A zoo of diffeomorphism groups on \(\mathbb{R }^{n}\)
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Abstract
We consider the groups \({\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)\), \({\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)\), and \({\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)\) of smooth diffeomorphisms on \(\mathbb{R }^n\) which differ from the identity by a function which is in either \(\mathcal{B }\) (bounded in all derivatives), \(H^\infty = \bigcap _{k\ge 0}H^k\), or \(\mathcal{S }\) (rapidly decreasing). We show that all these groups are smooth regular Lie groups.
Keywords
Diffeomorphism group Infinite dimensional regular Lie group Convenient calculusMathematics Subject Classification (1991)
58B20 58D151 Introduction

\({\mathrm{Diff }}_{\mathcal{B }}(\mathbb{R }^n)\), the group of all diffeomorphisms which differ from the identity by a function which is bounded together with all derivatives separately; see 3.3.

\({\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)\), the group of all diffeomorphisms which differ from the identity by a function in the intersection \(H^\infty \) of all Sobolev spaces \(H^k\) for \(k\in \mathbb{N }_{\ge 0}\); see 3.4.

\({\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)\), the group of all diffeomorphisms which fall rapidly to the identity; see 3.5.
2 Some words on smooth convenient calculus
Traditional differential calculus works well for finite dimensional vector spaces and for Banach spaces. For more general locally convex spaces, we sketch here the convenient approach as explained in [6, 8]. The main difficulty is that composition of linear mappings stops to be jointly continuous at the level of Banach spaces, for any compatible topology. We use the notation of [8] and this is the main reference for this section.
2.1 The \(c^\infty \)topology
 1.
\(C^\infty (\mathbb{R },E)\).
 2.
The set of all Lipschitz curves (so that \(\{\frac{c(t)c(s)}{ts}:t\ne s, t, s\le C\}\) is bounded in \(E\), for each \(C>0\)).
 3.
The set of injections \(E_B\rightarrow E\) where \(B\) runs through all bounded absolutely convex subsets in \(E\), and where \(E_B\) is the linear span of \(B\) equipped with the Minkowski functional \(\Vert x\Vert _B:= \inf \{\lambda >0:x\in \lambda B\}\).
 4.
The set of all Mackeyconvergent sequences \(x_n\rightarrow x\) (there exists a sequence \(0<\lambda _n\nearrow \infty \) with \(\lambda _n(x_nx)\) bounded).
2.2 Convenient vector spaces
 1.
For any \(c\in C^\infty (\mathbb{R },E)\) the (Riemann) integral \(\int _0^1c(t)dt\) exists in \(E\).
 2.
Any Lipschitz curve in \(E\) is locally Riemann integrable.
 3.
A curve \(c:\mathbb{R }\rightarrow E\) is smooth if and only if \(\lambda \,\circ \, c\) is smooth for all \(\lambda \in E^*\), where \(E^*\) is the dual consisting of all continuous linear functionals on \(E\). Equivalently, we may use the dual \(E^{\prime }\) consisting of all bounded linear functionals.
 4.
Any Mackey–Cauchy sequence (i. e. \(t_{nm}(x_nx_m)\rightarrow 0\) for some \(t_{nm}\rightarrow \infty \) in \(\mathbb{R }\)) converges in \(E\). This is visibly a mild completeness requirement.
 5.
If \(B\) is bounded closed absolutely convex, then \(E_B\) is a Banach space.
 6.
If \(f:\mathbb{R }\rightarrow E\) is scalarwise \(\mathrm{Lip }^k\), then \(f\) is \(\mathrm{Lip }^k\), for \(k\ge 0\).
 7.
If \(f:\mathbb{R }\rightarrow E\) is scalarwise \(C^\infty \), then \(f\) is differentiable at 0.
 8.
If \(f:\mathbb{R }\rightarrow E\) is scalarwise \(C^\infty \), then \(f\) is \(C^\infty \).
