 2γ}α$. We further define the bi-parametric potential of order $α$ by $$\mathfrak{S}_k^{(α,β)} = \left(I + (-Δ_k)^{β/2}\right)^{-α/β}$$ and establish its inverse along with a detailed description of the associated range space. Our approach employs a wavelet-based method that represents the inverse as the limit of truncated hypersingular integrals parameterized by $ε> 0$. To analyze the convergence of these approximations, we introduce the concept of $η$-smoothness at a point $x_0$ in the Dunkl setting. We show that if a function $f \in L_k^p(\mathbb{R}^n) \cap L_k^2(\mathbb{R}^n)$, for $1 \leq p \leq \infty$, possesses $η$-smoothness at $x_0$, then the truncated hypersingular approximations converge to $f(x_0)$ as $ε\to 0^+$. |
| title | Characterization of bi-parametric potentials and rate of convergence of truncated hypersingular integrals in the Dunkl setting |
| topic | Functional Analysis 41A35, 42A38, 44A35, 47G20 G.0 |
| url | https://arxiv.org/abs/2505.15748 |