Dorronsoro's theorem in Heisenberg groups
Katrin F\"assler, Tuomas Orponen

TL;DR
This paper extends Dorronsoro's theorem to Heisenberg groups, showing that horizontal Sobolev functions can be approximated by affine functions independent of the last variable, with applications to Poincaré inequalities.
Contribution
It proves a variant of Dorronsoro's theorem in Heisenberg groups, introducing a new approximation method for horizontal Sobolev functions and providing alternative proofs for Poincaré inequalities.
Findings
Horizontal Sobolev functions can be approximated by affine functions in Heisenberg groups.
New proofs for vertical vs. horizontal Poincaré inequalities are established.
The approximation is independent of the last variable in the Heisenberg group context.
Abstract
A theorem of Dorronsoro from the 1980s quantifies the fact that real-valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last variable. As an application, we deduce new proofs for certain vertical vs. horizontal Poincar\'e inequalities for real-valued functions on the Heisenberg group, originally due to Austin-Naor-Tessera and Lafforgue-Naor.
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Dorronsoro’s theorem in Heisenberg groups
Katrin Fässler and Tuomas Orponen
Department of Mathematics
University of Fribourg
Chemin du Musée 23, CH-1700 Fribourg, Switzerland
University of Helsinki, Department of Mathematics and Statistics
Abstract.
A theorem of Dorronsoro from the 1980s quantifies the fact that real-valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro’s theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last variable.
As an application, we deduce new proofs for certain vertical vs. horizontal Poincaré inequalities for real-valued functions on the Heisenberg group, originally due to Austin-Naor-Tessera and Lafforgue-Naor.
Key words and phrases:
Heisenberg group, Sobolev space, coarse differentiation
2010 Mathematics Subject Classification:
26B05 (Primary) 26A33, 42B35 (Secondary)
K.F. is supported by the Swiss National Science Foundation via the project Intrinsic rectifiability and mapping theory on the Heisenberg group, grant No. 161299. T.O. is supported by the Academy of Finland via the project Quantitative rectifiability in Euclidean and non-Euclidean spaces, grant No. 309365.
1. Introduction
We start with a word on general notation. For , we consider the -th Heisenberg group . Points in will typically be denoted by , and we write for the left-invariant vector fields with the property that is the standard -th basis vector, . We will make no notational distinction between vector fields and the associated differential operators. If the derivatives , , of a function exist in the distributional sense, we denote by the horizontal gradient of . The symbol stands for an open ball with center and radius with respect to the Korányi metric on .
Let be a locally integrable function, let , and let be the family of (real) polynomials of degree at most . Some of the definitions below would, formally, make sense for all , but only the cases will be considered in the paper. Slightly abusing notation, we often view the elements of as functions on depending only on the first variables. For , , and , we define the following quantity:
[TABLE]
Here "" refers to integration with respect to Lebesgue measure on , and is the unique map in with the property that
[TABLE]
If , then is simply the orthogonal projection in to the subspace , and for instance
[TABLE]
the -average of over . It is not hard to compute explicitly either, see (2.4), and this is one way to establish the existence of in the generality of . The uniqueness of is straightforward: if are two candidates satisfying (1.2), then is -orthogonal to , and hence the projection of to is zero. On the other hand, since , this projection equals , and so .
Here is the main result:
Theorem 1.3**.**
Let , and let be a function with . Define the following square function:
[TABLE]
Then,
[TABLE]
The theorem applies in particular to compactly supported Lipschitz functions on . It is an variant of a theorem of Dorronsoro [6, Theorem 2] from the 80’s, and we will simply implement his proof strategy in . As in [6], we will derive Theorem 1.3 from a more general statement, which concerns affine approximation of functions in Sobolev spaces of fractional order, as defined by Folland [8]. The definition is based on the fractional (sub-)Laplace operator
[TABLE]
see [8, p. 181,186] (where the same object is denoted by ). For the definition of Sobolev spaces below, we would only need to consider , but complex values of will make a brief appearance in the proof of Lemma 5.1.