2.3 Smooth mappings
Let \(E\), \(F\), and \(G\) be convenient vector spaces, and let \(U\subset E\) be \(c^\infty \)open. A mapping \(f:U\rightarrow F\) is called smooth or \(C^\infty \), if \(f\,\circ \, c\in C^\infty (\mathbb{R },F)\) for all \(c\in C^\infty (\mathbb{R },U)\).
 1.
For mappings on Fréchet spaces this notion of smoothness coincides with all other reasonable definitions. Even on \(\mathbb{R }^2\) this is nontrivial.
 2.
Multilinear mappings are smooth if and only if they are bounded.
 3.
If \(f:E\supseteq U\rightarrow F\) is smooth then the derivative \(\mathrm{d}f:U\times E\rightarrow F\) is smooth, and also \(\mathrm{d}f:U\rightarrow L(E,F)\) is smooth where \(L(E,F)\) denotes the space of all bounded linear mappings with the topology of uniform convergence on bounded subsets.
 4.
The chain rule holds.
 5.The space \(C^\infty (U,F)\) is again a convenient vector space where the structure is given by the obvious injectionwhere \(C^\infty (\mathbb{R },\mathbb{R })\) carries the topology of compact convergence in each derivative separately.$$\begin{aligned} C^\infty (U,F) \overset{C^\infty (c,\ell )}{\longrightarrow } \prod _{c\in C^\infty (\mathbb{R },U), \ell \in F^*} C^\infty (\mathbb{R },\mathbb{R }), \quad f\mapsto (\ell \,\circ \, f\,\circ \, c)_{c,\ell }, \end{aligned}$$
 6.The exponential law holds: For \(c^\infty \) open \(V\subset F\),is a linear diffeomorphism of convenient vector spaces.$$\begin{aligned} C^\infty (U,C^\infty (V,G)) \cong C^\infty (U\times V, G) \end{aligned}$$
 7.
A linear mapping \(f:E\rightarrow C^\infty (V,G)\) is smooth (by (2) equivalent to bounded) if and only if \(E \overset{f}{\longrightarrow } C^\infty (V,G) \overset{\mathrm{ev }_v}{\longrightarrow } G\) is smooth for each \(v\in V\). This is called the smooth uniform boundedness theorem [8, 5.26].
 (8)The following canonical mappings are smooth.$$\begin{aligned}&\mathrm{ev }: C^\infty (E,F)\times E\rightarrow F,\quad \mathrm{ev }(f,x) = f(x)\\&\mathrm{ins }: E\rightarrow C^\infty (F,E\times F),\quad \mathrm{ins }(x)(y) = (x,y)\\&(\quad )^\wedge :C^\infty (E,C^\infty (F,G))\rightarrow C^\infty (E\times F,G)\\&(\quad )^\vee :C^\infty (E\times F,G)\rightarrow C^\infty (E,C^\infty (F,G))\\&\mathrm{comp }:C^\infty (F,G)\times C^\infty (E,F)\rightarrow C^\infty (E,G)\\&C^\infty (\quad ,\quad ):C^\infty (F,F_1)\times C^\infty (E_1,E)\rightarrow C^\infty (C^\infty (E,F),C^\infty (E_1,F_1))\\&\qquad (f,g)\mapsto (h\mapsto f\,\circ \, h\,\circ \, g)\\&\prod :\prod C^\infty (E_i,F_i)\rightarrow C^\infty \left( \prod E_i,\prod F_i\right) \end{aligned}$$
Theorem 2.1
 1.
\(c\) is smooth
 2.
There exist locally bounded curves \(c^{k}:\mathbb{R }\rightarrow E\) such that \(\ell \,\circ \, c\) is smooth \(\mathbb{R }\rightarrow \mathbb{R }\) with \((\ell \,\circ \, c)^{(k)}=\ell \,\circ \, c^{k}\), for each \(\ell \in \mathcal{V }\).
This theorem is surprisingly strong: note that \(\mathcal{V }\) does not need to recognize bounded sets.
