Besov spaces and function series on Lie groups Leszek Skrzypczak

Besov spaces and function series on Lie groups

Leszek Skrzypczak

Abstract. In the paper we investigate the absolute convergence in the sup-norm of Harish- Chandra’s Fourier series of functions belonging to Besov spaces defined on non-compact connected Lie groups.

Keywords: Besov spaces, Harish-Chandra-Fourier series Classification: 46E35

1. Harish-Chandra’s Fourier series.

In this section we recall the definition and basic properties of Fourier series on a Lie group introduced by Harish-Chandra. For details we refer to [2] and [13,§4.4].

Let G be an n-dimensional Lie group countable at infinity and let K be a k- dimensional connected compact subgroup of G. Let Σ(K) denote the set of all equivalence classes of finite-dimensional irreducible representations ofK. For any δ∈Σ(K), letx_δ be the character of the classδ andd(δ) its degree. We define

(1) α_δ=d(δ)x_δ.

Let R be the Lie algebra ofK. SinceK is compact we can choose a positively- definite quadratic formQon R, which is invariant under the action of the adjoint representation of K. LetX₁, . . . , X_k be a base of R orthonormal with respect to Qand put

(2) Ω =I−(X₁²+· · ·+X_k²).

It is well known that Ω regarded as a differential operator commutes with both left and right translations onK, and that the functionsα_δ are eigenvectors of the operator Ω with eigenvaluesc(δ)≥1,

(3) Ω(α_δ) =c(δ)α_δ.

The group G is countable at infinity, therefore the space of smooth functions C^∞(G) and the space of smooth functions with compact supports C₀^∞(G) taken with their usual topologies are locally convex, complete and metrizable vector topo- logical spaces. LetG∋x→L(x) be the left regular representation ofGonC^∞(G)

(orC₀^∞(G)), i.e. (L(x)f) =f(x⁻¹y), andG∋x→R(x) be the right regular repre- sentation ofGonC^∞(G) (orC₀^∞(G)), i.e. (R(x)f)(y) =f(yx). Letf be a suitable function onG, then the functions

(α_δ∗f)(x) = Z

K

α_δ(y)f(y⁻¹x)dy, x∈G, (4)

(f∗α_δ)(x) = Z

K

α_δ(y⁻¹)f(xy)dy, x∈G, (5)

are called aδ-Fourier component of the functionf with respect to the representa- tion L(x) and R(x), respectively. Here dy denotes the normalized Haar measure onK. Identifyingα_δwith an element of the space of Radon measures with compact support onGwe can regard (4) and (5) as the convolutions onG.

Theorem 1 (Harish-Chandra). Let f ∈C^∞(G) (f ∈C₀^∞(K)), then the Fourier

series X

δ∈Σ(K)

α_δ∗f and X

δ∈Σ(K)

f∗α_δ

converge absolutely tof in C^∞(G) (C₀^∞(G)).

For every distributionT ∈ D^′(G) onGitsδ-Fourier component with respect to L(x) andR(x) can be defined respectively by

α_δ∗T, f∗α_δ (the convolution of distributions).

Using the notation of a contragradient representation it is not hard to see that the

series X

δ∈Σ(K)

α_δ∗T and X

δ∈Σ(K)

T∗α_δ

converge to T in D^′(G) equipped with the topology of uniform convergence on bounded subsets (cf. [13,§4.4.3]).

2. Function spaces on Lie groups.

Let e be the identity ofG and S =T_eGbe Lie algebra of G identified with the tangent space T_eG. We equip S with a scalar productg_e. For every x∈G, a scalar product inTxGis now defined by

gx=dxl(x⁻¹)ge (pull-back)

where dxl(x⁻¹) denotes the tangent mapping at xto l(x⁻¹) :y → xy. Furnished with this Riemannian metric g the Lie group G becomes a connected complete Riemannian manifold with a positive injectivity radius and a bounded geometry (cf. [3], [9]). Furthermore,g is left invariant, that is

g_yx(d_xl(y)X, d_xl(y)Y) =g(X, Y), x, y ∈G, Y ∈T_eG.

If r > 0 is a sufficiently small number, then the group G can be covered by a countable family of sets l(x_j)(B(r)), j = 1,2, . . ., B(r) = exp{X ∈ S : ge(X, Y)< r²}, such that each setl(x_j)(B(r)) has non-empty intersection with at mostN elements of the family. Moreover, there exists a resolution of unity{φ_j} corresponding to the above covering such that

φ_j ∈C^∞(G), 0≤φ_j ≤1, suppφ_j ⊆l(x_j)(B(r)), X φ_j = 1, (6)

for any multi-indexβ there is a positive number b_β with (7)

|D^β(φ_j◦l(x_j)◦exp)(X)| ≤b_β, j= 1,2, . . . (cf. [9, [11]]).

