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2015

Near equality in the Riesz-Sobolev inequality in higher dimensions

Christ, Michael

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The Riesz-Sobolev inequality provides an upper bound for a trilinear expression involving convolution of indicator functions of sets.

It is known that equality holds only for homothetic ordered triples of appropriately situated ellipsoids.
We characterize ordered triples of subsets of Euclidean space $R^d$ that nearly realize equality, for arbitrary dimensions $d$, extending a result already known for $d=1$.

Near equality in the Riesz-Sobolev inequality in higher dimensions · Around

Built on

F. Riesz, Sur une inégalité intégrale
1930
Earlier work this paper cites.
S. L. Sobolev, On a theorem of functional analysis
1963
Earlier work this paper cites.
by same author, Near equality in the two-dimensional Brunn-Minkowski inequality
Original
1965
Earlier work this paper cites.

Similar

A. Burchard, Cases of equality in the Riesz rearrangement inequality
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E. H. Lieb and M. Loss, Analysis
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by same author, On near-extremizers for Young’s inequality for ℝ d {\mathbb{R}}^{d}
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by same author, Near equality in the Brunn-Minkowski inequality
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by same author, Near equality in the Riesz-Sobolev inequality
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M. Christ and T. Flock, Cases of equality in certain multilinear inequalities of Hardy-Riesz-Brascamp-Lieb-Luttinger type
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Cited in the paper.

Then

T. Tao and V. Vu, Additive Combinatorics
Original
2006
Later among the works it cites.
A. Figalli and D. Jerison, On the addition of sets in ℝ n {\mathbb{R}}^{n} : a quantitative stability result
2013
Later among the works it cites.

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