2.4 Faa di Bruno formula
3 Groups of smooth diffeomorphisms
3.1 Model spaces for Lie groups of diffeomorphism
If we consider the group of all orientation preserving diffeomorphisms \({\mathrm{Diff }}(\mathbb{R }^n)\) of \(\mathbb{R }^n\), it is not an open subset of \(C^\infty (\mathbb{R }^n,\mathbb{R }^n)\) with the compact \(C^\infty \)topology. So it is not a smooth manifold in the usual sense, but we may consider it as a Lie group in the cartesian closed category of Frölicher spaces, see [8, Section 23] with the structure induced by the injection \(f\mapsto (f,f^{1})\in C^\infty (\mathbb{R }^n,\mathbb{R }^n)\times C^\infty (\mathbb{R }^n,\mathbb{R }^n)\). Or one can use the theory of smooth manifolds based on smooth curves instead of charts from [11, 12], which agrees with the usual theory up to Banach manifolds.
We shall now describe regular Lie groups in \({\mathrm{Diff }}(\mathbb{R }^n)\) which are given by diffeomorphisms of the form \(f = \mathrm{Id } + g\) where \(g\) is in some specific convenient vector spaces of bounded functions in \(C^\infty (\mathbb{R }^n,\mathbb{R }^n)\). Now, we discuss these spaces on \(\mathbb{R }^n\), we describe the smooth curves in them, and we describe the corresponding groups.
3.2 Regular Lie groups
 For each smooth curve \(X\in C^{\infty }(\mathbb{R },\mathfrak g )\) there exists a curve \(g\in C^{\infty }(\mathbb{R },G)\) whose right logarithmic derivative is \(X\), i.e.,where \(\mu :G\times G\rightarrow G\) is multiplication with \(\mu (g,h)=g.h = \mu _g(h)= \mu ^h(g)\). The curve \(g\) is uniquely determined by its initial value \(g(0)\), if it exists.$$\begin{aligned} {\left\{ \begin{array}{ll} g(0)\;\;\, = e \\ \partial _t g(t)= T_e(\mu ^{g(t)})X(t) = X(t).g(t) \end{array}\right. } \end{aligned}$$

Put \(\mathrm{evol }^r_G(X)=g(1)\) where \(g\) is the unique solution required above. Then \(\mathrm{evol }^r_G: C^{\infty }(\mathbb{R },\mathfrak g )\rightarrow G\) is required to be \(C^{\infty }\) also.
3.3 The group \(\text{ Diff }_\mathcal{B }(\mathbb{R }^{n})\)
The space \(\mathcal{B }(\mathbb{R }^n)\) (called \(\mathcal{D }_{L^\infty }(\mathbb{R }^n)\) by Schwartz [21]) consists of all smooth functions with all derivatives (separately) bounded. It is a Fréchet space. By [22], the space \(\mathcal{B }(\mathbb{R }^n)\) is linearly isomorphic to \(\ell ^\infty \hat{\otimes }\, \mathfrak s \) for any completed tensor product between the projective one and the injective one, where \(\mathfrak s \) is the nuclear Fréchet space of rapidly decreasing real sequences. Thus, \(\mathcal{B }(\mathbb{R }^n)\) is not reflexive and not nuclear.

For all \(k\in \mathbb{N }_{\ge 0},\,\alpha \in \mathbb{N }_{\ge 0}^n\) and each \(t\in \mathbb{R }\) the expression \(\partial _t^{k}\partial ^\alpha _x c(t,x)\) is uniformly bounded in \(x\in \mathbb{R }^n\), locally in \(t\).
\({\mathrm{Diff }}^+_{\mathcal{B }}(\mathbb{R }^n)=\bigl \{f=\mathrm{Id }+g: g\in \mathcal{B }(\mathbb{R }^n)^n, \det (\mathbb{I }_n + \mathrm{d}g)\ge \varepsilon > 0 \bigr \}\) denotes the corresponding group, see Theorem 3.1 below.