Definition 1 (cf. [9]). Let{φ_j} be the above resolution of unity.

(i) Let either 0< p <∞, 0< q≤ ∞orp=q=∞. Let−∞< s <∞. Then F_p,q^s (G) ={f ∈ D^′(G) :kf |F_p,q^s (G)k=

= ( X∞

j=1

φ_jf ◦l(x_j)◦exp|F_p,q^s (Rⁿ)k)<∞}

(with usual modification ifp=∞).

(ii) Let 0< p≤ ∞, 0< q≤ ∞. Let−∞< s₀ < s < s₁ <∞. Then B_p,q^s (G) = (F_p,p^s0(G), F_p,p^s1(G))_θ,q,

withs= (1−θ)s0+θs1, 0< θ <1.

Remarks. The definition of F_p,q^s (G) and B_p,q^s (G) is independent of the chosen resolution of unity and the chosen left invariant Riemannian metric. The groupG can be equipped with a right invariant Riemannian metric and on this base one can introduce “right” function spacesF^s_p,q(G),B^s_p,q(G). In general the “left” and

“right” spaces do not coincide, but the relation between them is not enough clear up to now. Various characterizations of the above spaces by the means of derivatives, differences and the like, can be found in [9]–[11].

A lot of properties of the scales F_p,q^s −B_p,q^s on Rⁿ have counterparts in the properties of the above defined function space on the Lie group G. For example, the following embeddings hold:

B^s_p,min(p,q)(G)⊂F_p,q^s (G)⊂B_p,max(p,q)^s (G) (8)

for 0< p <∞, 0< q≤ ∞ and − ∞< s <∞,

B^s_p,q(G)⊂B∞^σ,∞(G) for 0< p, q≤ ∞ and 0< σ < s−n p. (9)

Moreover one can prove the following interpolation property,

(B_p,q^s00(G), B_p,q^s11(G))_θ,q= (F_p,q^s00(G), F_p,q^s11(G))_θ,q=B^s_p,q(G)

for 0< p <∞, 0< q₀,q₁≤ ∞,−∞< s₀< s₁<∞, 0< θ <1,s= (1−θ)s₀+θs₁. The spacesF_p,2^s (G), 1< p <∞coincide with the Bessel-potential spaces for the corresponding Beltrami-Laplace operator (cf. [11]).

3. Absolute convergence of Fourier series.

Harish-Chandra’s result presented in Section 1 says that on the one hand Fourier series converge in a very strong sense for very smooth functions, on the other, the Fourier series of any distribution converges in the topology of uniform convergence on bounded sets. In this section, to fill the gap between these two convergences, we investigate the absolute convergence in the sup-norm of Fourier series of functions belonging to the Besov spacesB^s_p,q(G).

LetxK, x∈ K, denote a left-coset ofK. We will use the notation introduced in the foregoing sections. In particular, l(x) : y → x⁻¹y is an isometry of the Riemannian manifold (G, g) andxKis a compact submanifold ofGfor everyx∈G.

We need the following version of the trace theorem for the scales F_p,q^s (G)− B_p,q^s (G) (cf. [7]).

Lemma 1. Let1≤p <∞,1≤q≤ ∞orp=q=∞(1≤p,q≤ ∞in the case of theB_p,q^s (G)-scale). Lets > ⁿ⁻_p^k,n= dimG, k= dimK. Let R_x, x∈G, be the restriction operator andE_xthe extension operator

R_x :F_p,q^s (G)→F_p,p^s1(xK) (R_x:B_p,q^s (G)→B^s_p,q¹(xK)), Ex:F_p,p^s1(xK)→F_p,q^s (G) (Ex:B_p,q^s1(xK)→B_p,q^s (G))

described in Theorem1in [7],s−s₁ =^n−k_p . Then there are normsk· |F_p,q^s (xK)k (k· |B_p,q^s (xK)k)such that:

(i) kf |F_p,q^s (xK)k=kxf |F_p,q^s (K)k(kf |B^s_p,q(xK)k=kxf |B^s_p,q(K)k), where

xf(y) =f(x⁻¹y),

(ii) there are constantsc₁ =c₁(p, q, s,K)andc₂ =c₂(p, q, s,K)dependent on p, q, s,Kbut independent ofxsuch that

kRxk ≤c₁ and kExk ≤c₂.

