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FOURIER TRANSFORM OF LIPSCHITZ FUNCTIONS ON RIEMANNIAN SYMMETRIC SPACES OF RANK ONE
S. S. Platonov
Petrozavodsk State University, Russia
1 Introduction and statement of main results
Among Riemannian manifolds the symmetric spaces of rank 1 form an important class. On these spaces we can study many problems of geometry, theory of functions and mathematical physics (see [HI], [H2], [H3]). Examples of symmetric spaces of rank 1 are n-dimensional sphere Sn and n-dimensional Lobachevsky space (hyperbolic space) Hn. On symmetric spaces there are analogs of the Fourier series (for compact spaces) and of the Fourier transform (for noncompact spaces), and many problems of classical harmonic analysis have a natural analogue for symmetric spaces. In what follows we will consider only symmetric spaces of noncompact type. Our main result is an analogue of one classical result of E. Titchmarsh connected with the Fourier transform of L2-functions satisfying certain Lipschitz conditions. »
Let f(x) € L2(K), || • ||l2(k) be the norm on L2(R), a € (0,1).
Definition 1. A function f(x) belongs to the Lipschitz class Lip{a,2) if
\\f(x + t)~ f(x)||L2(r) = 0{ta)
as t —> 0.
Theorem 1 ([T, Theorem 85]). Let f(x) € L2(K) and /(A), Л £ I, be the Fourier
transform of f. Then the conditions
f 6 Lip(a, 2), 0 < a < 1,
and
|A|
as r —» oo, are equivalent.
Recall some standard definitions connected with symmetric spaces ([HI]). Any Riemannian symmetric space X can be realized as the quotient space G/K, where G is a semisimple connected Lie group with finite center and K is a maximal compact subgroup of G. The group G acts transitively on X = G/K by left translations, and K coincides with the stabilizer of the point o = eK ( e is the unity of G). Let G = NAK be the Iwasawa decomposition of G, and let g, p, a, n be the Lie algebras of the groups G, K, A, N, respectively. We denote by M be the centralizer of the subgroup A in K and put B = K/M. Let dx be a G-invariant measure on X; the symbols db and dk will denote the normalized if-invariant measures on B and K, respectively.
We denote by a* the real dual space to a, and by W the finite Weyl group acting on a*. Let £ be the set of restricted roots (S C a* ), £+ be the set of restricted positive roots, and
a+ = {h G a : 7(h) >0, 7 £ E+}
be the positive Weyl chamber. Let p denote the half-sum of the positive roots (with multiplicity), then p G a*. Let (-, •) be the Killing form on the Lie algebra g. This form is positive definite on a. For A 6 a*, let H\ denote a vector in a such that A(H) = (H\, H) for all H E a. For A ,/i € a* we put (A, //} := (H\, H^). The correspondence A H\ enables us to identify a* and a. Via this identification, the action of the Weyl group W can be transferred to a. Let
a; = {A G a* : Hx G a+}.
If X is a symmetric space of rank 1, then dim a* = 1, and the set consists of the roots 7 and 27 with some multiplicities m7 and m27 depending on X (see [H2]). In this case we identify the set a* with R via the correspondence A K> A7, A 6 K, and the positive numbers will correspond to the set a+. The numbers m7 and m2T often arise in various formulas related to symmetric spaces of rank 1. For instance, the area of the sphere of radius t in X is equal to
S(t) = c(sinh t)™1 (sinh 2i)m27, (1.1)
where c is a constant, and for the dimension of X we have
dimX = m7 + m27 + 1.
We return to an arbitrary symmetric space X.
For g G G, let A(g) G a be a unique element for which
g = n ■ exp A(g) • u, where u G K, n G N. For x - gK G X = G/K and b = kM G B = K/M we put A(x,b) := A{k~lg).
Let T>(X) and V(G) denote the sets of infinitely differentiable complex-valued functions with compact support in X and in G, respectively. We note that the functions on X = G/K can be identified in a natural way with the functions f(g) on G satisfying
f(gu) = f(g), uEK.