Definition 1.4**.**
Let and . The (inhomogeneous) Sobolev space is equipped with the norm
[TABLE]
In the sequel, we omit the subscript “” in the notation for the operator whenever the meaning is clear from the context (in particular, if acts on a function ).
Remark 1.5*.*
The properties of the spaces are discussed in detail in [8, Section 4] and, in a more general setting, in [7, Section 4.4.1]. In particular, is a Banach space for and .
Here is the generalised version of Theorem 1.3:
Theorem 1.6**.**
Let and . For , define the following square function:
[TABLE]
Here stands for the integer part of . Then, with
[TABLE]
Remark 1.8*.*
We note that Theorem 1.3 follows from the case of Theorem 1.6, because by [4, (52)].
1.1. Extensions and applications
Generalising the "-based" numbers defined in (1.1), one can consider the -variants
[TABLE]
It is then possible to ask if and when Theorems 1.3 and 1.6 continue to hold for these -numbers. We do not here pursue the most general results: we only show in Section 6 that Theorem 1.3 holds for the numbers if and
[TABLE]
where . In particular, this range covers the case which appears to be relevant for applications of Dorronsoro’s theorem – at least in Euclidean space, see for instance [5, Section 10]. The argument required for the extension is virtually the same as employed by Dorronsoro in [6, Section 5]: one can literally reduce matters to Theorem 1.3. We will repeat the details in Section 6.
An explicit application in Heisenberg groups where the cases come handy are certain horizontal vs. vertical Poincaré inequalities, first established by Austin, Naor, and Tessera [1], later extended by Lafforgue and Naor [14] and Naor and Young [15]. It turns out that many "non-endpoint" cases of these inequalities can be obtained as corollaries of Dorronsoro’s theorem in the Heisenberg group. The matter will be further discussed in Section 7.
We conclude the introduction with a few words on the proof structure of Theorem 1.6. Section 2 is mostly preparatory; notably, it reduces the "homogeneous" inequality (1.7) to its "inhomogeneous" analogue, see Lemma 2.6. In Section 3, we prove Theorem 1.6 in the regime . Then, in Section 4, we prove the case by a reduction to the case . Finally, in Section 5, we derive the case by complex interpolation.
Our proof strategy of Theorem 1.6 – hence Theorem 1.3 – is exactly the same as in Dorronsoro’s original work [6]. The main point here is to check that the use of horizontal Sobolev spaces in produces no serious complications. The case of Theorem 1.6 is essentially contained in [4, p. 291 ff]; the cases involve approximation by polynomials of degree , and these are not discussed in [4].
2. Preliminaries
We start by verifying that is always a near-optimal choice for the -approximation of by functions in :
Lemma 2.1**.**
Let , and . Then, for ,
[TABLE]
For later use, we separately mention the following immediate corollary:
Corollary 2.2**.**
Let and be such that and . Then
[TABLE]
Proof.
Apply the lemma with and . ∎
Proof of Lemma 2.1.
We first claim that
[TABLE]
One easily reduces to the case by scalings and (left) translations, and using the uniqueness of the element satisfying (1.2). Further, the equation
[TABLE]
follows from the definition (1.2) of , by choosing . Consequently, using also the equivalence of all norms on the finite-dimensional space , we infer that
[TABLE]
Rearranging the terms gives (2.3). We learned this quick argument from a paper of Prats, see [16, Remark 2.4].
To complete the proof of the lemma, we write for the function in corresponding to , as in (1.1)-(1.2). In particular, is linear, and for all . It follows that,
[TABLE]
using (2.3) in the last inequality. ∎
Before the next remark, we record that (as in the proof above) has the form
[TABLE]
where the coefficients , , and are
[TABLE]
Remark 2.5*.*
We discuss the sufficiency to prove Theorem 1.6 for in , the space of smooth compactly supported functions on . By [8, Theorem (4.5)], the functions in are dense in for all and . So, if , with and , we may choose a sequence such that . In particular, , which easily implies that
[TABLE]
as . To see this for , use the explicit expression for the maps obtained above (for the claim is trivial, as ). Then, if Theorem 1.6 has already been proved for some fixed , and for all functions in , we infer from Fatou’s lemma that
[TABLE]
Hence, Theorem 1.6 follows for general .
