The Laplace and Leray transforms on some (weakly) convex domains in $\mathbb{C}^2$
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| Format: | Preprint |
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2024
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| _version_ | 1866918043449294848 |
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| author | Chatterjee, Agniva |
| author_facet | Chatterjee, Agniva |
| contents | The space of Laplace transforms of holomorphic Hardy-space functions have been characterized as weighted Bergman spaces of entire functions in two cases: that of planar convex domains (Lutsenko--Yumulmukhametov, 1991), and that of strongly convex domains in higher dimensions (Lindholm, 2002). In this paper, we establish such a Paley--Weiner result for a class of (weakly) convex Reinhardt domains in $\mathbb{C}^2$ that are well-modelled by the so-called egg domains. We consider Hardy spaces on these domains with respect to a canonical choice of boundary Monge--Ampere measure. This class of domains was introduced by Barrett--Lanzani (2009) to study the $L^2$-boundedness of the Leray transform in the absence of either strongly convexity or $\mathcal{C}^2$-regularity. The boundedness of the Leray transform plays a crucial role in understanding the image of the Laplace transform. As a supplementary result, we expand the known class of convex Reinhardt domains for which the Leray transform is $L^2$-bounded (with respect to the aforementioned choice of boundary measure). Finally, we also produce an example to show that the Lutsenko--Yumulmukhametov result cannot be expected to generalize to all convex domains in higher dimensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12753 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Laplace and Leray transforms on some (weakly) convex domains in $\mathbb{C}^2$ Chatterjee, Agniva Complex Variables Functional Analysis 32A26, 44A10, 30H10, 32A36 The space of Laplace transforms of holomorphic Hardy-space functions have been characterized as weighted Bergman spaces of entire functions in two cases: that of planar convex domains (Lutsenko--Yumulmukhametov, 1991), and that of strongly convex domains in higher dimensions (Lindholm, 2002). In this paper, we establish such a Paley--Weiner result for a class of (weakly) convex Reinhardt domains in $\mathbb{C}^2$ that are well-modelled by the so-called egg domains. We consider Hardy spaces on these domains with respect to a canonical choice of boundary Monge--Ampere measure. This class of domains was introduced by Barrett--Lanzani (2009) to study the $L^2$-boundedness of the Leray transform in the absence of either strongly convexity or $\mathcal{C}^2$-regularity. The boundedness of the Leray transform plays a crucial role in understanding the image of the Laplace transform. As a supplementary result, we expand the known class of convex Reinhardt domains for which the Leray transform is $L^2$-bounded (with respect to the aforementioned choice of boundary measure). Finally, we also produce an example to show that the Lutsenko--Yumulmukhametov result cannot be expected to generalize to all convex domains in higher dimensions. |
| title | The Laplace and Leray transforms on some (weakly) convex domains in $\mathbb{C}^2$ |
| topic | Complex Variables Functional Analysis 32A26, 44A10, 30H10, 32A36 |
| url | https://arxiv.org/abs/2405.12753 |