Pompeiu's problem on symmetric spaces

Berenstein, Carlos A.; Zalcman, Lawrence

doi:10.1007/bf02566709

Cited by 102 publications

(58 citation statements)

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“…We note that if R > π (i.e., B R = S n ), then, in general, statements 3a) and 3b) are false (see [7,Theorem 4] and also §1 of the present paper).…”

Section: §8 Uniqueness Theoremsmentioning

confidence: 80%

“…This fact follows easily from the mean-value theorem for eigenfunctions of the LaplaceBeltrami operator on X (see [1, Chapter 4, Proposition 2.4]) (for Euclidean spaces, this fact was first mentioned in the paper [6]). Some necessary and sufficient conditions for the injectivity of P F for the family F = B r 1 , B r 2 (where the symbol B r i , i = 1, 2, stands for the open ball of radius r i and centered at the origin of X, and B r i is the closure of B r i ) were obtained in [7]. The results of this type are called "two-radii theorems".…”

Section: Problem (See [2]) Is the Transformation P F ;U Injective? Imentioning

confidence: 99%

“…Some radically new effects arise when we pass to compact two-point homogeneous spaces X (which means that X is isometric to the sphere S n or to one of the projective spaces P n (R), P n (C), P n (H), or P 16 (Cay); see [ where F is the hypergeometric function and a is the diameter of X (see [7,Theorem 4]). This implies that the set of all r such that P Br is injective, as well as the complement of this set, is dense in [0, a].…”

Section: §1 Introductionmentioning

confidence: 99%

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A local two-radii theorem on the sphere

Volchkov

2005

St. Petersburg Math. J.

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Abstract. Various classes of functions with vanishing integrals over all balls of a fixed radius on the sphere S n are studied. For such functions, uniqueness theorems are proved, and representations in the form of series in special functions are obtained. These results made it possible to completely resolve the problem concerning the existence of a nonzero function with vanishing integrals over all balls on S n the radii of which belong to a given two-element set. §1. Introduction Let X be a Riemannian two-point homogeneous manifold (see [1, Chapter 1]), let G be the isometry group of X, and let dx be Riemannian measure on X. We consider a collectionIf U = X, we call P F ;U the global Pompeiu transformation and denote it by P F .For given F and U, the following questions arise. Problem (see [2]). Is the transformation P F ;U injective? If not, what is its kernel?A set E ⊂ X for which P E is injective is called a Pompeiu set. The injectivity of the Pompeiu transformation and related questions for some X, F, and U were studied in many papers (see the surveys [2]-[5]). It turned out that there is a qualitative difference between the results for noncompact spaces X and their analogs for compact spaces. Here, we consider some of them.Suppose X is a noncompact two-point homogeneous space, i.e., X is isometric either to the Euclidean space R n , or to one of the hyperbolic spaces H n (R), H n (C), H n (H), or H 16 (Cay) (see [1, Chapter 1,§4, Subsection 3]. Then every ball in X is not a Pompeiu set. This fact follows easily from the mean-value theorem for eigenfunctions of the LaplaceBeltrami operator on X (see [1, Chapter 4, Proposition 2.4]) (for Euclidean spaces, this fact was first mentioned in the paper [6]). Some necessary and sufficient conditions for the injectivity of P F for the family F = B r 1 , B r 2 (where the symbol B r i , i = 1, 2, stands for the open ball of radius r i and centered at the origin of X, and B r i is the closure of B r i ) were obtained in [7]. The results of this type are called "two-radii theorems". Very general conditions ensuring the injectivity of P E were obtained by Williams [8] for X = R n , and by Berenstein and Shahshahani [9] for hyperbolic spaces. In these papers, it was proved that if a bounded open set E with connected complement and with Lipschitz 2000 Mathematics Subject Classification. Primary 26B15, 44A15, 49Q15.

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“…We note that if R > π (i.e., B R = S n ), then, in general, statements 3a) and 3b) are false (see [7,Theorem 4] and also §1 of the present paper).…”

Section: §8 Uniqueness Theoremsmentioning

confidence: 80%

Section: Problem (See [2]) Is the Transformation P F ;U Injective? Imentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

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A local two-radii theorem on the sphere

Volchkov

2005

St. Petersburg Math. J.

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“…Although the result is most striking in the context of the unit ball, an analogous theorem holds when B" is replaced by C" and 91L by the group M(2n) of Euclidean motions of R2". In fact, the same proof applies verbatim, the result from [3] being replaced by the earlier [6,4].…”

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confidence: 90%

A test for holomorphy in the unit ball of 𝐶ⁿ

Berenstein¹

1984

Proc. Amer. Math. Soc.

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“…The work of Berenstein and Zaicman [5], [6], has been very helpful. Although none of their results are used directly, their papers suggested that spherical means might be the right tool to deal with Case (I).…”

Section: {0} C H(b) Conjh(b) C(b)mentioning

confidence: 99%

Moebius-invariant algebras in balls

Rudin¹

1983

Annales De L’institut Fourier

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Pompeiu's problem on symmetric spaces

Cited by 102 publications

References 19 publications

A local two-radii theorem on the sphere

A local two-radii theorem on the sphere

A test for holomorphy in the unit ball of 𝐶ⁿ

Moebius-invariant algebras in balls

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