Remarks. The above lemma is true in a more general situation, when we replace G by any complete connected Riemannian manifold with bounded geometry, K by a compact submanifold, the translationsl(x) by a family of isometries and the left-cosets by images of the compact submanifold under the action of the family of the isometries. In particular one can take the right-invariant Riemannian metric onG, the right-cosets ofK and the spaces F^s_p,q(G), B^s_p,q(G). We recall that the operatorsRx andExsatisfy the identityRx◦ Ex= id .

Proof: The proof is similar to the proof of Theorem 1 in [7], therefore the details are omitted. The only difference is that now we have to pay some attention to the norms of the restriction and extension operators. Since the groupK is compact, there is a finite family{(B(y_i, ρ_i),Φ_i)}^m_i=1 of charts ofGsuch that:

–B(yi, ρi) is a geodesic ball inGcentered atyi ∈Kwith radiusρi,ρi < i(G)/4, i(G) being the injectivity radius ofG,

– Φ_i(y_i) = 0, Φ_i(B(y_i, ρ_i)∩K)⊂R^k={(t₁, . . . t_n)∈R^k:t_k+1=· · ·=t_n= 0},

– setsV_i=B(y_i, ρ_i/4)∩K,i= 1, . . . , m, form an open covering ofK,

–B(y_i, ρ_i)∩K ⊆ BK(y, i(K)/2) for every y ∈ B(y_i, ρ_i)∩K, BK(y, r) being a geodesic ball inK,

(cf. [7,§4.2]).

The sets _xV_i = B(xy_i, ρ_i/4)∩xK, i = 1, . . . , m, cover xK. Let {ψ_i}^m_i=1 be a smooth resolution of unity corresponding to the covering {V_i}^m_i=1. Then the functionsxψ_i(y) =ψ_i(x⁻¹y),i= 1, . . . , m, form a resolution of unity corresponding to the covering{_xV_i}ofxK. The expression

(10) kf |F_p,q^s (xK)k= Xm i=1

k(xψ_if)◦l(x⁻¹)◦Φ⁻_i ¹|F_p,q^s (G)k

is a norm inF_p,q^s (xK). It should be clear that

kf |F_p,q^s (xK)k=k_xf |F_p,q^s (K)k, wherek· |F_p,q^s (K)kis given by (10) withx=e.

Let a function β_i ∈ C^∞(G) be such that suppβ ⊆ B(y_i, ρ_i), 0 ≤ β_i ≤ 1, βi(B(yi, ρi/4)) ={1}. Let f ∈F_p,q^s (G), then we define the restriction off onxK by

R_x(f)(y) = Xm i=1

(_xβ_if_i◦Φ_i◦l(x))(y)

where f_i is the restriction of the function (β_if)◦ l(x⁻¹)◦Φ⁻¹ on R^k (cf. [12, Theorem 2.7.2]). Theorem 1 in [7] asserted that the operatorRx is a continuous linear operator fromF_p,q^s (G) ontoF_p,q^s1(xK),s₁=s−ⁿ⁻_p^k. Moreover, according to the proof of this theorem, the norm ofR_x depends on:

– the numberm,

– the norm of the restriction operatorRx:F_p,q^s (Rⁿ)→F_p,p^s1(R^k), – the cardinal number of the setsxJ_i,x∈G,i= 1, . . . , m,

xJ_i={j:l(x_j)(B(r))∩B(xy_i, ρ_i)6=∅},

– the norms of the pointwise multiplier operator in F_p,p^s (R^k) defined by the functions _xψ_i◦l(x⁻¹)◦Φ⁻_i ¹ and_xβ_i◦l(x⁻¹)◦Φ⁻_i ¹,

– the norms of the isomorphism of F_p,q^s (Rⁿ) defined by the diffeomorphisms (Φ_j◦l(x))◦(l(x⁻¹)◦Φ_j) and exp⁻_xj¹◦l(x⁻¹)◦Φ⁻_i ¹, exp_ybeing the Riemannian exponential mapping at a pointy, (cf. [7,§4.1 and 4.2]). But,

– there is a constant C such that for every x and i card (xJ_i) ≤ C (cf. [7, Lemma 2]),

– the norms of the pointwise multiplier operators are bounded by the constant independent ofxbecause

k_xψ_i◦l(x⁻¹)◦Φ⁻_i ¹|C^m(R^k)k=kψ_i◦Φ⁻_i ¹|C^m(R^k)k

and

kxβ_i◦l(x⁻¹)◦Φ⁻_i ¹|C^m(R^k)k=kβ_i◦Φ⁻_i ¹|C^m(R^k)k (cf. [8, Theorem 1]),

– a norm of an isomorphism of F_p,q^s (Rⁿ) given by a diffeomorphism of Rⁿ depends on a constant which bounds from below a determinant of a Jacobian matrix of the given diffeomorphism. We have

exp⁻_xj¹◦l(x⁻¹)◦Φ⁻_i ¹= (exp⁻_xj¹◦exp_xyi)◦(exp⁻_xy¹_i◦l(x⁻¹)◦Φ⁻_i ¹).

