# On how Poincaré inequalities imply weighted ones

## Abstract

*w*and are concerned with the question whether a Poincaré inequality holds on \(\varOmega \), i.e., if there exists a finite constant

*C*independent of

*f*such that It turns out that it is essentially sufficient that on all superlevel sets of

*w*there hold Poincaré inequalities w.r.t. the constant weight 1 and that the corresponding Poincaré constants satisfy an integrability condition. Furthermore we provide an explicit bound of the constant

*C*in the weighted inequality (1) in terms of the Poincaré constants of the superlevel sets. A similar statement holds true in the more general asymmetric case where we allow for certain weights \(\rho \) different from

*w*on the right hand side of (1).

## Keywords

Weighted Poincaré inequality Poincaré constant Sobolev inequality Superlevel sets## Mathematics Subject Classification

26D10 35A23 46E35## 1 Introduction

*Poincaré*type inequalities bound the \(L^q\)-norm of a function

*f*on a domain \(\varOmega \subset \mathbb {R}^d\) in terms of the \(L^p\)-norm of its gradient, i.e.,

*C*in (2) may depend on \(\varOmega , w, p, q\) and \(\mathcal {C}\) but cannot depend on

*f*.

- (i)
*Dirichlet*boundary conditions, i.e., \(\mathcal {C}\) consisting of functions vanishing on the boundary \(\partial \varOmega \) or - (ii)
*vanishing mean*, i.e.,$$\begin{aligned} \int _\varOmega f w = 0\quad \text {for all } f\in \mathcal {C}. \end{aligned}$$

*p*,

*q*and

*d*, in the literature Poincaré inequalities are also linked to the names of

*Wirtinger*(\(p=q=2\) and \(d=1\)) and

*Sobolev*(\(p<d\) and \(q=dp/(d-p)\)).

Due to its importance in the theory of partial differential equations a vast amount of work and effort has been put in the study of Poincaré type inequalities. The following overview is by far not a complete collection of the available research on this topic.

*Muckenhoupt*weights as introduced in [21], compare [12, 17]. For certain weight functions, \(q=p\) and arbitrary bounded domains a sufficient condition for a Poincaré to hold is provided in [3, 14]. However—depending on

*w*—these criteria may be very difficult to verify.

*C*in (3) are known only under very specific restrictions on the parameters

*d*,

*p*,

*q*, the geometry of the domain \(\varOmega \) and the weight

*w*: In case

*w*is constant, \(p=q\) and \(\varOmega \) is convex and bounded it is known that

For star shaped domains—again under the assumptions that *w* is constant and \(p=q\)—an explicit bound on the Poincaré constant is given in [18].

In the onedimensional situation [9] provides a bound on the Poincaré constant for arbitrary *p*, *q* and *w*, which in certain cases is sharp in some sense. Computing the bound amounts to determining the supremum of a possibly complicated expression and therefore—depending on *w*—may not be feasible. In [10] the same authors give explicit bounds on the Poincaré constant when \(q\le p\), \(\varOmega \) is bounded and convex and *w* is a positive power of a concave function.

Gaussian or more generally so called *log-concave* weights on the full space \(\varOmega =\mathbb {R}^d\) are considered in [5, 7] for the case \(p=q=2\).

For arbitrary weight *w* and domain \(\varOmega \) the Poincaré constant can be estimated from above in terms of the so called *Cheeger* constant—a well studied concept in Riemannian geometry—see [8] and [19, Appendix]. Again computation of the Cheeger constant may not be feasible depending on *w*.

Explicit bounds for the constant in weighted Poincaré inequalities are only known in very specific scenarios. The main result of this paper shows that the Poincaré constant w.r.t. a weight *w* can be controlled in terms of the Poincaré constants w.r.t. the constant weight 1 on the superlevel sets of *w*. Combined with knowledge of Poincaré constants for the unweighted case this yields a powerful tool for estimating the Poincaré constant on a general domain equipped with a weight whose superlevel sets only have to be connected. The key ideas are inspired by the work of Dyda and Kassmann in [13], where radially symmetric weights were considered. As we will see a similar approach can be taken in a much more general setting.