Let dg be the Haar measure on G. We assume that dg is normalized in such a way that
J f{x) dx = j f(go) dg, V/ G V(X), (1.3)
x G
where o — eK G X = G/K.
For any functions f(x) G V(X), its Fourier transform, introduced by S. Helgason [H4], is defined by the formula
/(A, b) := J f(x) el-a+MM**)) dx, A G a*, b E B — K/M. (1.4)
(1.2)
The measure dx on X can be normalized in such a way that the inversion formula for the above Fourier transformation on X will look like this:
/(*) = |^F| / f(\b)e^+^A^\c(X)\-2d\db, (1.5)
a*xB
where \W\ is the order of the Weyl group, dX is the element of the Euclidean measure on a*, and c(A) is the Harish-Chandra function. In what follows, for brevity we put
dfi(X) := |c(A)|-2 dX.
We have the following Plancherel formula:
J\f(x)\2dx = -~ I |/(A, b)\2 dfi(X) db = I \f(X,b)\2dfi(X)db. (1.6)
X a*xB a*+xB
By continuity the map f(x) f(X,b) extends from T>(X) to an isomorphism of the Hilbert
space L2(X) = L2(X, dx) onto the Hilbert space L2(a+ x B, d[i(\)db). This extended map, also denoted /(x) /(A, b), keeps the name of the Fourier transformation, and the relations (1.5) and (1.6) remain valid.
In what follows, X is a Riemannian symmetric space of noncompact type of rank 1, n = dimX. By d(x,y) we denote distance from x to y, where x,y € X. Let
a(x; t) = {y EX : d(x, y) = t}
be the sphere in X centered at x of radius t > 0. We denote by dax(y) the (n — l)-dimensional element of area and by \a{t)\ the area of entire sphere a(x; t) (the latter quantity is independent of a;).
Let Cc(X) be the set of all continuous complex-valued functions on X with compact support. For / £ Cc(X) we define a function Stf by the formula
(Stf)(x) = J f(y)dax(y), ¿>0, (1.7)
a(x;t)
The operator Sl is called the shift operator or the spherical averaging operator. It can be proved (see the next section) that the operator 5* extends by continuity from Cc(X) to L\X).
Definition 2. A function f(x) belongs to the Lipschitz class Lipx{a, 2), 0 < a < 1, if f e L2(X) and
W&f-fhnv^oin
as t —> 0.
The next theorem is an analogue of Theorem 1 for symmetric spaces.
Theorem 2. Let X be a Riemannian symmetric space of noncompact type of rank 1, n = dimAT. For any function f(x) G L2(X) the conditions
f E Lipx(oc,2), 0 < a < 1, (1.8)
and
J J\f(\,b)\2d\db = 0(r-2a-n+l)
(1.9)
|Л|^г В
as r —> +oo, are equivalent.
The proof of this theorem is the main purpose of the paper. The other analogue of Theorem 1 on the Lobachevsky plane H2 was considered by Younis [Yo]. This author considered the other shift operator which depends on model of the Lobachevsky plane, whilst the shift Sl has geometrical origin.
2 Auxiliary propositions
Here we collect the auxiliary results of the Harmonic analysis on symmetric spaces. For any h £ G and f(x) E Cc{X) we put
K
where x = go, g £ G. If a; has another representation x = g\o7 g\ € G, then g\ = g5 for some S € K. Using the i(T-invariance of the Haar measure on K, we obtain
Consequently, the formula (2.1) is well-defined.
The operator Th is the other form of the shift operator 5*. It follows from the next lemma.
Lemmal. If h E G and d(ho,o) = t, then
(2.1)
к
к
к
(2.2)
(2.3)
(2.4)
(Thf)(x) = (Thf)(go) = (Lg(Thf))(o) = (:Th{Lgf)){o), (S^Kz) = (S‘/)M = (L^mo) = (S^Ljmo).