We conclude this section with one more reduction in the proof of Theorem 1.6. The proof given below for Theorem 1.6 will initially produce the estimate
[TABLE]
which is seemingly weaker than (1.7). (To be precise, this phenomenon will only occur in the case , but that is the case most relevant for Theorem 1.3.) This is precisely the result Dorronsoro proves in [6, Theorem 2]. However, the following homogeneity considerations allow us to remove the term from the estimate.
Lemma 2.6**.**
Let and . If
[TABLE]
then also
[TABLE]
Proof.
Given and , set , where denotes the usual Heisenberg dilation. Since the transformation has Jacobian , with , one has
[TABLE]
Below, we will moreover argue that
[TABLE]
Consequently, by (2.7),
[TABLE]
and (2.8) then follows by letting . To establish (2.10), we first need to compute , and for this purpose, we need expressions for the numbers , with and . We observe that
[TABLE]
Again applying the change-of-variables formula, we find that
[TABLE]
Thus, for all and , one has
[TABLE]
and now the first part of (2.10) follows from (2.9).
Finally, to compute , we have to use the definition of the fractional sub-Laplacian [8, p. 181], namely
[TABLE]
where denotes convolution with the heat kernel and . We recall from [8, p.184 and Theorem 4.5] that for all , and hence for and . Consequently, the limit in (2.12) exists in for any choice of . Exploiting the homogeneity
[TABLE]
we derive
[TABLE]
Therefore,
[TABLE]
by the homogeneity of , and so,
[TABLE]
Then we introduce a new variable , which yields
[TABLE]
Then, the second part of (2.10) follows again from (2.9). ∎
3. The case
We start by defining an auxiliary square function. Fix and , and let . Write
[TABLE]
where we have abbreviated . Then, a special case of [4, Theorem 5] states that
[TABLE]
With (3.1) in hand, the case of Theorem 1.6 will (essentially) follow once we manage to control by . To see this, first note that
[TABLE]
for all and . Consequently,
[TABLE]
cf. [4, p. 291]. Combining this estimate with (3.1) gives for all , and the case of Theorem 1.6 follows form Remark 2.5.
4. The case
This case will be reduced to the case . We start by recording the following result, which is a special case of [7, Theorem 4.4.16(2)]:
Proposition 4.1**.**
Let and . Then,
[TABLE]
In addition, we will need the following -analogue of [6, Theorem 5]:
Proposition 4.2**.**
Let and . Then,
[TABLE]
We can now complete the proof of Theorem 1.6 in the case as follows. Fix and . Combining the two propositions above with the case of Theorem 1.6, we infer that
[TABLE]
as desired. So, it remains to establish Proposition 4.2.
Proof of Proposition 4.2.
Let , , and recall that
[TABLE]
where
[TABLE]
Now, we define a function so that the average of equals the average of on , and the average of equals the average of in a larger ball , where will be chosen momentarily. Formally,
[TABLE]
where , and
[TABLE]
Then, noting that
[TABLE]
and using Lemma 2.1 and the weak -Poincaré inequality, see [13], we obtain
[TABLE]
The preceding holds if was chosen large enough. This implies that
[TABLE]
as claimed. ∎
5. Interpolation and the case
To handle the case , we use complex interpolation, see for instance [2]. In order to get the machinery started, we first observe that , is a compatible couple (or interpolation pair in the sense of Calderón [3]). That is, and are both Banach spaces and they embed continuously in the space of tempered distributions on , see [7, Theorem 4.4.3(4)]. Thus we can define the complex interpolation space for .
Lemma 5.1**.**
Let , with , fix , and let . Then, every satisfies
[TABLE]
The proof is otherwise the same as in [12, Lemma 34], except that the domain is in place of , so we need to use a few results from [8], and we work with non-homogeneous Sobolev spaces. It is convenient to use a norm on that is different from, but equivalent to, the norm in Definition 1.4. According to [8, Proposition 4.1], we have for and that
[TABLE]
where denotes the identity operator on .
Proof of Lemma 5.1.
