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See detailSolvability of systems of invariant differential equations on H2 and beyond
Palmirotta, Guendalina UL; Olbrich, Martin UL

E-print/Working paper (2022)

We show how the Fourier transform for distributional sections of vector bundles over symmetric spaces of non-compact type G/K can be used for questions of solvability of systems of invariant differential ... [more ▼]

We show how the Fourier transform for distributional sections of vector bundles over symmetric spaces of non-compact type G/K can be used for questions of solvability of systems of invariant differential equations in analogy to Hörmander’s proof of the Ehrenpreis-Malgrange theorem. We get complete solvability for the hyperbolic plane H2 and partial results for products H^2 × · · · × H^2 and the hyperbolic 3-space H^3. [less ▲]

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See detailDelorme’s intertwining conditions for sections of homogeneous vector bundles on two and three dimensional hyperbolic spaces
Palmirotta, Guendalina UL; Olbrich, Martin UL

E-print/Working paper (2022)

The description of the Paley-Wiener space for compactly supported smooth functions C_c^∞(G) on a semi-simple Lie group G involves certain intertwining conditions that are difficult to handle. In the ... [more ▼]

The description of the Paley-Wiener space for compactly supported smooth functions C_c^∞(G) on a semi-simple Lie group G involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for G = SL(2, R)^d (d ∈ N) and G = SL(2, C). Our results are based on a defining criterion for the Paley-Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley-Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces. [less ▲]

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See detailA topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on G/K
Palmirotta, Guendalina UL; Olbrich, Martin UL

E-print/Working paper (2022)

We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type X = G/K. We prove a characterisation of their ... [more ▼]

We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type X = G/K. We prove a characterisation of their range. In fact, from Delorme’s Paley-Wiener theorem for compactly supported smooth functions on a real reductive group of Harish-Chandra class, we deduce topological Paley-Wiener and Paley-Wiener- Schwartz theorems for sections. [less ▲]

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See detailConference 'SL2R Days'
Palmirotta, Guendalina UL; Voglaire, Yannick; Olbrich, Martin UL

Report (2019)

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