Lp-Fourier transforms on nilpotent Lie groups and solvable Lie groups acting on Siegel domains

Inoue, Junya

doi:10.2140/pjm.1992.155.295

Cited by 12 publications

(9 citation statements)

References 14 publications

(18 reference statements)

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“…1. It is important to note that since the best constant for Young's inequality on locally compact topological groups is always less than 1 [41], Corollary 2 does not offer any improvement to the theory, however it is nonetheless included in this paper for the sake of context; we refer the reader to [23,35,39,41] for further details on Young's inequality in abstract settings. We remarked earlier on that Theorem 2 is an example of an affine-invariant inequality, in the sense that the left-hand side is invariant under the natural action A :…”

Section: Resultsmentioning

confidence: 99%

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An Algebraic Brascamp–Lieb Inequality

Duncan

2021

J Geom Anal

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The Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions $$f_j\in L^{q_j}(\mathbb {R}^{n_j})$$ f j ∈ L q j ( R n j ) , for $$j=1,\ldots ,m$$ j = 1 , … , m , under some corresponding linear maps $$B_j$$ B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where $$B_j$$ B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

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Section: Resultsmentioning

confidence: 99%

“…BL(L, p) det(L m+1 L * m+1 ) − 1Proof By duality, (6) is equivalent to the boundG φ(x) ˚m j=1 f j j−1 l=1 1 p l (x)dμ(x) deg(G) φ L r (G) m j=1 f j L p j(G) (35). For 1 ≤ j ≤ m, define the nonlinear maps B j : G m → G, B j (x 1 , .…”

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confidence: 99%

An Algebraic Brascamp–Lieb Inequality

Duncan

2021

J Geom Anal

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“…We give a proof of this corollary in section 5. It is important to note that since the best constant for Young's inequality on locally compact topological groups is always less than one [40], Corollary 1.7 does not offer any improvement to the theory, however it is nonetheless included in this paper for the sake of context; we refer the reader to [23,34,38,40] for further details on Young's inequality in abstract settings. We remarked earlier on that Theorem 1.5 is an example of an affine-invariant inequality, in the sense that the left-hand side is invariant under the natural action A : B j Þ Ñ B j ˝A of GL n pRq on the class of quasialgebraic data, however this inequality in fact exhibits a more general diffeomorphism-invariance property, as described by the following proposition.…”

Section: Resultsmentioning

confidence: 99%

An Algebraic Brascamp-Lieb Inequality

Duncan¹

2020

Preprint

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“…Furthermore, Fournier [4] showed that the norm kF p .G/k D 1 if and only if G has a compact open subgroup. Now, suppose G D exp g is a connected and simply connected nilpotent Lie group with Lie algebra g. Some estimates have been achieved for various examples and cases in this class by earlier works of Russo [9], [10], and [11], Klein and Russo [8], Inoue [6], Baklouti, Smaoui and Ludwig [1]. Let us recall that the unitary dual b G of a connected and simply connected nilpotent Lie group G is parameterized by the coadjoint orbits of G on the dual space g of g by the orbit method.…”

Section: Introductionmentioning

confidence: 99%