Let , and fix a parameter to be specified later. Fix , and define the following map :
[TABLE]
To justify that maps into , we observe that , the Schwartz class in , since (see for instance the last paragraph of the proof of [7, Corollary 4.3.16]). Then it suffices to note that ; this is a special case of [7, Lemma 4.4.1].
By [8, Theorem (3.15)(iv)], is an analytic -valued function on , satisfying , and hence
[TABLE]
by the definition of the norm in the complex interpolation space . It remains to estimate the norms on the right hand side of (5.2). Using the equation , see [8, Theorem (3.15)(iii)], we find that
[TABLE]
It follows that
[TABLE]
Next, we recall from [8, Proposition (3.14)] the bound
[TABLE]
where the latter estimate follows from Stirling’s formula (see also [12, (79)]). We have now reached a point corresponding to [12, (107)]; the remainder of the proof no longer uses (Heisenberg specific) results from [8] and can be completed as in [12]. ∎
The second piece of information we need is a standard result from complex interpolation of Banach-space valued functions. Here we follow [6] almost verbatim. Let denote the Korányi unit ball centered at . For , we first define the Banach space of functions with
[TABLE]
Then, for , we denote by the space of functions with
[TABLE]
To apply complex interpolation, we have to verify that if , then is a compatible couple. Indeed, it follows from Hölder’s inequality that
[TABLE]
for every compact set , and for all and . This shows that the Banach spaces and both embed continuously into .
As in [6], we infer from the [2, p. 107, 121] that if and , then
[TABLE]
In the proof of (5.3), there is no difference between and . We will use (5.3) for any parameters and such that . We fix such parameters for the rest of the argument. Then, we consider the linear map
[TABLE]
where , , and as in (1.2). We already know that is a bounded operator , since
[TABLE]
by the case of Theorem 1.6. In fact, we also know that is a bounded operator . This follows from the calculation above with "" in place of "", and also plugging in the estimate
[TABLE]
after line (5.4) (this is immediate from Lemma 2.1).
Now, it follows by complex interpolation that is a bounded operator
[TABLE]
recalling (5.3). Repeating once more the calculation around (5.4), and finally using Lemma 5.1, we obtain
[TABLE]
This finishes the case of Theorem 1.6, recalling Remark 2.5 and Lemma 2.6. The proof of Theorem 1.6 is complete.
6. Extension to -mean -numbers
In this section, we consider an extension of Theorem 1.3, briefly mentioned in Section 1.1, which is analogous to the one discussed at the end of Dorronsoro’s paper, [6, Section 5]. To avoid over-indexing, we slightly re-define our notation for this last section. For , we write
[TABLE]
where
[TABLE]
So, corresponds to in the previous notation.
Theorem 6.1**.**
Write . Let and with
[TABLE]
Let with . Then,
[TABLE]
Remark 6.3*.*
The case is just Theorem 1.3. The general case follows from the argument given in [6, Section 5], and there is virtually no difference between and here: the idea is to demonstrate that the left hand of (6.2) is bounded by , which is then further bounded by the -norm of by Theorem 1.3. We will give the details for the reader’s convenience.
We begin by claiming that if , , then
[TABLE]
To see this, it suffices to establish that
[TABLE]
for all in the open ball , and for all sufficiently large (depending on ). Then (6.4) will follow by Lebesgue’s differentiation theorem. To derive (6.5), pick and so large that . Then, start with the following estimate:
[TABLE]
Here
[TABLE]
applying Corollary 2.2. To treat , recall that . In general, we will write for the element of corresponding to . In particular, since , we have
[TABLE]
Hence, for ,
[TABLE]
using (2.3) in the final inequality. Then, to complete the treatment of , it remains to note that
[TABLE]
using Corollary 2.2 again. Finally, to estimate , note that
[TABLE]
and hence
[TABLE]
In the application of (2.3), we used the assumption that . Finally,
[TABLE]
by Corollary 2.2. Summing the estimates above for completes the proof of (6.5).
As a corollary of (6.4), we infer the following inequality, which is an analogue of [6, (11)]: for and ,
[TABLE]
for Lebesgue almost all . To obtain the second inequality, use Corollary 2.2 once more.
