The convolution equation of Choquet and Deny on nilpotent groups

Chu, Cho-Ho; Hilberdink, Titus

doi:10.1007/bf01229501

Cited by 14 publications

(10 citation statements)

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“…The above proof is analogous to the one given in [9] for groups which is an extension of Choquet and Deny's abelian arguments in [6]. Similar method has also been used in [25] to prove a Liouville theorem for Gelfand pairs which is a special case of the following theorem and can be applied to obtain Furstenberg's characterisation in [14] of harmonic functions of the Laplacian on symmetric spaces.…”

Section: Liouville Theoremmentioning

confidence: 78%

“…Throughout, C ru (G) denotes the Banach space of all bounded right uniformly continuous complex functions on G, where a bounded continuous function f on G is called right uniformly continuous if the map R f : y ∈ G → f * δ y ∈ C b (G) is continuous where C b (G) denotes the Banach space of bounded complex continuous functions on G. To avoid confusion, we remark that such a function on a group is called left uniformly continuous in [8,9,11].…”

Section: Nilpotent Hypergroupsmentioning

confidence: 98%

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Harmonic functions on hypergroups

Amini

Chu

2011

Journal of Functional Analysis

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We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions.

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Section: Liouville Theoremmentioning

confidence: 78%

Section: Nilpotent Hypergroupsmentioning

confidence: 98%

Harmonic functions on hypergroups

Amini

Chu

2011

Journal of Functional Analysis

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“…If G is non-compact, then there is no non-zero constant function in L p (G) for p < ∞, in other words, the only constant function in H p σ (G) is the zero function. Hence a uniqueness result will be expressed as H p σ (G) ⊂ C1 for arbitrary groups G. In [6,8], various Liouville-type theorems were proved for right uniformly continuous bounded σ-harmonic functions where harmonicity was defined in terms of the convolution equation f = σ * f . We first improve these results by removing the uniform continuity condition.…”

Section: Uniqueness Resultsmentioning

confidence: 99%

Harmonic Function Spaces on Groups

Chu¹

2004

J. London Math. Soc.

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“…For this, we introduce a useful device, namely, the determinant of a matrix-valued measure which enables us to reduce some arguments to the scalar case. Naturally several directions can be followed, an obvious next step is to examine other classes of groups and to extend, for instance, the results in [7], [9], [11] to the matrix-valued case. Finally in Section 6, we extend Choquet and Deny's method in [4], [14] to show that the (unbounded) positive matrixvalued s-harmonic functions on abelian groups, with range commuting with that of s, are integrals of matrix-valued exponential functions.…”

Section: Introductionmentioning

confidence: 99%

Matrix-valued harmonic functions on groups

Chu¹

2002

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We study the basic structures of matrix-valued harmonic functions on locally compact groups. We show that the bounded matrix-valued harmonic functions on a group form a Jordan triple system and we determine its structure. We also show that Liouville property implies amenability of the group. We characterize the unbounded matrix-valued harmonic functions on abelian groups.which motivates the following definition. Given a probability measure s on a locally compact group G, a Borel function f :Harmonic functions on groups play an important role in many areas of mathematics. Recently, matrix-valued harmonic functions on groups have been studied in [8], [10], [33], with some applications to problems concerning the L p -dimension of vector-valued selfsimilar measures. As in the scalar case, the matrix-valued harmonic functions on groups arise naturally in the following way. Brought to you by | University of Connecticut Authenticated Download Date | 5/29/15 2:29 AM f A L 2 w ðG; M n Þ 7 ! H t f ðxÞ A M n is bounded and H t f is continuous, then there exists C t; x A L 2 w ðG; M n Þ (cf. [20], Proposition 4.4) such that H t f ðxÞ ¼ Ð G f ðyÞw 2 ðyÞC t; x ðyÞ dlðyÞ:Since L u L ¼ LL u , we have ðL u H t Þ f ðxÞ ¼ H t f ðu À1 xÞ which gives, for l-almost all y A G, w 2 ðyÞC t; u À1 x ðyÞ ¼ w 2 ðuyÞC t; x ðuyÞ ðu A GÞ:Then h t ðx À1 yÞ ¼ w 2 ðx À1 yÞC t; e ðx À1 yÞ ¼ w 2 ðyÞC t; x ðyÞ. Chu, Matrix-valued harmonic functions on groups Brought to you by | University of Connecticut Authenticated Download Date | 5/29/15 2:29 AM

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The convolution equation of Choquet and Deny on nilpotent groups

Cited by 14 publications

References 13 publications

Harmonic functions on hypergroups

Harmonic functions on hypergroups

Harmonic Function Spaces on Groups

Matrix-valued harmonic functions on groups

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