3.4 The group \({\mathrm{Diff }}_{H^{\infty }}(\mathbb{R }^n)\)
The space \(H^\infty (\mathbb{R }^n)=\bigcap _{k\ge 1}H^k(\mathbb{R }^n)\) is the intersection of all Sobolev spaces which is a reflexive Fréchet space. It is called \(\mathcal{D }_{L^2}(\mathbb{R }^n)\) by Schwartz in [21]. By [22], the space \(H^{\infty }(\mathbb{R }^n)\) is linearly isomorphic to \(\ell ^2\hat{\otimes }\, \mathfrak s \). Thus it is not nuclear, not Schwartz, not Montel, but still smoothly paracompact.

For all \(k\in \mathbb{N }_{\ge 0}\), \(\alpha \in \mathbb{N }_{\ge 0}^n\) the expression \(\Vert \partial _t^{k}\partial ^\alpha _xf(t,\quad )\Vert _{L^2(\mathbb{R }^n)}\) is locally bounded near each \(t\in \mathbb{R }\).
\({\mathrm{Diff }}^+_{H^{\infty }}(\mathbb{R }^n)=\bigl \{f=\mathrm{Id }+g: g\in H^\infty (\mathbb{R }^n), \det (\mathbb{I }_n + \mathrm{d}g)>0\bigr \}\) denotes the correponding group, see Theorem 3.1 below.
3.5 The group \({\mathrm{Diff }}_\mathcal{S }(\mathbb{R }^n)\)
The algebra \(\mathcal{S }(\mathbb{R }^n)\) of rapidly decreasing functions is a reflexive nuclear Fréchet space.

For all \(k,m\in \mathbb{N }_{\ge 0}\) and \(\alpha \in \mathbb{N }_{\ge 0}^n\), the expression \((1+x^2)^m\partial _t^{k}\partial ^\alpha _xc(t,x)\) is uniformly bounded in \(x\in \mathbb{R }^n\), locally uniformly bounded in \(t\in \mathbb{R }\).
3.6 The group \({\mathrm{Diff }}_{c}(\mathbb{R }^n)\)

For each compact interval \([a,b]\) in \(\mathbb{R }\) there exists a compact subset \(K\subset \mathbb{R }^n\) such that \(f(t,x)=0\) for \((t,x)\in [a,b]\times (\mathbb{R }^n\setminus K)\).
3.7 Ideal properties of function spaces
Theorem 3.1
The case \({\mathrm{Diff }}_c(\mathbb{R }^n)\) is well known, see for example, [8, 43.1]. The onedimensional version \({\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R })\) was treated in [13, 6.4].
Proof
Let \(\mathcal{A }\) denote any of \(\mathcal{B }\), \(H^\infty \), \(\mathcal{S }\), or \(c\), and let \(\mathcal{A }(\mathbb{R }^n)\) denote the corresponding function space as described in 3.3–3.6. Let \(f(x)= x+g(x)\) for \(g\in \mathcal{A }(\mathbb{R }^n)^n\) with \(\det (\mathbb{I }_n + \mathrm{d}g)>0\) and for \(x\in \mathbb{R }^n\). We have to check that each \(f\) as described is a diffeomorphism. By the inverse function theorem, \(f\) is a locally a diffeomorphism everywhere. Thus, the image of \(f\) is open in \(\mathbb{R }^n\). We claim that it is also closed. So let \(x_i\in \mathbb{R }^n\) with \(f(x_i)=x_i + g(x_i)\rightarrow y_0\) in \(\mathbb{R }^n\). Then \(f(x_i)\) is a bounded sequence. Since \(g\in \mathcal{A }(\mathbb{R }^n)\subset \mathcal{B }(\mathbb{R }^n)\), the \(x_i\) also form a bounded sequence, thus contain a convergent subsequence. Without loss let \(x_i\rightarrow x_0\) in \(\mathbb{R }^n\). Then \(f(x_i)\rightarrow f(x_0)=y_0\). Thus, \(f\) is surjective. This also shows that \(f\) is a proper mapping (i.e., compact sets have compact inverse images under \(f\)). By [14, 17.2], a proper surjective submersion is the projection of a smooth fiber bundle. In our case here \(f\) has discrete fibers, so \(f\) is a covering mapping and a diffeomorphism since \(\mathbb{R }^n\) is simply connected. In each case, the set of \(g\) used in the definition of \({\mathrm{Diff }}_{\mathcal{A }}\) is open in \(\mathcal{A }(\mathbb{R }^n)^n\).
