Abstract
Here we deal with the theme of a-priori estimates, in suitable Sobolev spaces, for Hörmander’s operators. This involves the concept of homogeneous group, the construction of fundamental solutions, the use of abstract singular integral theories, and the development of suitable algebraic and differential geometric tools. The chapter surveys three fundamental papers of the middle 1970’s on this subject, which introduced a number of ideas and techniques which are still part of the indispensable background in this area.
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Notes
- 1.
The exact form of these a priori estimates will be made precise in the following, dealing with each of the three papers.
- 2.
The group law of \(\mathbb {H}^{n}\) has already been defined in Sect. 2.2.4.
- 3.
This computation is not trivial, and here we will skip it. See for instance the book [2, Sect. 3.2].
- 4.
The rotations we are considering are just those around the \(t\)-axis.
- 5.
- 6.
It is possible to prove a more precise result about the integration of radial functions in homogeneous groups (see [9, Prop. 1.5]), which allows to prove an equality, and not just an inequality. I prefer to present this argument because it is simpler, can be extended to more general situations, and is enough for our purposes.
- 7.
- 8.
Rigorously speaking, there is an annoying further assumption which must be done: one has to assume that the balls are open with respect to the topology they induce, a property which is not automatic if \(d\) is not a distance. However, in any reasonable example this assumption is fulfilled.
- 9.
If \(D\left( \lambda \right) \) is a family of dilations defined by an \(N\)-tuple of positive real numbers \(\left( \alpha _{1},\alpha _{2},...,\alpha _{N}\right) \), then for any \(k>0\) also the exponents \(\left( k\alpha _{1},k\alpha _{2},...,k\alpha _{N}\right) \) will define a family of dilations. Hence we can always normalize the exponents so that \(\alpha _{1}=1\). We will see later that, under the further assumptions that we will make, if \(\alpha _{1}\) is an integer then all the other exponents are integers.
- 10.
For the reason explained in the previous footnote, in actual applications of the theory this number will be an integer.
- 11.
In particular this means that in this case the homogeneity exponents \(\alpha _{j}\), and therefore the homogeneous dimension \(Q\), are integers.
- 12.
This \(X_{0}\) is actually one of the initial vector fields \(\left\{ X_{i}\right\} _{i=1}^{N}\), which just for convenience we relabel \(X_{0}\).
- 13.
The free Lie algebra of step \(r\) on \(q\) generators is defined as the quotient of the free Lie algebra on \(q\) generators with the ideal spanned by the commutators of length at least \(r+1\).
- 14.
We will illustrate some examples of this fact in Chap. 4.
- 15.
Just to give a glimpse of the kind of difficulty, we can say that the problem is analogous to the one described in Sect. 3.3.5 dealing with homogeneous groups, but in a more abstract context.
- 16.
We have written that the theory is adaptable, not directly applicable to our situation. Rothschild-Stein [19] just sketch a proof of this adaptation. By now, however, this point is clearly established on the basis of more recent theories of singular integrals.
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Bramanti, M. (2014). A Priori Estimates in Sobolev Spaces for Hörmander’s Operators. In: An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-02087-7_3
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