## 2 Preliminaries

In the following \(\varOmega \) will be called a *domain* in \(\mathbb {R}^d\), if \(\varOmega \) is a nonempty, open and connected subset of \(\mathbb {R}^d\). We call a function \(w:\varOmega \rightarrow \mathbb {R}\) a *weight* on \(\varOmega \subset \mathbb {R}^d\) if *w* is measurable and nonnegative.

*d*-dimensional Lebesgue measure of

*A*will be denoted by \(\left| A \right| \). For \(v\in \mathbb {R}^d\) we write \(\left| v \right| \) for the euclidean length of

*v*.

*Lipschitz*functions on \(\varOmega \), i.e. the set of functions that are Lipschitz continuous on every compact subset of \(\varOmega \) will be denoted by \(Lip(\varOmega )\). For \(q\in [1,\infty )\) we define

*weighted mean*of

*f*on

*D*is denoted by

### Remark 1

*w*is integrable on \(\varOmega \), \(D\subset \varOmega \) and that \(f\in L^p(\varOmega ,w)\) for \(p\ge 1\). Using Hölder’s inequality for \(\frac{1}{p}+\frac{1}{p'}=1\) we obtain

The central topic of this work are so called Poincaré inequalities:

### Definition 1

*w*an integrable weight on \(\varOmega \), \(\rho \) a weight on \(\varOmega \) and let \(p,q\in [1,\infty )\). Then a weighted

*(q, p)-Poincaré inequality*holds if there exists a finite constant

*C*such that

*C*in (7) is called

*Poincaré constant*and denoted by \(C_q^p(\varOmega ,w,\rho )\).

*superlevel sets*defined by

## 3 Main theorem

### Lemma 1

*w*be an integrable weight on \(\varOmega \) and \(p\in [1,\infty )\). Then for any \(f\in L^p(\varOmega ,w)\) it holds that

The following lemma provides a formula on how to write a weighted integral as a double integral.

### Lemma 2

*w*be an integrable weight on \(\varOmega \) and let \(g\in L^1(\varOmega ,w)\) then

### Proof

*A*let \(\chi _A\) denote the characteristic function of

*A*, then

*g*is replaced by \(\left| g \right| \). Since \(g\in L^1(\varOmega ,w)\) we can apply Fubini’s theorem and change order of integration. Using \(\chi _{[0,w(x))}(t)=\chi _{\varOmega _t^w}(x)\) we obtain

Next we show a simple relation between Lipschitz function spaces as defined in (5).

### Lemma 3

*w*be a weight on a domain \(\varOmega \) and \(q\ge 1\). Then for any \(t>0\)

### Proof

*f*is a Lipschitz function on \(\varOmega _t:=\varOmega _t^w\) for arbitrary \(t>0\). We can now estimate

We are set to state the main result:

### Theorem 1

*w*be a bounded and integrable weight on a domain \(\varOmega \subset \mathbb {R}^d\), let \(\rho \) be a weight on \(\varOmega \) and let \(1\le q\le p\). Then

^{1}since this is precisely what the Cheeger constant captures - and vice versa. In this spirit Theorem 1 can be qualitatively read as follows:

- 1.
If none of the superlevel sets \(\varOmega _t^w\) possesses a bottleneck w.r.t. to the constant weight 1 then neither will \(\varOmega \) w.r.t.

*w*. In that case \(C_q^p(\varOmega ,w,w)\) is small. - 2.
However if \(C_q^p(\varOmega ,w,w)\) is large then some of the superlevel sets \(\varOmega _t^w\) will have large Poincaré constants w.r.t. the constant weight 1 and therefore a bottleneck, see Fig. 1.

### Proof of Theorem 1

*w*cut off at level \(\tau \), i.e.,

*a*,

*b*the inequality \(\left| a+b \right| ^q\le 2^{q-1} \left( \left| a \right| ^q+\left| b \right| ^q\right) \) holds we estimate

*c*is an arbitrary real number. The choice \(c=g(x)\) yields

Finally we consider a concrete example. Our aim here is not to find the smallest possible bound on the Poincaré constant but rather to indicate how the result of Theorem 1 can be applied in practice. We choose a weight that is neither a power of a concave function nor log-concave in order to emphasize that our method yields results in situations where existing techniques are not applicable.

### Example 1

*w*vanishes at the origin and on the boundary of the domain,

*w*is not equivalent to a constant weight. Thus we can not resort to results for the unweighted case in order to find a bound for the Poincaré constant.