Let us check next that multiplication is smooth on \({\mathrm{Diff }}_{\mathcal{A }}(\mathbb{R }^n)\). Suppose that the curves \(t\mapsto \mathrm{Id }+f(t,\quad )\) and \(t\mapsto \mathrm{Id }+g(t,\quad )\) are in \(C^\infty (\mathbb{R },\mathrm{Diff }_\mathcal{A }(\mathbb{R }^n))\) which means that the functions \(f,g\in C^\infty (\mathbb{R }^{n+1},\mathbb{R }^n)\) satisfy condition \(\bullet \) of either 3.3, 3.4, 3.5, or 3.6. We have to check that \(f(t,x+g(t,x))\) satisfies the same condition \(\bullet \). For this, we reread the proof that composition preserves \({\mathrm{Diff }}_{\mathcal{A }}(\mathbb{R }^n)\) and pay attention to the further parameter \(t\).
Next, we check that inversion is smooth on \({\mathrm{Diff }}_{\mathcal{A }}(\mathbb{R }^n)\). We retrace the proof that inversion preserves \({\mathrm{Diff }}_{\mathcal{A }}\) assuming that \(g(t,x)\) satisfies condition \(\bullet \) of either 3.3, 3.4, 3.5, or 3.6. We see again that \(f(t,x+g(t,x))=g(t,x)\) satisfies the condition 3.1 as a function of \(t,x\), and we claim that \(f\) then does the same. We reread the proof paying attention to the parameter \(t\) and see that the same condition \(\bullet \) is satisfied.
If \(\mathcal{A }=\mathcal{S }\), we already know that \(f(s,x)\) is globally bounded in \(x\), locally in \(s\). Thus, we may insert \(X(s,x+f(s,x))=O(\frac{1}{(1+x+f(s,x)^2)^k})=O(\frac{1}{(1+x^2)^k})\) into (8) and can conclude that \(f(t,x)=O(\frac{1}{(1+x^2)^k})\) globally in \(x\), locally in \(t\), for each \(k\). Using this argument, we can repeat the proof for the case \(\mathcal{A }=\mathcal{B }\) from above and conclude that \({\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)\) is a regular Lie group.
Finally, we prove that \({\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)\) is a normal subgroup of \({\mathrm{Diff }}_{\mathcal{B }}(\mathbb{R }^n)\). We redo the last proof under the assumption that \(s\in H^\infty (\mathbb{R }^n)^n\). By the argument in (3), we see that \(s(x+g(x))\) is in \(H^\infty \) as a function of \(x\). The rest is as above, since \(H^\infty \) is an ideal in \(\mathcal{B }\) as noted in 3.7. \(\square \)
Corollary 3.2
\({\mathrm{Diff }}_{\mathcal{B }}(\mathbb{R }^n)\) acts on \(\Gamma _c\), \(\Gamma _{\mathcal{S }}\) and \(\Gamma _{H^\infty }\) of any tensorbundle over \(\mathbb{R }^n\) by pullback. The infinitesimal action of the Lie algebra \(\mathfrak X _{\mathcal{B }}(\mathbb{R }^n)\) on these spaces by the Lie derivative thus maps each of these spaces into itself. A fortiori, \({\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)\) acts on \(\Gamma _{\mathcal{S }}\) of any tensor bundle by pullback.