Therefore, the identity
(Thf)(x) = (SVKz) V/ G Cc(X), Vxex,
will be proved if we show that
(Thf)(o) = (S'/Ho) V/ G Cc(X). (2.5)
Let G(a(o; t)) be the set of all continuous functions on the sphere a(o; t). For <p(x) G C(a(o; t)) we put
h(<p) := (TV)(o), I2(<p) := (SV)(o),
assuming that </?(x) is extended in some way from <r(o; t) to a function of class Cc(X); the values of Ii(<p) and hi’f) do not depend on specific way of extension. Obviously, I\ and I2 are positive linear functionals invariant with respect to the action of if, that is,
Ii(Lk<p) = /1 (</?), h(Lk<p) = I2(v) VkeK.
The functionals I\ and /2 give rise to if-invariant measures on the sphere 0(0; I). Since any symmetric space of rank 1 is a two-point homogeneous manifold (see [W]), it follows that K acts transitively on cr(o;i). Hence, a if-invariant measure on a(o;t) is uniquely determined up to a coefficient. If we take <po(x) = 1, then
■^1(^0) = h(<Po) = l;
therefore, Ii and /2 coincide. This implies (2.5) and (2.2).
Let LP(X) = LP(X, dx), 1 ^ p < 00, and || • ||p be the norm in the Banach space ^(X).
Lemma 2. For any function f G Cc(X) and any g G G we have
\\Thf\\P < 11/Up. (2.6)
Proof. Let p > 1. If (p(k) is a continuous function on the group K then using fK dk = I and the Holder inequality, we obtain
f ip{k)dkP ^ f \ip{k)\pdk. (2.7)
Jk Jk
The inequality (2.7) is obvious for the case p = 1.
Let f(x) G Cc(X). Using (1.3) and (2.7) we obtain
\\Thf\\pp = [ \Thf(x)\pdx = / |Thf(go)\pdg =
Jx Jg
= f I f f{gkho)dk\pdg ^ f f \f{gkho)\pdkdg -
Jg Jk JgJk
= [ ([ I f(gkho)\p dg)dk = [ {[ \f{go)\p dg) dk = \\f\\p .
Jk Jg Jk Jg
Here we have used the invariance of the Haar measure dg on G with respect to the right shifts.
Since Cc(X) is dense in LP(X) for 1 ^ p < oo, Lemma 2 shows that the operator Th (and the operator Sl) can be extended to a continuous operator on the entire space LP(X), and
(2.3) remains true for any / E LP(X). In particular
W&fh < ll/lb, / € L\X). (2.8)
In harmonic analysis on symmetric spaces the central role plays spherical functions (see
[H2], [GW], [ST]). For A € a*, let y>\(g) denote the spherical function on G defined by the
Harish-Chandra formula
[ e(iA+p)(A(fes))^, geG, (2.9)
Jk
where all notations is defined in §1. We list some properties of the spherical functions to be used later on:
V\{uigu2) = <P\(g), ui,u2 G K; (2.10)
¥>a(s-1) = <P\{g), jeG; (2.11)
<px(e) = 1. (2.12)
Lemma 3. Let $(/)(A,6) := /(A, b) be the Fourier transform of f(x) € L2(X). Then $(Th/)(A,b) = ip\(h) ■ /(A, b), heG. (2.13)
Proof. Since V(X) is dense in L2(X), it is sufficient to prove (2.13) for / G V(X). We recall that the element A(g) G a is defined from the Iwasawa decomposition
g = n-expA{g) ■ u, (2.14)
where u G K, n G N. Let g,h G G, k G K. Since
kgh = n ■ exp(A(kg)) ■ u(kg) ■ h
for n G N, u(kg) G K, and since the subgroup A involved in an Iwasawa decomposition normalizes the subgroup N, we obtain
A(kgh) = A(kg) + A(u(kg)h). (2-15)
For brevity we put
ex(g):=e^iX+^A^\ geG.
