Cesàro-type operators on mixed norm spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915414170140672 |
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| author | Blasco, Óscar Mas, Alejandro |
| author_facet | Blasco, Óscar Mas, Alejandro |
| contents | Given a positive Borel measure $μ$ on $[0,1)$ and a parameter $β>0$, we consider the Cesàro-type operator $\mathcal C_{μ,β}$ acting on the analytic function $f(z)=\sum_{n=0}^\infty a_n z^n$ on the unit disc of the complex plane $\mathbb D$, defined by \[ \mathcal C_{μ,β}(f)(z)= \sum_{n=0}^\infty μ_n \left( \sum_{k=0}^n \frac{Γ(n-k+β)}{(n-k)! Γ(β)} a_k \right) z^n = \int_0^1 \frac{f(tz)}{(1-tz)^β} dμ(t), \] where $μ_n=\int_0^1 t^n dμ(t)$. This operator generalizes the classical Cesàro operator (corresponding to the case where $μ$ is the Lebesgue measure and $β=1$) and includes other relevant cases previously studied in the literature. In this paper we study the boundedness of $\mathcal C_{μ,β}$ on mixed norm spaces $H(p,q,γ)$ for $0<p,q\leq\infty$ and $γ>0$. Our results extend and unify several known characterizations for the boundedness of Cesàro-type operators acting on spaces of analytic functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_20586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cesàro-type operators on mixed norm spaces Blasco, Óscar Mas, Alejandro Complex Variables 47B38, 30H20 Given a positive Borel measure $μ$ on $[0,1)$ and a parameter $β>0$, we consider the Cesàro-type operator $\mathcal C_{μ,β}$ acting on the analytic function $f(z)=\sum_{n=0}^\infty a_n z^n$ on the unit disc of the complex plane $\mathbb D$, defined by \[ \mathcal C_{μ,β}(f)(z)= \sum_{n=0}^\infty μ_n \left( \sum_{k=0}^n \frac{Γ(n-k+β)}{(n-k)! Γ(β)} a_k \right) z^n = \int_0^1 \frac{f(tz)}{(1-tz)^β} dμ(t), \] where $μ_n=\int_0^1 t^n dμ(t)$. This operator generalizes the classical Cesàro operator (corresponding to the case where $μ$ is the Lebesgue measure and $β=1$) and includes other relevant cases previously studied in the literature. In this paper we study the boundedness of $\mathcal C_{μ,β}$ on mixed norm spaces $H(p,q,γ)$ for $0<p,q\leq\infty$ and $γ>0$. Our results extend and unify several known characterizations for the boundedness of Cesàro-type operators acting on spaces of analytic functions. |
| title | Cesàro-type operators on mixed norm spaces |
| topic | Complex Variables 47B38, 30H20 |
| url | https://arxiv.org/abs/2507.20586 |