Proof
Since \({\mathrm{Diff }}_c(\mathbb{R }^n)\), \({\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)\), and \({\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)\) are normal subgroups in \({\mathrm{Diff }}_{\mathcal{B }}(\mathbb{R }^n)\), their Lie algebras \(\mathfrak X _{\mathcal{A }}(\mathbb{R }^n)=\Gamma _{\mathcal{A }}(T\mathbb{R }^n)\) are all invariant under the adjoint action of \({\mathrm{Diff }}_{\mathcal{B }}(\mathbb{R }^n)\). This extends to all tensor bundles. The Lie derivatives are just the infinitesimal versions of the adjoint actions. \(\square \)
References
 1.Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. Ann. Global Anal. Geom. (2012). doi: 10.1007/s104550129353x
 2.Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdVequation. Ann. Global Anal. Geom. 41(4), 461–472 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Bauer, M., Bruveris, M., Peter W.M.: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group II (2013). doi: 10.1007/s1045501393704
 4.Bauer, M, Bruveris, M., Peter W.M.: The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line. arXiv:1209.2836 (2012)Google Scholar
 5.Faà di Bruno, C.F.: Note sur une nouvelle formule du calcul différentielle. Quart. J. Math. 1, 359–360 (1855)Google Scholar
 6.Frölicher A., Kriegl A.: Linear spaces and differentiation theory. Pure and Applied Mathematics (New York). Wiley, Chichester (1988)Google Scholar
 7.Gerald A.G: Lectures on diffeomorphism groups in quantum physics. In: Contemporary problems in mathematical physics, pp 3–93. World Science Publication, Hackensack (2004)Google Scholar
 8.Kriegl, A., Michor, P.W.: The convenient setting of global analysis, volume 53 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997)CrossRefGoogle Scholar
 9.Kriegl, A., Michor, P.W.: Regular infinitedimensional Lie groups. J. Lie Theory 7(1), 61–99 (1997)MathSciNetzbMATHGoogle Scholar
 10.Micheli, M., Michor, P.W., Mumford, D.: Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds. Izvestiya: Mathematics, to appear (2012)Google Scholar
 11.Michor, P.W.: A convenient setting for differential geometry and global analysis. Cahiers Topologie Géom. Différentielle 25(1), 63–109 (1984)MathSciNetzbMATHGoogle Scholar
 12.Michor, P.W.: A convenient setting for differential geometry and global analysis. II. Cahiers Topologie Géom. Différentielle 25(2), 113–178 (1984)MathSciNetGoogle Scholar
 13.Michor, P.W.: Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach. In: Phase space analysis of partial differential equations, vol. 69 of Programme. Nonlinear Differential Equations Appl., pp. 133–215. Birkhäuser Boston, Boston (2006)Google Scholar
 14.Michor, P.W.: Topics in differential geometry, volume 93 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2008)Google Scholar
 15.Michor, P.W.: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Orpington (1980)Google Scholar
 16.Michor, P.W.: Manifolds of smooth maps II: The Lie group of diffeomorphisms of a non compact smooth manifold. Cahiers Topologie Géom. Différentielle 21, 63–86 (1980)MathSciNetzbMATHGoogle Scholar
 17.Milnor, J.: Remarks on infinitedimensional Lie groups. In: Relativity, groups and topology, II (Les Houches, 1983), pp. 1007–1057. NorthHolland, Amsterdam (1984)Google Scholar
 18.Mumford, D., Michor, P. W.: On Euler’s equation and ‘EPDiff’. arXiv:1209.6576 (2012)Google Scholar
 19.Omori, H., Maeda, Y., Yoshioka, A.: On regular Fréchet Lie groups IV. Definitions and fundamental theorems. Tokyo J. Math. 5, 365–398 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
 20.Omori, H., Maeda, Y., Yoshioka, A.: On regular Fréchet Lie groups V. Several basic properties. Tokyo J. Math. 6, 39–64 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
 21.Schwartz, L.: Théorie des distributions. Nouvelle édition, Hermann, Paris (1966)Google Scholar
 22.Vogt, D.: Sequence space representations of spaces of test functions and distributions. In: Functional analysis, holomorphy, and approximation theory (Rio de Janeiro, 1979), volume 83 of Lecture Notes in Pure and Applied Mathematics, pp. 405–443. Dekker, New York (1983)Google Scholar
 23.Walter, B.: Weighted diffeomorphism groups of Banach spaces and weighted mapping groups. Diss. Math. (Rozprawy Mat.) 484, 128 (2012)Google Scholar