From definition of the Fourier transformation we deduce that
Thf{X,b) = J (^J f(gvho)dvj e\(k~lg) dg, (2.16)
G K
where b — kM, k G K. Since uo = o and A{gu) = A(g) for u G K, the right-hand side in (2.16) can be reshaped as follows:
Thf(X,b) = J j figvhv^o) e\(k~lg) dg dv =
K G
= J J f{go) e\(k~lgvh~lv~l) dg dv =
K G
= J/(30) e\{k~1gvh~1) dv'j dg. (2.17)
G К
Then, by (2.15) we have
ex(k~lgvhTl) = e\{k~lg) ■ ex{uvh~l), where u = u(k~1g). Substituting this in (2.17), we obtain
Thf(X,b) = J f(go) e\(k~1g) (J ex{uvhTl)dv) dg =
G K
= <px(h'l)f(\,b) = <px(h)-f(\,b).
This completes the proof.
Let hi and /12 be elements of the group G such that d(h\o,o) = d(h,20,o) = t. Since X is a two-point homogeneous space (see [W]), it follows that K acts transitively on a(o;t). Hence, /120 = uh\o for some u G K. Since K is the stabilizer of the point o, it follows from h,20 = uh\o that /¿2 = uh\v for some v G K. By (2.10) we have (px(h\) = (px(h2), consequently, the function ipx{h) depends only on the distance t — d(ho, o), and we will write often y>x(t) instead of tp\{h) for t — d(ho,o).
Using Lemma 1 we can rewrite (2.13) in the form
$(5V)(A,6) = ¥*(*)•/(A,6), t GK+ = [0;+oo). (2.18)
We recall that the set a* is identified with M ( see §1). Since ipx{t) = <p~x{t), we will propose
A G M_(_.
Lemma 4 (Estimates for spherical functions) For any spherical function (px{t), A ^ 0, t ^ 0, we have:
1) Iv»a(<)| < l;
2) 1 -vxit) < ^2(A2 + p2);
3) there is a constant c > 0 depending only on X such that if Xt ^ 1, then
1~<P\ (*)>c. (2.19)
Proof. See [PI, Lemmas 3.1, 3.2 and 3.3].
3 Proof of Theorem 2
Proof of implication (1.8) (1.9)
Let f(x) E L2(X) and the condition (1.8) is true, that is
\\Stf-fh = 0(t°‘) (3.1)
as t —» 0. From the Plancherel formula and (2.18) it follows that
IIS'*/- /111 = / \Stf(x)-f(x)\2dx =
X
oo
= 1111 - v’aWI2 I/(a> 6)l2 dv(x)db- (3-2)
о в
For brevity we denote
F(A):= I \f(X,b)\2db. (3.3)
B
It follows from (3.2) and (3.3) that we can rewrite (3.1) in the form
\l-iPx(t)\2F(X)dfi(\) = 0(t2a). (3.4)
/'
We recall that
dft( A) = |c(A)|-2 dX,
where c(A) is the Harish-Chandra function of the symmetric space X. For any functions A(A) > 0 and B(A) >0 we say that
A(A) ^ B(A)
as A —> oo if
сі B(A) ^ Л(А) < c2 B(A)
for some constants ci > 0 and c2 > 0. Below, c,ci,c2,... are positive constants that may depend on X and a and independent on / and A.
The Harish-Chandra function c(A) can be expressed in terms of the Г-function of Euler (see [H2, Chapter 4, §6]). When X is a symmetric space of rank 1, we have
/ /хчч-і Г(|т7 + ^ + |A)Г(jm7 + 2^27 + 2ґп r\
(с(л)) = c°---------------г(іа'Т|Гг(|л) ’ (3'5)
where co > 0 is a constant, m7 and шг7 are the multiplicities of the roots 7 and 27 in E+ (see §1).
