Spherical rearrangements, subharmonic functions, and ∗-functions in n-space

Abstract. These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere. We shall address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.

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“…The following symmetry result is kind of folklore in the literature and we can quote [6,34,12] for various related results.…”

Section: Using (5) It Follows Thatmentioning

confidence: 99%

Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

Dolbeault

Esteban²,

Kowalczyk³

et al. 2013

Chin. Ann. Math. Ser. B

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“…We shall also need some basic facts concerning spherical rearrangements; see for example [Baernstein and Taylor 1976;Sperner 1973]…”

Section: It Follows Thatmentioning

confidence: 99%

Some sharp Hardy inequalities on spherically symmetric domains

Chiacchio¹,

Ricciardi²

2009

Pacific J. Math.

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We prove several sharp Hardy inequalities for domains with a spherical symmetry. In particular, we prove an inequality for domains of the unit ndimensional sphere with a point singularity, and an inequality for functions defined on the half-space ‫ޒ‬ n+1 + vanishing on the hyperplane {x n+1 = 0}, with singularity along the x n+1 -axis. The proofs rely on a one-dimensional Hardy inequality involving a weight function related to the volume element on the sphere, as well as on symmetrization arguments. The one-dimensional inequality is derived in a general form.

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“…Our proof is similar to that presented in [FS02]. We start by reviewing the two-point symmetrization procedure introduced in [BT76]. Let r ∈ S d−1 and H be the hyperplane passing through the origin with normal r. The hyperplane H partitions S d−1 into three sets: points above, below, and on the hyperplane.…”

Section: Theorem 35 (Restated) Let 1 ≥ a > 0 And Let A And B Be Twmentioning

confidence: 93%

“…We first prove that the sets in G k which share the least amount of edge are caps with the same center. The proof of Theorem 3.5 is based on symmetrization techniques similar to those presented in [BT76,FS02]. We prove Theorem 3.5 in section 9.…”

Section: The Continuous Graph Let D Be a Large Constant And Letmentioning

confidence: 95%

Graphs with Tiny Vector Chromatic Numbers and Huge Chromatic Numbers

Feige¹,

Langberg²,

Schechtman³

2004

SIAM J. Comput.

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Abstract. Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246-265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly ∆ 1−2/k colors. Here ∆ is the maximum degree in the graph and is assumed to be of the order of n δ for some 0 < δ < 1. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets.We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than n/∆ 1−2/k (and hence cannot be colored with significantly fewer than ∆ 1−2/k colors). For k = O(log n/ log log n) we show vector k-colorable graphs that do not have independent sets of size (log n) c , for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn.As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653-750] for this problem.

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Spherical rearrangements, subharmonic functions, and ∗-functions in n-space

Cited by 147 publications

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Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

Some sharp Hardy inequalities on spherically symmetric domains

Graphs with Tiny Vector Chromatic Numbers and Huge Chromatic Numbers

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