Butl(x⁻¹) is an isometry of the Riemannian manifoldG, therefore exp_xyi◦ l(x⁻¹)◦Φ⁻_i ¹ =dxyil(x⁻¹)◦exp_yi◦exp_yi◦Φ⁻_i ¹. Thus, det(exp_xyi◦Φ⁻_i ¹) = det(exp_yi◦Φ⁻_i ¹) by the properties of normal coordinates. In consequence, we can find a common below bound for the determinants of the Jacobian matrices of the diffeomorphism we are interested in (cf. [8, Theorem 2], [11,

§4.1]).

This proves the lemma for theF_p,q^s -scale and the restriction operator. The proof for the extension operator is similar. The statement for theB_p,q^s -scale follows from

the properties of real interpolation method.

It was proved by Harish-Chandra that for sufficiently largem X

δ∈Σ(K)

d(δ)²c(δ)^−m<∞ (cf. [4, Lemma 7]).

Thus for everyr ∈ R, 0 < r ≤2, there is the smallest non-negative number mr

such that

(11) sup

δ∈Σ(K)

d(δ)^rc(δ)^−m<∞, for every m > mr.

Proposition 1. Let 1 ≤ r ≤ 2, w ∈ R, w+mr + ^k₂(1− ^r₂) > 0. Let s >

2r(w+mr) +k(¹_r−¹₂) +ⁿ⁻₂^k. Then there is a positive number Csuch that

(12) X

δ∈Σ(K)

c(δ)^wkα_δ∗fk ≤Ckf |B^s_2,r(G)k^r

holds for allf ∈B^s_2,r(K).

Proof: First we show that there is a constantC such that

(13) X

δ∈Σ(K)

c(δ)^v|hx_δ, fi|^r≤Ckf |B^s_2,r⁰(K)k^r

holds for 1≤r≤2,v∈R,v+^k₂(1−^r₂)>0,s₀= 2^v_r+k(¹_r−¹₂) and allf ∈B_p,q^s0(K).

Hereh·,·idenotes the scalar product inL₂(K).

The above inequality is known for abstract Besov spaces (cf. [6, Theorem 6.5.3]).

So, the only thing we have to do is to show that for a compact Lie group the Riemannian approach and the abstract approach lead to the same space.

First let us recall that in case of compact manifolds the definition of the scales F_p,q^s −B_p,q^s is independent of a chosen Riemannian metric. In particular the “left”

scalesF_p,q^s −B_p,q^s and the “right” scalesF^s_p,q−B^s_p,qon compact Lie group coincide.

The operator Ω (cf. (2)) with the domain D(Ω) = {f ∈ L₂(K) : Ωf ∈ L₂(K)}

is a self-adjoint, positively defined operator in L₂(K) with pure point spectrum.

Therefore, there are abstract potential spaces D(Ω^s), D(Ω^s) ⊂ L2(K), and ab- stract Besov spaces B_q^s, B_q^s ⊂ L2(K), s > 0, 1 ≤ q ≤ ∞, corresponding to the operator Ω — cf. [6, Definition 6.2.2 and 6.3.1] — the relevant properties of the Riesz means follow easily from the spectral theorem. The operator Ω coincides with the Laplace-Beltrami operator corresponding to the bi-invariant Riemannian metric onK. Therefore Theorem 4 in [11] assures us thatD(Ω^s) =F_2,2^2s(K),s >0.

Now the identity

B^s_q=B_2,q^2s(K), s >0, 1≤q≤ ∞,

follows by interpolation (cf. [6, Theorem 6.3.2] and [11, Theorem 5]). Moreover, we have the following estimate for the numberN(λ) of eigenvalues of Ω, counted with multiplicity, ≤ λ : N(λ) ≤ λ^k/2. The last inequality is nothing more than the famous Weyl asymptotic formula. Thus (13) follows from Theorem 6.4.3 in [6], because the functionsx_δform an orthogonal system of eigenvectors of the operator Ω with eigenvaluesc(δ) (cf. (1), (3)).

From (11) and (13) it follows that

(14)

X

δ∈Σ(K)

c(δ)^w|hα_δ, fi|^r≤

≤ sup

δ∈Σ(K)

(d(δ)^rc(δ)^−m) X

δ∈Σ(K)

c(δ)^w+m|hα_δ, fi|^r≤

≤Ckf |B_2,r^s1(K)k^r, s₁> 2

r(w+mr) +k(1 r −1

2).

