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1 Nonlinear Analysis 74 (0) Contents lists available at ScienceDirect Nonlinear Analysis ournal homepage: wwwelseviercom/locate/na Subelliptic estimates on compact semisimple Lie groups András Domokos, Roland Esquerra, Bob Jaffa, Tom Schulte Department of Mathematics and Statistics, California State University at Sacramento, 6000 J Street, Sacramento, CA, 9589, USA a r t i c l e i n f o a b s t r a c t Article history: Received 8 August 00 Accepted 6 April 0 Communicated by Ravi Agarwal Keywords: Semisimple Compact Lie group Cartan subalgebra Root space decomposition Subelliptic analysis In this paper, we consider a natural subelliptic structure in semisimple, compact and connected Lie groups, and estimate the constant in the so-called subelliptic Friedrichs Knapp Stein inequality, which has implications in the regularity theory of p-energy minimizers 0 Elsevier Ltd All rights reserved Introduction In this paper, we propose to obtain the best constant for the subelliptic Friedrichs Knapp Stein inequality: X f L () C X f L (), () where the subelliptic structure is generated by an orthonormal basis X of the orthogonal complement of the Cartan subalgebra of a semisimple, compact and connected Lie group In inequality () X f L () = (X,Y) X X XYf dµ is the norm of the matrix of second order horizontal derivatives and X f = X X XXf is the subelliptic Laplacian A look at the generators of the Lie algebra of SO(n) for n 4 (see (35)) shows that a semisimple compact Lie group can be endowed with a variety of subelliptic structures, and thus, the estimates of C can vary Once our method becomes clear, we will be able to make further comments on the advantages and disadvantages of a certain choice We will consider a subelliptic structure that naturally appears as the result of the root space decomposition associated with a Cartan subalgebra This is a generalization of the subelliptic structure on SU() and implicitly on SO(3) considered in [], which uses two out of the three Pauli matrices as generators Moreover, the proposed subelliptic structure satisfies the assumptions of [], and therefore, we already have some regularity results available for quasilinear subelliptic PDEs, which constitute the basis of a good application of () Corresponding author address: domokos@csusedu (A Domokos) X/$ see front matter 0 Elsevier Ltd All rights reserved doi:006/na00405

2 A Domokos et al / Nonlinear Analysis 74 (0) The existence of C has been proved for general subelliptic structures (see [3 5]), but its estimation was done ust in certain nilpotent cases, nice as the rušin plane, Heisenberg group and free nilpotent Lie groups of step two on an even number of generators [6 8] or on strictly pseudo-convex pseudo-hermitian manifolds [9] Our paper proposes to use the tools of the noncommutative harmonic analysis and the representation theory of semisimple Lie groups to estimate C in non-nilpotent subelliptic structures and by this, extend the results from [6 8] Harmonic analysis on compact Lie groups We start with a short review of the essential results we need regarding the harmonic analysis on compact groups As references for the following facts we can give [0,,] Throughout this paper will denote a semisimple, compact and connected Lie group We fix a normalized Haar measure µ on Compact groups are unimodular, so µ is both left and right invariant Let be the set of equivalence classes of irreducible unitary representations of Elements of will be denoted by [π] For compact groups irreducible representations are finite dimensional, so we can choose a finite dimensional vector space V π such that π : L(V π ) To every pair u, v V π we associate the following entry function of the representation π: Φ u,v : C, Φ u,v (g) = π(g)u, v We collect the entry functions into a vector space E π = span{φ u,v u, v V π }, which is a subspace of L () Let us choose an orthonormal basis {e i : i d π } of V π, to which we associate a basis of E π made by the functions π i (g) = π(g)e, e i The Schur orthogonality relations say that if [π] [λ] then E π and E λ are orthogonal subspaces of L () Moreover, { d π π i, i, d π } forms an orthonormal basis of E π By the Peter Weyl theorem we have that L () = E π Therefore, for every f L () we have f (g) = i,= dπ c π π i i(g), where the convergence is in the L norm and c π i = d π f (g) π i (g) dµ(g) Using the Fourier transform f (π) = f (g) π(g ) dµ(g) we get f (g) = d π trace ( f (π)π(g)) With the aid of the character of the representation π, χ π (g) = trace π(g), we can change (3) to f (g) = d π f χ π (g), where denotes the group convolution The fact that d π f χ π is the orthogonal proection on E π implies the following form of Plancherel s Theorem on compact Lie groups: f L () = d π f χ π L () (5) () () (3) (4)

