Skip to main content

Advertisement

Log in

Uncertainty Inequalities on Groups and Homogeneous Spaces via Isoperimetric Inequalities

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ciatti, P., Cowling, M.G., Ricci, F.: Hardy and uncertainty inequalities on stratified Lie groups (2013). arXiv:1308.2373

  2. Ciatti, P., Ricci, F., Sundari, M.: Heisenberg–Pauli–Weyl uncertainty inequalities and polynomial volume growth. Adv. Math. 215(2), 616–625 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Coulhon, T., Saloff-Coste, L.: Isopérimétrie pour les groupes et les variétés. Revista Matemática Iberoamericana 9(2), 293–314 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. D’Angelo, J.P., Tyson, J.T.: An invitation to Cauchy-Riemann and sub-Riemannian geometries. Notices Am. Math. Soc. 57(2), 208–219 (2010)

    MATH  MathSciNet  Google Scholar 

  5. Federer, H.: Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, vol. 153. Springer-Verlag, New York (1969)

    Google Scholar 

  6. Folland, G.B.: A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)

    Google Scholar 

  7. Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–238 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kasso, A., Okoudjou, K.A., Saloff-Coste, L., Teplyaev, A.: Weak uncertainty principle for fractals, graphs and metric measure spaces. Trans. Am. Math. Soc. 360(7), 3857–3873 (2008)

    Article  MATH  Google Scholar 

  9. Martini, A.: Generalized uncertainty inequalities. Math. Z. 265(4), 831–848 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Miranda Jr., M.: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. 82(8), 975–1004 (2003)

  11. Misha, G.: Metric structures for Riemannian and non-Riemannian spaces, progress in mathematics, Based on the 1981 French original, with appendices by Katz, M., Pansu, P., Semmes, S. Translated from the French by Sean Michael Bates, vol. 152. Birkhäuser Boston Inc., Boston (1999)

  12. Montgomery, R.: Mathematical Surveys and Monographs. A Tour of Subriemannian Geometries, Their Geodesics and Applications. American Mathematical Society, Providence (2002)

    Google Scholar 

  13. Ricci, F.: Uncertainty inequalities on spaces with polynomial volume growth. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 29(1), 327–337 (2005)

    MathSciNet  Google Scholar 

  14. Saloff-Coste, L.: Pseudo-Poincaré inequalities and applications to Sobolev inequalities. Around the Research of Vladimir Maz’ya. I, pp. 349–372. Springer, New York (2010)

    Chapter  Google Scholar 

  15. Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and geometry on groups, Cambridge tracts in mathematics, vol. 100. Cambridge University Press, Cambridge (1992)

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gian Maria Dall’Ara.

Additional information

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

$$\begin{aligned} \pi _m: G&\longrightarrow M\\ x&\longmapsto \phi _x(m):=x\cdot m, \end{aligned}$$

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

$$\begin{aligned} \frac{1}{\nu (B)}\int \nolimits _B\nu (Ab\setminus A)d\nu (b)\ge \frac{\nu (A)}{2}. \end{aligned}$$

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 \(G\times G\):

$$\begin{aligned} C:=\{(a,b)\in A\times B: ab\notin A\}. \end{aligned}$$

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

$$\begin{aligned} \int \nolimits _A \nu (b\in B: ab\notin A)d\nu (a)=\int \nolimits _B \nu (a\in A: ab\notin A)d\nu (b), \end{aligned}$$

which can be rewritten as follows

$$\begin{aligned} \int \nolimits _A \nu (B\setminus a^{-1}A)d\nu (a)=\int \nolimits _B \nu (A\setminus Ab^{-1})d\nu (b). \end{aligned}$$

Now the assumptions on the measures \(A\) and \(B\) and left invariance give

$$\begin{aligned} \nu (B\setminus a^{-1}A)\ge \nu (B)-\nu (A)\ge \frac{\nu (B)}{2}. \end{aligned}$$

This, combined with the equality above, gives

$$\begin{aligned} \frac{1}{\nu (B)}\int \nolimits _B \nu (Ab\setminus A)d\nu (b)=\frac{1}{\nu (B)}\int \nolimits _B \nu (A\setminus Ab^{-1})d\nu (b)\ge \frac{\nu (A)}{2}, \end{aligned}$$

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

$$\begin{aligned} \int \nolimits _G \left| \widetilde{f}(yx) - \widetilde{f}(y) \right| d\nu (y) \le d_{CC}(0,x\cdot 0)\int \nolimits _M|\nabla _Hf(m)|d\mu (m). \end{aligned}$$
(19)

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,

$$\begin{aligned} |\widetilde{f} (yx) - \widetilde{f}(y)|&= |f\circ \phi _y(\gamma (T))-f\circ \phi _y(\gamma (0))|\\ \left| \int \nolimits _0^T g(\nabla _H(f\circ \phi _y)(\gamma (t)), \dot{\gamma }(t)) dt \right|&\le \int \nolimits _0^T |\nabla _H f |(y\cdot \gamma (t))\cdot |\dot{\gamma }(t)| dt\mathrm . \end{aligned}$$

Integrating in \(y\) with respect to \(\nu \) and using Fubini, we obtain

$$\begin{aligned} \int \nolimits _G |\widetilde{f}(yx)- \widetilde{f}(y)|d\nu (y)&\le \int \nolimits _0^T|\dot{\gamma }(t)| \left( \int \nolimits _G |\nabla _H f(\pi _{\gamma (t)}(y))| d\nu (y)\right) dt\\&= L\int \nolimits _M |\nabla _H f(m)| d\mu (m), \end{aligned}$$

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 \(x\in G\),

$$\begin{aligned} \nu (\widetilde{E}x\triangle \widetilde{E})\le d_{CC}(0,x\cdot 0)\left\| \partial _H E \right\| . \end{aligned}$$

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

$$\begin{aligned} \int \nolimits _G \left| \widetilde{f}_n(y\cdot x^{-1}) - \widetilde{f}_n(y) \right| d\nu (y) \le d_{CC}(0,x\cdot 0)\int \nolimits _M|\nabla _Hf_n(m)|d\mu (m), \end{aligned}$$

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

$$\begin{aligned} \nu (\widetilde{E}x\triangle \widetilde{E})=\int \nolimits _G \left| 1_{\widetilde{E} x}- 1_{\widetilde{E}} \right| d\nu \le d_{CC}(0,x\cdot 0)\liminf _{n\rightarrow +\infty }\int \nolimits _M|\nabla _Hf_n(m)|d\mu (m). \end{aligned}$$

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

$$\begin{aligned} \mu (E)\le r\left\| \partial _HE \right\| . \end{aligned}$$

Let \(x\in B:=\pi _0^\leftarrow (B(0,r))\): by the right-invariance of \(\nu \) and (C) we have

$$\begin{aligned} \nu (\widetilde{E}x\setminus \widetilde{E})+\nu (\widetilde{E}x^{-1}\setminus \widetilde{E})=\nu (\widetilde{E}x\setminus \widetilde{E})+\nu (\widetilde{E}\setminus \widetilde{E}x)=\nu (\widetilde{E}x\triangle \widetilde{E})\le r\left\| \partial _H E \right\| . \end{aligned}$$

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

$$\begin{aligned} \mu (E)\le 2r\left\| \partial _H E \right\| . \end{aligned}$$

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-014-9512-3

Keywords

Mathematics Subject Classification (2010)

Navigation