Now, letf ∈B^s_2,r(G). Letfx, x∈G, denote the functionfx(y) =f(xy). Then

(15)

(α_δ∗f)(x) = Z

K

α_δ(y)f(y⁻¹x)dy= Z

K

α_δ(y)f_x(y⁻¹)dy=

= Z

K

α_δ(y)f_x(y)dy=hR_e(f_x), α_δi,

becausex_δ(y⁻¹) =x_δ(y). Applying Lemma 1 to the spacesB_2,r^s (G) we have (16) kRe(fx)|B_2,r^s2(K)k=kRx(f)|B^s_2,r^1(Kx)k ≤Ckf |B^s_2,r(G)k, s−s₁= ⁿ⁻₂^k,C being independent ofx.

It follows from (14)–(16) that for everyx∈G X

δ∈Σ(K)

c(δ)^w|(α_δ∗f)(x)|^r≤Ckf |B^s_2,r(G)k^r,

which completes the proof.

LetC(G) denote the Banach space of bounded continuous functions onGwith the standard norm.

Theorem 2. Let 1 ≤ p≤ ∞, 1 ≤ q ≤ ∞ and s > 2m₁+ ⁿ_p +kmax(0,¹₂ − ¹_p).

Letf ∈B^s_p,q(G). Then the Fourier seriesP

δ∈Σ(K)α_δ∗f converges absolutely in C(G)to the functionf. Moreover, there is a constantC such that

X

δ∈Σ(K)

kα_δ∗fk∞≤Ckf |B^s_p,q(G)k.

Proof: Leth∈ B_2,1^2m+k/2(K). The formula (13) withw= 0 and r= 1 reads as follows:

(17) X

δ∈Σ(K)

|hh, α_δi| ≤Ckh|B_2,1^2m+k/2(K)k.

Since

k(ψ_ih)◦Φ⁻_i ¹|F_p,q^s (R^k)k=k(ψ_ih)◦Φ⁻_i ¹ |F_p,q^s (U_i)k, U_i= Φ_i(B(y_i, δ_i)∩K), it follows from (10) that if−∞< s1< s0<∞,s0−_p^k

0 > s1−_p^k

1, 1≤p0, p1≤ ∞, 1≤q₀, q₁ ≤ ∞or p₀=q₀=∞,p₁ =q₁=∞then

F_p^s₀⁰_,q0(K)⊂F_p^s₁¹_,q1(K) (cf. [12, Theorem 3.3.1]).

By the real method of interpolation we have

(18) B_p^s₀⁰_,q0(K)⊂B_p^s1_1,q1(K),

for−∞< s₁< s₀ <∞,s₀−_p^k₀ > s₁−_p^k₁, 1≤p₀, p₁≤ ∞, 1≤q₀, q₁ ≤ ∞.

Let f ∈ B^s_p,q(G). Then f ∈ C(G) (cf. [11, Theorem 5]), α_δ ∗f ∈ C(G) for any δ∈Σ(K) and Re(fx)∈Bp,q^s0(K), s₀ =s−ⁿ⁻_p^k. But (18) and the inequality s₀>2m₁+ max(^k₂,^k_p) imply

kR_e(f_x)|B^2m+k/2_2,1 (K)k ≤CkR_e(f_x)|B_p,q^s0(K)k.

Thus,

X

δ∈Σ(K)

|hRe(fx), α_δi| ≤CkRe(fx)|B^s_p,q⁰(K)k cf. (17).

Now, by Lemma 1

X

δ∈Σ(K)

kα_δ∗Fk∞≤Ckf |B^s_p,q(G)k.

So the Fourier series of f converges in C(G). But this series converges to f in D^′(G) in the topology of uniform convergence on bounded sets, therefore it

converges tof inC(G).

Remarks.

1. Sincef∗α_δ(x) =h_xf, α_δiboth Proposition 1 and Theorem 2 are true if we replaceα_δ∗f byf∗α_δandB^s_p,q(G) byB^s_p,q(G).

2. One also can regard the series X

δ1,δ2∈Σ(K)

α_δ1 ∗f ∗α_δ2.

It was proved by Harish-Chandra that the above series converges tof absolutely inC^∞(G) (C₀^∞(G)) iff ∈C^∞(G) (f ∈C₀^∞(G)). One can ask whether the series converges absolutely tof in the sup-norm iff ∈B_p,q^s (G)∩B^s_p,q(G).

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Institute of Mathematics, A. Mickiewicz University, Pozna´n, Poland (Received May 29, 1992)