Generalized Helgason-Fourier transforms associated to variants of the Laplace-Beltrami operators on the unit ball in $\mathbb{R}^{n}$

Liu, Congwen; Peng, Lizhong

doi:10.1512/iumj.2009.58.3588

Cited by 27 publications

(19 citation statements)

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“…We begin by quoting some preliminary facts on Fourier analysis on hyperbolic spaces which will be needed in the sequel and refer to [1,2,3,13,15,27,28,30,31,32,39,49,51,54] for more information about this subject.…”

Section: Notations and Preliminariesmentioning

confidence: 99%

Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz'ya inequalities on half spaces

Lu¹,

Yang²

2019

American Journal of Mathematics

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Using the Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the n−1 2 -th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n coincides with the best n−1 2 -th order Sobolev constant when n is odd and n ≥ 9 (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the k−th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of dimension n and k-th order derivatives (see Theorem 1.7). Precise expressions and optimal bounds for Green's functions of the operator −∆ H − (n−1) 2 4 on the hyperbolic space B n and operators of the product form are given, where (n−1) 2 4 is the spectral gap for the Laplacian −∆ H on B n . Finally, we give the precise expression and optimal pointwise bound of Green's function of the Paneitz and GJMS operators on hyperbolic space, which are of their independent interest (see Theorem 1.10).

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Section: Notations and Preliminariesmentioning

confidence: 99%

Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz'ya inequalities on half spaces

Lu¹,

Yang²

2019

American Journal of Mathematics

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“…We begin by quoting some preliminary facts which will be needed in the sequel and refer to [2,9,12,13,15,22] for more information about this subject.…”

Section: Preliminaries On Fourier Transform and Fractional Laplaciansmentioning

confidence: 99%

“…Also using the Möbius transformations, we can define the convolution of measurable functions f and g on B n by (see e.g. [22])…”

Section: Preliminaries On Fourier Transform and Fractional Laplaciansmentioning

confidence: 99%

Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four

Yang

2017

Advances in Mathematics

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We establish sharp Hardy-Adams inequalities on hyperbolic space B 4 of dimension four. Namely, we will show that for any α > 0 there exists a constant C α > 0 such thatAs applications, we obtain a sharpened Adams inequality on hyperbolic space B 4 and an inequality which improves the classical Adams' inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two given by Wang and Ye [37] and on any convex planar domain by the authors [26]. The tools of fractional Laplacian, Fourier transform and the Plancherel formula on hyperbolic and symmetric spaces play an important role in our work.2000 Mathematics Subject Classification. Primary 35J20; 46E35.

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“…For a function

f \in C_{0}^{\infty} (B^{m})

, it is defined by

\hat{f} (λ, ζ) = \int_{B^{m}} f (x) e_{- λ, ζ} (x) d μ_{v} (x),

with

e_{λ, ζ} (x) = \frac{{(1 - | x |^{2})}^{(1 - v + i λ) / 2}}{| x - ζ |^{m - 1 + i λ}}, x \in B^{m}

for

λ \in double-struckC

and ζ ∈ S m − 1 . For more details, we refer to .…”

Section: Preliminariesmentioning

confidence: 99%

Clifford–Fourier transform on hyperbolic space

Lian

Bao

Bie

et al. 2016

Math Methods in App Sciences

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In this paper, we introduce a new generalization of the Helgason–Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role. Copyright © 2016 John Wiley & Sons, Ltd.

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Generalized Helgason-Fourier transforms associated to variants of the Laplace-Beltrami operators on the unit ball in $\mathbb{R}^{n}$

Cited by 27 publications

References 0 publications

Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz'ya inequalities on half spaces

Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz'ya inequalities on half spaces

Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four

Clifford–Fourier transform on hyperbolic space

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