3 4644 A Domokos et al / Nonlinear Analysis 74 (0) Subelliptic structures on semisimple, compact Lie groups The Killing form on the Lie algebra g of a semisimple compact Lie group is negative definite, so we can define a scalar product X, Y = ρ trace (ad X ad Y), where the positive constant ρ can be chosen according to our normalization needs This scalar product makes the mappings ad X : g g, ad X(Y) = [X, Y] skew symmetric Fix a maximal torus T in and its Lie algebra t, which is a Cartan subalgebra of g and which gives the rank of the Lie algebra by its dimension r Consider an orthonormal basis T = {T,, T r } of t and extend it by X = {X,, X k, Y,, Y k } to an orthonormal basis of g in such a way that T span[x, X] We will consider the subelliptic structure generated by a system of left invariant vector fields X = {X,, X k, Y,, Y k }, as an orthonormal basis of the orthogonal complement of a Cartan subalgebra This construction is in accordance with the root decomposition of the complexified Lie algebra g C as g C = t C α 0 g α, (3) where the summation has finitely many terms and it is done over the real roots α from the dual space t of t for which g α = {Z g C : ad T(Z) = iα(t)z, T t} The root decomposition is based on the fact that the extensions of ad T to g C can be simultaneously diagonalized with purely imaginary eigenvalues We have the orthonormal basis T of t fixed, so we can adopt the following setup Identifying T with is own dual basis, we can suppose that α t and therefore ad T(Z) = i α, T Z, if T t and Z g α For the following important properties of the subspaces g α we quote [ 5]: [g α, g β ] g α+β ; dim C g α = ; 3 g α = g α ; 4 If E α g α and E α g α then [E α, E α ] = i E α, E α α; 5 g α, g β = {0} if β ±α We say that a root is positive if its first non-zero coordinate is positive This introduces an ordering on the set of roots The real counterpart of the root decomposition (3) is g = t α P Re(g α g α ), (3) where Re means the real part of vectors of complex numbers, and P means the set of positive roots We consider a new orthonormal basis of g T,, T r, X,, Xk, Ỹ,, Ỹ k, for which each linear mapping ad T has a diagonal matrix By (3) we can suppose each pair ( X, Ỹ ) belongs to one subspace Re (g α g α ) and E ±α = Xα ± iỹ α g ±α Therefore, E α, E α =, and by property (4) from above we have that [ Xα, Ỹ α ] = α The Laplacian operator is invariant under orthogonal changes of variables, so we have k k X = X + Y = X + Ỹ = = We will use the notation r k = T + (X + Y = = for the elliptic Laplacian on )