From this point on, one can follow the proof presented after [6, (11)] quite literally. Fix, first,
[TABLE]
Then, choose some and such that
[TABLE]
We will apply the fact that the fractional Hardy-Littlewood maximal function
[TABLE]
maps boundedly, when are related as in (6.8). This fact holds generally in -regular metric measure spaces, see for example [11, Theorem 4.1]. It now follows from (6.7), Minkowski’s integral inequality, and the boundedness of that
[TABLE]
where stands for the maximal function . Consequently, recalling from (6.8) that , we arrive at
[TABLE]
Next, noting that and using Hardy’s inequality in the form
[TABLE]
see [10, Theorem 330], we obtain
[TABLE]
Finally, following [6] verbatim, the estimate
[TABLE]
can be inferred from the Fefferman-Stein vector-valued maximal function inequality, namely
[TABLE]
Here is a family of functions , and . The inequality (6.11) is valid generally in -regular metric measure spaces, see [9, Theorem 1.2]. To infer (6.10) from (6.9), one needs to apply (6.11) to functions of the form
[TABLE]
and exponents and , noting (by Corollary 2.2) that
[TABLE]
Once (6.10) has been established, the case of Theorem 6.1 is a corollary of Theorem 1.3 (since the right hand side of (6.10) is precisely ).
The case is similar. Indeed, if and , we may choose and such that (6.8) holds. Then, the arguments after (6.8) can be repeated. Finally, the choices of exponents and also remain valid in the application of the Fefferman-Stein inequality. The proof of Theorem 6.1 is complete.
7. Application: Vertical vs. horizontal Poincaré inequalities
As a corollary of Theorem 6.1, we derive the following vertical vs. horizontal Poincaré inequality originally due to Lafforgue and Naor [14, Theorem 2.1] (the case was earlier obtained by Austin, Naor, and Tessera [1]; see also [15, Remark 43]):
Theorem 7.1**.**
Let , and let with . Then,
[TABLE]
Here we have denoted by the point with . In [14], the target of is allowed to be a much more general Banach space than , and the "" in (7.2) can also be a more general exponent .
Proof of Theorem 7.1.
Fix and . We first claim that
[TABLE]
if is a sufficiently large constant. Here is the -based -number, as defined at the head of the previous section. Let be a collection of balls "" of radius whose union covers , and such that the concentric balls "" of radius have bounded overlap for some constant to be determined shortly. For , let be the affine function from the definition of . Since
[TABLE]
we may deduce (7.3) as follows, assuming that are appropriately chosen:
[TABLE]
In the last inequality we implicitly used Corollary 2.2. Now, the inequality (7.2) follows from Minkowski’s integral inequality and Theorem 6.1 with :
[TABLE]
making the change of variables before applying Theorem 6.1. The proof of Theorem 7.1 is complete. ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Tim Austin, Assaf Naor, and Romain Tessera. Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces. Groups Geom. Dyn. , 7(3):497–522, 2013.
- 2[2] Jöran Bergh and Jörgen Löfström. Interpolation spaces. An introduction . Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223.
- 3[3] A.-P. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math. , 24:113–190, 1964.
- 4[4] Thierry Coulhon, Emmanuel Russ, and Valérie Tardivel-Nachef. Sobolev algebras on Lie groups and Riemannian manifolds. Amer. J. Math. , 123(2):283–342, 2001.
- 5[5] G. David and S. Semmes. Singular integrals and rectifiable sets in 𝐑 n superscript 𝐑 𝑛 {\bf R}^{n} : Au-delà des graphes lipschitziens. Astérisque , (193):152, 1991.
- 6[6] José R. Dorronsoro. A characterization of potential spaces. Proc. Amer. Math. Soc. , 95(1):21–31, 1985.
- 7[7] Veronique Fischer and Michael Ruzhansky. Quantization on nilpotent Lie groups , volume 314 of Progress in Mathematics . Birkhäuser/Springer, [Cham], 2016.
- 8[8] G. B. Folland. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. , 13(2):161–207, 1975.
