Abstract
We prove a new family of \(L^p\) uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The novelty of our technique consists in the observation that the \(L^1\) endpoint can be proved by means of appropriate isoperimetric inequalities.
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Acknowledgments
The authors would like to express their gratitude to F. Ricci, who introduced them to the subject of uncertainty inequalities and read very carefully a draft of the present paper. We also thank L. Ambrosio, for his interesting comments on the isoperimetric side of these matters, and S. Di Marino, who pointed us to the work of M. Miranda.
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Communicated by Loukas Grafakos.
Appendix
Appendix
1.1 Proofs of Theorem 4 and Theorem 8
In this Appendix we assume that \(M\) satisfies the assumptions of Sect. 2.1. In particular, \(M\) is not assumed to be compact, unless otherwise specified. Before starting, we fix a Haar measure \(\nu \) on \(G\) and observe that if \(m\in M\) and
then \(\mu _m(E):=\nu (\pi _m^\leftarrow (E))\) (\(E\subseteq M\) Borel) defines a \(G\)-invariant measure which is finite on compact sets, since isotropy subgroups are assumed to be compact. Unimodularity of \(G\) easily implies that \(\mu _m\) does not depend on \(m\) and we accordingly suppress the subscript \(m\) from the notation. Notice that the \(G\)-invariant measure on \(M\) is unique up to multiplicative constants (see Theorem \(2.49\) of [6]), so there is no loss of generality in working with this measure \(\mu \).
(A) If \(A,B\subseteq G\) are Borel subsets such that \(\nu (A)\le \nu (B)/2<+\infty \), then
Notice that this statement has a probabilistic interpretation: if \(B\) is significantly larger than \(A\), the right-translates of \(A\) by a random element of \(B\) are on average significantly disjoint from \(A\). Its proof is essentially combinatorial, being a continuous double-counting argument. We proceed with the details.
Consider the following subset of \(G\times G\):
If \(m:G\times G\rightarrow G\) denotes group multiplication, then \(C=m^\leftarrow (G\setminus A)\cap (A\times B)\), and hence \(C\) is a Borel subset of \(G\times G\) with respect to the product topology. Recall that the Borel \(\sigma \)-algebra of \(G\times G\) equals the product of the Borel \(\sigma \)-algebras of the two factors, since \(G\) is second countable. If we endow \(G\times G\) with the product measure \(\nu \times \nu \), we can then apply Fubini’s theorem, obtaining
which can be rewritten as follows
Now the assumptions on the measures \(A\) and \(B\) and left invariance give
This, combined with the equality above, gives
where we used the right invariance of \(\nu \) in the first equality.
We now fix \(0\in M\) and if \(f\) is a function on \(M\) and \(E\subseteq M\), we let \(\widetilde{f}:=f\circ \pi _0\) and \(\widetilde{E}:=\pi _0^\leftarrow (E)\). From now on, every set is assumed to be Borel.
(B) If \(f\in C^\infty _c(M)\) and \(x\in G\), it holds that
Let \(\gamma : [0,T] \rightarrow M\) be a piecewise \(C^1\) horizontal curve of length \(L\) connecting \(0\) to \(x\cdot 0\). Then it holds that for every \(y\in G\), by the Fundamental Theorem of Calculus, the \(G\)-invariance of \(\nabla _H\), and Cauchy–Schwarz,
Integrating in \(y\) with respect to \(\nu \) and using Fubini, we obtain
where the last identity follows from our observation that \(\mu \) is the push-forward of \(\nu \) with respect to \(\pi _m\) for any \(m\). Taking the \(\inf \) over all the horizontal curves connecting \(0\) to \(x\cdot 0\), we find the thesis. Estimate (19) also appears in Sect. \(6\) of [14], where the author deals with pseudo-Poincaré and Sobolev inequalities.
(C) For any Borel set \(E\subseteq M\) of finite \(\mu \) -measure and \(x\in G\),
Here \(\triangle \) denotes the symmetric difference of sets. Notice that this is the version for sets of the previous step.
Take any sequence \(f_n\in C^\infty _c(M)\) converging in \(L^1(M)\) to \(1_E\). Now (B) gives the inequalities
where we used \(d_{CC}(0,x\cdot 0)=d_{CC}(0,x^{-1}\cdot 0)\), a consequence of \(G\)-invariance of the distance. Taking the \(\liminf \) of both sides, we find
Taking the \(\inf \) with respect to the choice of the sequence \(f_n\), we find what we wanted.
(D) If \(\mu (E)\le \Gamma _M(r)/2\), then
Let \(x\in B:=\pi _0^\leftarrow (B(0,r))\): by the right-invariance of \(\nu \) and (C) we have
Since \(\nu (\widetilde{E})=\mu (E)\le \mu (B(0,r))/2= \nu (B)/2\), we can average the left-hand side with respect to \(x\in B\) and conclude by an application of (A).
What we said until now gives a proof of Theorem 8.
(E) Theorem 4 holds, i.e., if \(M\) is non-compact, \(E\subseteq M\) and \(\mu (E)\le \Gamma _M(r)\), then
By the super-additivity of \(\Gamma _M\) (see Sect. 4.1), \(\Gamma _M(r)\le \Gamma _M(2r)/2\). We can then apply (D) and conclude.
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Dall’Ara, G.M., Trevisan, D. Uncertainty Inequalities on Groups and Homogeneous Spaces via Isoperimetric Inequalities. J Geom Anal 25, 2262–2283 (2015). https://doi.org/10.1007/s12220-014-9512-3
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DOI: https://doi.org/10.1007/s12220-014-9512-3
Keywords
- Uncertainty inequalities
- Isoperimetric inequalities
- Homogeneous spaces
- Sub-Riemannian manifolds
- Cayley graphs