4 A Domokos et al / Nonlinear Analysis 74 (0) Example Let us consider the special orthogonal group SO(n) = {M M n (R) : M T M = I, det M = }, (33) and its Lie algebra so(n) = {X M n (R) : X T = X, trace X = 0} (34) The group SO(n) acts on R n by rotations and its generators can be considered the rotations of the dimensional subspaces The corresponding infinitesimal rotations, which form a basis of the Lie algebra so(n) can be organized in a matrix form 0 X X 3 X n 0 0 X 3 X n X m X k X mk X n,n which helps us to visualize the commutation relations [X mk, X k ] = X m, [X k, X m ] = X mk, [X m, X mk ] = X k A vector field X k, < k corresponds to a n n matrix with all the elements being zero, with the exception of the k th element which equals and the k th elements which equals X k has its representation over the space of homogeneous polynomials in n variables and of degree m equal to x x k x k x As the basis of the Cartan subalgebra we can choose T = {X, X 34, } and the remaining vector fields forming X There is a slight difference between the cases with n = ν + and n = ν, but T always has ν elements Let us show how our construction looks in so(3) and so(4) The rank of so(3) is r = and we can choose ρ =, T = X, X = X 3 and Ỹ = X 3 In so(4) we start with ρ = 4, T = {X, X 34 } and X = {X 3, X 4, X 3, X 4 } For the new orthonormal basis we choose T = X, T = X 34, X = (X 3 + X 4 ), Ỹ = (X 3 X 4 ), X = (X 3 X 4 ) and Ỹ = (X 3 + X 4 ) The two positive roots are α = X X 34 and α = X + X 34, or with other notations, the vectors (, ) and (, ) In the following π always denotes an irreducible unitary representation of the semisimple, compact and connected group As any representation of a group, π induces a representation also named π of its Lie algebra g, defined by π(x)u = d dt t=0 π(e tx )u For the definition of weights we adopt a similar identification of t with t as in the case of roots The linear mappings π(t) : V π V π, T t, can be simultaneously diagonalized, so we can consider the weights λ t such that the weight spaces V λ = {v V π : π(t)v = i λ, T v, for all T t} (35) are not empty These weight spaces give an orthogonal direct sum decomposition of V as V π = Vλ π λ weights (36) For each root α and E α g α we have π(e α ) : Vλ π V λ+α π, which means that if λ + α is not a weight, then π(e α)(vλ π) = 0 Among the weights we consider an ordering similar to the one we mentioned for roots, and therefore we can talk about lowest and highest weights In particular, if λ is a highest weight and α is a positive root, then π(e α )(Vλ π) = 0 The decomposition (36) of V π into direct sum of weight spaces leads to the decomposition of the space of entry functions as E π = Eλ π, (37) λ weights

5 4646 A Domokos et al / Nonlinear Analysis 74 (0) where Φ u,v Eλ π if u V λ π This decomposition is also orthogonal because (see [0, Theorem 633]) Φ u,v (g) Φ u,v (g) dµ(g) = u, u v, v d π By incorporating our settings into the ideas of [, Theorem 35] we get the following lemma: Lemma 3 Let π be an irreducible, unitary representation of on a finite dimensional vector space V π with highest weight λ π and let δ be half of the sum of positive roots Then Φ = ( λ π + λ π, δ ) Φ, for all Φ E π, (38) and for all weights λ X Φ λ = ( λ π + λ π, δ λ )Φ λ, for all Φ λ E π λ (39) Proof Let u Vλ π and T t Then TΦ u,v (g) = d ds π(g e st )u, v s=0 = d π(g) π(e st )u, v ds s=0 = π(g)(π(t)u), v = i λ, T Φ u,v (g) By Schur s theorem Φ = c Φ for all Φ E π So, for determining the constant c it is enough to use a highest weight vector u V π λ π, for which E α Φ u,v = 0 if α is a positive root Therefore, Φ u,v = r = T Φ u,v + k ( X + Ỹ )Φ u,v = r = T Φ u,v + (E α E α + E α E α )Φ u,v = α P r = T Φ u,v + [E α, E α ] + E α E α Φ u,v = α P r = T Φ u,v + i α Φ u,v + E α E α Φ u,v = α P α P r = λ π, T λ π, α Φ u,v = α P = ( λ π + λ π, δ ) Φ u,v To prove (39) consider u V π λ for an arbitrary weight λ Then, Φ u,v E π λ X Φ u,v = Φ u,v r = T Φ u,v = ( λ π + λ π, δ λ )Φ u,v and we obtain By Lemma 3 and the fact that T Φ λ = i λ, T Φ λ holds for Φ λ Eλ π, we obtain the following corollary Corollary 3 For r, weight λ and Φ λ E π λ we have the following sharp estimate: T Φ λ L () λ λ π + λ π, δ λ XΦ λ L () (30) We are ready to prove one of the important results of this paper:

6 A Domokos et al / Nonlinear Analysis 74 (0) Theorem 3 For r and f C () we have the following sharp estimate: where T f L () c X f L (), c = sup λ weights λ λ π + λ π, δ λ (3) (3) Proof We will use the fact that the character χ π of the representation π belongs to E π and is a central function, so f χ π = χ π f and that by the orthogonal direct sum decomposition (37) we can write χ π = Φλ π, λ weights where Φλ π E λ π Using the Plancherel formula (5) we have that T f L () = d π T f χ π L () = d π f T χ π L () = f T Φλ π L () d π λ weights c d π λ weights d π λ weights λ λ π + λ π, δ λ f X Φ π λ L () = c d π f Xχ π L () = c d π Xf χ π L () = c X f L () f X Φ π λ L () Using the properties that for any weight λ and highest weight λ π we have λ λ π, and λ π, δ > 0, we obtain the following corollary: Corollary 3 For r and f C () we have the following estimate: T f L () sup λ π λ π, δ Xf L () (33) Until this stage, our results depended mainly on the root decomposition and implicitly on the properties of the Cartan subalgebra However, in the remaining part of this paper the commutation relations inside the orthogonal complement of t become increasingly important and this requires a specialized study of the different compact semisimple Lie groups To show how this works we will continue with the special orthogonal group SO(n) 4 Subelliptic analysis on SO(n) There are slight differences between n even and odd, so let us discuss them separately For details regarding roots and weights of SO(n) we refer to [4, Chapter 4] The case n = ν The rank of SO(ν) is ν, so T = {X, X 34,, X ν,ν } The set of positive roots is P = {e ± e k, < k ν} R ν

7 4648 A Domokos et al / Nonlinear Analysis 74 (0) Therefore, δ = (ν, ν,,, 0) The highest weight for the representation over the homogeneous polynomials of degree m is λ m alternating tensor representation over Λ m C n we have = me, while for the λ m = e + + e m, if m ν Hence, the constant λ π λ π, δ can take one of the values (ν ) or m (ν (m + )) (4) The maximum of these values are: in SO(4) in SO(6) 3 in SO(ν) if ν 4 (ν ) (4) We can estimate now the constant C from inequality () Theorem 4 For all f C (SO(ν)) we have X f L () C X f L () Xf L (), (43) where 3 if ν = 5 C = if ν = 3 3 ν if ν 4 ν Proof Let us remember that and T = {X, X 34,, X ν,ν } X = {X 3,, X,ν, X 3,, X,ν,, X ν,ν } To estimate the integral of X f = X k X lm f X k,x lm X in terms of the L norm of X, we have to use integration by parts to evaluate X k X lm f dµ To simplify the notations for the integration by parts, let us suppose that X, Y, Z are three left invariant vector fields with the commutation relations [X, Y] = Z, [Y, Z] = X, [Z, X] = Y

8 A Domokos et al / Nonlinear Analysis 74 (0) Then Similarly, Therefore, XYf XYf dµ = XYf (YXf + Zf ) dµ = XYf YXf dµ + XYf Zf dµ = Yf XYXf dµ + XYf Zf dµ = Yf (YX + Z)Xf dµ + XYf Zf dµ = Yf YXXf dµ Yf ZXf dµ + XYf Zf dµ = YYf XXf dµ Yf (XZf + Yf ) dµ + XYf Zf dµ = YYf XXf dµ + XYf Zf dµ Yf dµ YXf YXf dµ = XYf + YXf dµ = We have two cases: Case [X k, X lm ] T XXf YYf dµ XXf YYf dµ + YXf Zf dµ Zf dµ Xf dµ Xf dµ Yf dµ (44) In the second line of (44) each element of T appears as commutator of ν pairs from X and each element of X is used twice Case [X k, X lm ] X In this case, for each triplet X, Y, Z the formula (44) is repeated three times, one for each pair, and by addition this will cancel the second line of (44) In conclusion, we have r X f L () Xf L () + (ν ) T f L () Xf L () (45) = We ust have to take into account (3) and (4) to finish the proof Corollary 4 For all f C (SO(ν)) we have X f L () C X f L (), (46) where 3 if ν = 5 C = if ν = 3 3 ν if ν 4 ν The case n = ν + The rank of SO(ν + ) is ν, so as in the previous case we have T = {X, X 34,, X ν,ν } The set of positive roots is P = {e ± e k, e l : < k ν, l ν} R ν