Using well known limiting relation
л->+оо Г(А) Att
(see [BE]) we get from (3.5) that
|c(A)|-2 x
Since m7 + rri27 + 1 = n = dimX, it follows that
|c(A)|-2 x A71“1, n = dimX. If A G [j, |] then At > 1, and we get from (2.19) that
tr
Then
2/t 2/t
I F(X)dn(\) < ^ J |1 — <P\{t)\2 F(X) dn(X) ^
l/t
oo
i 111 — <p\(t)\2 F(X) d^(X) = 0(t2a).
l/t l/t
oo
C“
It follows from (3.8) and (3.6) that
2/i
I F(X) A"'1 dX = 0{t2a)
as t -> 0 or, equivalently,
l/t
2 r
j F{A) An_1 dA = 0(r~2Q)
T
as r +oo. We can rewrite (3.10) in the equivalent form
2 r
j F{X)dX = 0(r~2a~n+l)
as r —> -foo.
It follows from (3.11) that
2r
2a—n+1
j F(A) dA < ci r
Г
where ci > 0 is a constant. Using this inequality, we get
°° oo 2^Ir OO
/ F(A) dA = / F(A) dA < J] ci (2fcr)-2Q-n+1 < c2 r"
fc=o ¿ fc=o
2a—n+1
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
and the condition (1.9) is true. We obtain that (1.8) implies (1.9).
Proof of implication (1.9) => (1.8)
Let f(x) € L2(X) and the condition (1.9) is true, that is
OO
J F{A) dA = 0(
r~2a-n+l\
as r -> oo. It follows from (3.13) that
2 r
J F(\)d\ = 0(r-2a-n+1),
hence,
2 r 2 r
J F(A) A”-1 dX ^ 2n~1rn~1 J F(A) dX ^ c3 r“2".
Now
2h+1r
OO Ç OO
jP(A) An-1dA = ^ / F(A)An~xdA ^c3^2 r fc=o 2{r fc=o
2akr 2a ^ C4r 2a
Hence
and, by (3.6),
00
j F{X) A”'1 dA = 0(r~2a)
r
OO
1 F(X) d[i(X) = 0(r~2a).
We can rewrite (3.2) in the form
W&f-n^h + h,
where
l/t
/! = J \l-<px(t)\2F(\)d»(\),
0
OO
h = J \l-vx(t)\2F(X)dn(\).
1/4
Let us obtain the upper bounds of 7i and 12■ Using (3.14) and |c^>^(i)| ^ 1, we get
OO OO
h = j\l-Mt)\2F(X)d^(X)^4 j F(X) dfj.(X) = 0(t2a).
l/t i/t
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
where
Using the inequalities 1) and 2) of Lemma 4, we obtain
l/t l/t
h = I\l-<px(t)\2F(\)df*(\)^21\l-<px(t)\F(\)dp(\)< 0 0
l/t < 2t2 J(A2 + p2) F(A) rf/x(A) = /3 + /4,
0
l/t h = 2p2t2 J F(A)d/i(A),
0
l/t
74 = 212 J X2F(X)dfi(X).
0
Using the Plancherel formula, we get
OO
73 ^ 2p2t2 j F(A) d/x(A) = 2p2i2ll/Hl = 0(i2Q),
since 2a < 2.
Temporarily we denote
OO
<Kr) := j F(X) dp(\).
Using integration by parts, we get
і/t l/t
/4 = (-r2i)'(r)) dr = 2 +2/ * =
0 0
l/t
= — 2i/> + 4i2 J гф(г) dr.
0
Since i/)(r) = 0(r_2a) (see (3.14)), we have гт/К7") = 0(r1_2a) and
l/t l/t
j rtp(r) dr = o(^J rl 2a drj = 0(t2a 2) 0 0
Hence,
J4 = 0{t2a).
Finally, by (3.15), (3.18), (3.19), (3.20) and (3.21), we obtain
IIS'V -/111 = 0(i2a)
as t —>• 0, i.e. (1.9) implies (1.8). This completes the proof of Theorem 2.
(3.19)
(3.20)
(3.21)
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