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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2509.07536 |
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| _version_ | 1866915486334189568 |
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| author | Moreno, Álvaro Miguel Peláez, José Ángel |
| author_facet | Moreno, Álvaro Miguel Peláez, José Ángel |
| contents | For $0<p,q<\infty$ and $ω$ a radial weight, the space $L^{p,q}_ω$ consists of complex-valued measurable functions $f$ on the unit disk such that
$$
\| f\|_{L^{p,q}_ω}^q = \int_0^1 \left (\frac{1}{2π}\int_0^{2π}|f(re^{iθ})|^pdθ\right )^{\frac{q}{p}}rω(r)\,dr,
$$
and the mixed norm space $A^{p,q}_ω$ is the subset of $L^{p,q}_ω$ consisting of analytic functions.
We say that a radial weight $ω$ belongs to $\widehat{\mathcal{D}}$ if there exists $C=C(ω)>0$ such that $$\int_r^1ω(s)ds \leq C \int_{\frac{1+r}{2}}^1ω(s)\,ds \,\, \text{for every}\,\, 0\leq r <1.$$
We describe the dual space of $A^{p,q}_ω$ for every $0<p,q<\infty$ and $ω\in\widehat{\mathcal{D}}$. Later on,
we apply the obtained description of the dual space of $A^{p,q}_ω$ to prove that the Bergman projection induced by $ω$, $P_ω$, is bounded on $L^{p,q}_ω$ for $1<p,q<\infty$ and
$ω\in \widehat{\mathcal{D}}$. Besides, we also prove that $P_ω$ and the corresponding maximal Bergman projection $P_ω^+$ are not simultaneously bounded on $L^{p,q}_ω$ for $1<p,q<\infty$ and
$ω\in \widehat{\mathcal{D}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07536 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Duality of mixed norm spaces induced by radial one-sided doubling weight Moreno, Álvaro Miguel Peláez, José Ángel Complex Variables Functional Analysis For $0<p,q<\infty$ and $ω$ a radial weight, the space $L^{p,q}_ω$ consists of complex-valued measurable functions $f$ on the unit disk such that $$ \| f\|_{L^{p,q}_ω}^q = \int_0^1 \left (\frac{1}{2π}\int_0^{2π}|f(re^{iθ})|^pdθ\right )^{\frac{q}{p}}rω(r)\,dr, $$ and the mixed norm space $A^{p,q}_ω$ is the subset of $L^{p,q}_ω$ consisting of analytic functions. We say that a radial weight $ω$ belongs to $\widehat{\mathcal{D}}$ if there exists $C=C(ω)>0$ such that $$\int_r^1ω(s)ds \leq C \int_{\frac{1+r}{2}}^1ω(s)\,ds \,\, \text{for every}\,\, 0\leq r <1.$$ We describe the dual space of $A^{p,q}_ω$ for every $0<p,q<\infty$ and $ω\in\widehat{\mathcal{D}}$. Later on, we apply the obtained description of the dual space of $A^{p,q}_ω$ to prove that the Bergman projection induced by $ω$, $P_ω$, is bounded on $L^{p,q}_ω$ for $1<p,q<\infty$ and $ω\in \widehat{\mathcal{D}}$. Besides, we also prove that $P_ω$ and the corresponding maximal Bergman projection $P_ω^+$ are not simultaneously bounded on $L^{p,q}_ω$ for $1<p,q<\infty$ and $ω\in \widehat{\mathcal{D}}$. |
| title | Duality of mixed norm spaces induced by radial one-sided doubling weight |
| topic | Complex Variables Functional Analysis |
| url | https://arxiv.org/abs/2509.07536 |