9 4650 A Domokos et al / Nonlinear Analysis 74 (0) Therefore, δ = ν, ν 3,, The highest weight for the representation over the homogeneous polynomials of degree m is λ m alternating tensor representation over Λ m C ν+ we have λ m = e + + e m, for m ν Hence, the constant λ π λ π, δ can take one of the values ν, m(ν m) The maximum of these values are: in SO(3) in SO(5) in SO(ν + ) if ν 3 ν = me, while for the (47) (48) Theorem 4 For all f C (SO(ν + )) we have X f L () C X f L () Xf L (), (49) where 3 if ν = 7 C if ν = = 4 ν + if ν 3 ν Proof The proof is very similar to that of Theorem 4 We have the same set T, while X has additional terms: X = {X 3,, X,ν, X 3,, X,ν,, X ν,ν+ } The integration by parts is identical and ust some minor differences appear in the first case for commutators: The second line of (44) will show that each element of T appears as commutator of ν pairs from X and each element of X is used twice, except the elements of the last column of (35), which appear once In conclusion, we have r X f L () Xf L () + (ν ) T f L () Xf L () (40) We ust have to take into account (3) and (48) to finish the proof = Corollary 4 For all f C (SO(ν + )) we have X f L () C X f L (), where 3 if ν = 7 C = if ν = ν + if ν 3 ν (4) As an application of Corollaries 4 and 4 we can prove local regularity of minimizers of the p-energy functional on a domain Ω SO(n) The interesting fact is that the olden Ratio φ = + 5 appears in the estimates on SO(3) A similar result appeared in the Heisenberg group H (see [7]) and might suggest connections of φ with quantum mechanics

10 A Domokos et al / Nonlinear Analysis 74 (0) The subelliptic p-energy functional is defined as: Φ(f ) = X(f ) p dµ Ω Define the positive number γ in the following way: (4) - In SO(3) let - In SO(4) let - In SO(5) let - In SO(6) let γ = + 5 γ = γ = 4( + 9) 45 γ = 3( + 09) 5 - In SO(ν + ), for ν 3, let + γ = - In SO(ν), for ν 4, let + γ = ν (ν ) ν+ ν + (ν ) ν+ ν (ν ν) (ν ν ) ν ν + (ν ν ) ν ν Theorem 43 In SO(n) for γ given above and for p < + γ, (43) minimizers of the subelliptic p-energy functional have locally square integrable second order horizontal derivatives Proof For the proof we need the difference quotient method from [] developed for non-nilpotent cases and the Cordes technique from [7,8,6] The proof is similar in every aspect to what is found in the above cited papers, so we invite the interested reader to check the details We would like ust to highlight the fact that from [] it follows that in SO(n), for p < 4, the ε-regularized minimizers f ε have the property that (ε + Xf ε ) p 4 X f ε L loc (Ω), but with L -norms depending on ε Similar to formula [6, (48)] we can use the constant + d((d )C ) +, (d )C where d is the number of vector fields in X, to determine γ As in the proof of [7, Theorem 3] the Cordes technique allows us to let ε 0 Final Remarks Our methods show that in order to obtain estimates of C in inequality () we have to find connections between the group and subelliptic structure In SO(3) we have only one type of choice, but in higher dimensional spaces it is not the case One natural role of a subelliptic setting could be controlling the topology of a manifold with as few vector fields as possible In the case of SO(n) this means using the vector fields from ust the last column of (35) as generators of the subelliptic structure Such a choice would make Section 4 easier, but Section 3 more difficult In any case, a number of interesting problems seem to emerge

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