*w*are given by

## Footnotes

- 1.
We say a bottleneck is present if the domain can be partitionated into two subdomains of roughly equal measure w.r.t.

*w*such that the weight is small on the seperating boundary of the two subdamains.

## Notes

### Acknowledgements

Open access funding provided by University of Vienna. I would like to thank Philipp Grohs for inspiring discussions.

### Funding

Funding was provided by Universität Wien.

## References

- 1.Acosta, G., Durán, R.G.: An optimal Poincaré inequality in \(L^1\) for convex domains. Proc. Am. Math. Soc.
**132**(1), 195–202 (2004)CrossRefzbMATHGoogle Scholar - 2.Alaifari, R., Daubechies, I., Grohs, P., Yin, R.: Stable phase retrieval in infinite dimensions. Found. Comput. Math. (2018). https://doi.org/10.1007/s10208-018-9399-7 Google Scholar
- 3.Amick, C.J.: Some remarks on Rellich’s theorem and the Poincaré inequality. J. Lond. Math. Soc.
**18**(3), 319–328 (1978)MathSciNetzbMATHGoogle Scholar - 4.Bebendorf, M.: A note on the Poincaré inequality for convex domains. Z. Anal. Anwend.
**22**(4), 751–756 (2003)CrossRefzbMATHGoogle Scholar - 5.Bobkov, S.G.: Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab.
**27**(4), 1903–1921 (1999)MathSciNetzbMATHGoogle Scholar - 6.Bojarski, B.: Remarks on Sobolev imbedding inequalities. In: Laine, I., Rickman, S., Sorvali, T. (eds.) Complex Analysis Joensuu 1987, pp. 52–68. Springer, Berlin (1988)CrossRefGoogle Scholar
- 7.Brandolini, B., Chiacchio, F., Henrot, A., Trombetti, C.: An optimal Poincaré–Wirtinger inequality in Gauss space. Math. Res. Lett.
**20**(3), 449–457 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Gunning, R.C. (ed.) Problems Analysis, p. 195. Princeton University Press, Princeton (1970)Google Scholar
- 9.Chua, S.K., Wheeden, R.L.: Sharp conditions for weighted 1-dimensional Poincaré inequalities. Indiana Univ. Math. J.
**49**(1), 143–175 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Chua, S.K., Wheeden, R.L.: Estimates of best constants for weighted Poincaré inequalities on convex domains. Proc. Lond. Math. Soc.
**93**(1), 197 (2006)CrossRefzbMATHGoogle Scholar - 11.Deny, J., Lions, J.L.: Les espaces du type de beppo levi. Annales de l’institut Fourier
**5**, 305–370 (1954)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Drelichman, I., Durán, R.G.: Improved Poincaré inequalities with weights. J. Math. Anal. Appl.
**347**(1), 286–293 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Dyda, B., Kassmann, M.: On weighted Poincaré inequalities. Ann. Acad. Sci. Fenn. Math.
**38**, 721–726 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Edmunds, D.E., Opic, B.: Weighted Poincaré and Friedrichs inequalities. J. Lond. Math. Soc.
**47**(2), 79–96 (1992)zbMATHGoogle Scholar - 15.Ern, A., Guermond, J.: Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
- 16.Esposito, L., Nitsch, C., Trombetti, C.: Best constants in Poincaré inequalities for convex domains. J. Convex Anal.
**20**(1), 253–264 (2011)zbMATHGoogle Scholar - 17.Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ.
**7**(1), 77–116 (1982)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Farwig, R., Rosteck, V.: Note on Friedrichs’ inequality in n-star-shaped domains. J. Math. Anal. Appl.
**435**, 1514–1524 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Grohs, P., Rathmair, M.: Stable Gabor phase retrieval and spectral clustering. Commun. Pure Appl. Math.
**(To appear)**Google Scholar - 20.Martio, O.: John domains, bilipschitz balls and Poincaré inequalitiy. Rev. Rom. Math. Pures Appl.
**33**, 107–112 (1988)zbMATHGoogle Scholar - 21.Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc.
**165**, 207–226 (1972)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech Anal.
**5**, 286–292 (1960)CrossRefzbMATHGoogle Scholar

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