Spherical harmonics and spherical averages of Fourier transforms

Spherical Harmonics and Spherical Averages of Fourier Transforms.

PER SJO¨LIN(*)

ABSTRACT - We give estimates for spherical averages of Fourier transforms of functions which are linear combinations of products of radial functions and spherical harmonics. This generalizes the case of radial functions.

1. Introduction.

We shall here study Fourier transforms inRⁿand we shall always as- sume nF2 . Let u denote the area measure on Sⁿ²¹ and set

s(f)(R)4

Sⁿ²¹

Nf×(Rj)N²du(j) , RD1 ,

wheref×denotes the Fourier transform of a function fL¹(Rⁿ). We are interested in estimates of the type

s(f)(R)GCR^2b

Rⁿ

Nf×(j)N² 1 NjN^n2a

dj, RD1 . (1)

For 0EaGn we shall consider the statement .

/´

there exists a constant C4C_a,b such that ( 1 ) holds for all fC₀^Q(Rⁿ) with supp f%B₁and fF0 . (2)

Here B₁ denotes the unit ball in Rⁿ.

(*) Indirizzo dell’A.: Department of Mathematics, Royal Institute of Techno- logy, S-100 44 Stockholm, Sweden. E-mail: persHmath.kth.se

We set b₁(a)4sup]b; ( 2 ) holds(. The number b₁(a) has been studied in Mattila [2], Sjölin [4], Bourgain [1], and Wolff [9]. In the case n42 it is known that

b₁(a)4

. /

´

a, 1 /2 , a/2 ,

0EaG1 /2 1 /2EaG1 1EaG2

(see [2] and [9]). FornF3 one knows thatb₁(a)4a for 0EaG(n2 21 ) /2 , max((n21 ) /2 ,a21)_Gb₁(a)Gmin (a,a/21n/221 ) for (n2 21 ) /2EaEn, and b₁(n)4n21 (see [2] and [4]).

Results of this type have applications in geometric measure theory in the study of distance sets.

In Sjölin and Soria [6], [7],u is replaced by general measures and in these papers one also studies the case when the conditionfF0 in (2) is removed.

The case whenfis also assumed to be radial is studied in Sjölin [5].

We shall here generalize the case of radial functions. We recall that L²(Rⁿ)4

!

k40 Q

5H_k, where H_k is the space of all linear combinations of functions of the formfP, wherefranges over the radial functions andP over the solid spherical harmonics of degree k, so that fP belongs to L²(Rⁿ) (see Stein and Weiss [8], p. 151).

Now fixkF0 and letP₁,P₂,R,P_akbe an orthonormal basis for the space of solid spherical harmonics of degreek (where we use the inner product in L²(Sⁿ²¹)). The elements in H_k can be written in the form

f(x)4

!

j41 a_k

fj(r)Pj(x) (here r4 NxN) (3)

and

Rⁿ

Nf(x)N²dx4

!

1 ak

0 Q

Nf_j(r)N²r^n12k21dr.

We letR denote the class of all functionsgon [ 0 ,Q), which satisfy the

following conditions:

g(r)F0 g is C^Q g(r)40

for rF0 , on ( 0 , Q) , for rD1 , and

there exists eD0 such that g(r)40 for 0GrGe.

We say that fSk, k40 , 1 , 2 ,R, if f is given by (3) with all f_jR^.

For 0EaGn we shall consider the statement:

there exists C4Ca,b,k such that (1) holds for all fSk. (4)

We then set b(a)4bk(a)4bn,k(a)4sup]b; (4) holds(. We have the following result.

THEOREM. For k40 , 1 , 2 ,R,we haveb_k(a)4afor0EaGn21 , and b_k(a)4n21 for n21EaGn.

We shall first give a proof of the theorem which works directly for all kF0 . Another possibility is to first treat the case k40 (i.e. the case of radial functions), and then use the casek40 to study the casekF1 . We shall also say something about this second approach.

2. Proofs.

Iffis a function on [ 0 ,Q) we shall also use the notationffor the cor- responding radial function in Rⁿ. We also let Fn denote the Fourier transformation in Rⁿ.

PROOF OF THE THEOREM. Assume that f is given by (3) with all f_j belonging to R. It then follows from [8], p. 158, that

f×(x)4

!

1 ak

F_j(r)P_j(x) (here r4 NxN) ,

where

F_j(r)4c_kr^12n/22k

0 Q

f_j(s)J_n/21k21(rs)s^n/21kds, rD0 , and J_m denotes Bessel functions.

For NjN41 we obtain f×(Rj)4

!

j

F_j(R)P_j(Rj)4R^k

!

j

F_j(R)P_j(j) and hence

s(f)(R)4R^2k

Sⁿ²¹

N ^!

N

^!

We also have

Rⁿ

Nf×(j)N²NjN^a2ndj4

0 Q

u

Sⁿ²¹

Nf×(rj8)N²du(j8)

v

4

0 Q

r^2k

g ^!

h

^!

0 Q

NF_j(r)N²r^2k1a21dr

4

!

j

Rⁿ

NF_j(r)N²NjN^2k1a2ndj (where r4 NjN in the last integral).

It follows that the statement (4) is equivalent to the statement:

if fR and

F(r)4c_kr^12n/22k

0 Q

f(s)J_n/21k21(rs)s^n/21kds, rD0 , (5)

then

R^2kNF(R)N²GCkR^2b

Rⁿ

NF(r)N²NjN^2k1a2ndj, RD1 . (6)

Now assume that fR ^{and that F} is given by (5). It is then clear that

F(r)4c_kFn12kf(r) (7)

and since fC₀^Q(R^n12k) it follows that FS^{(Rn12k), where} S ^denotes the Schwartz class. Assume that 22n22kGbG1 . Thenn/21k21F F2b/2 and it follows that

NJ_n/21k21(s)N GCs^2b/2, sD0 (cf. [8], p. 158).

Inserting this estimate in (5) we obtain

NF(r)N GCr^12n/22k

0 1

f(s)(rs)^2b/2s^n/21kds

4Cr^12n/22k

0 Q

f(s)W(s)(rs)^2b/2s^n/21kds,

where WC₀^Q( 0 , Q) , WF0 and W(s)4^./

´ 1 , 0 ,

0EsG1 sF2 . The Fourier inversion formula implies that

f(s)4c_ks^12n/22k

0 Q

F(t)J_n/21k21(st)t^n/21kdt and hence

NF(r)N GCr^12n/22k

0

Q

u

0 Q

F(t)J_n/21k21(st)t^n/21kdt

v

QW(s)(rs)^2b/2s^n/21kds

4Cr^12n/22k2b/2

0 Q

F(t)t^n/21k

u

0 Q

W(s)s^12b/2J_n/21k21(st)ds

v

We conclude that

r^kNF(r)N GCr^12n/22b/2

0 Q

F(t)t^n/21kI(t)dt, (8)

where

I(t)4

0 Q

W(s)s^12b/2J_n/21k21(st)ds, tD0 .

We shall now estimate I(t). Setting g4b/21n/21k21 we obtain I(t)4t^n/21k21t^12n/22k

0 Q

W(s)J_n/21k21(st)s^n/21ks2n/22k112b/2ds

4c_kt^n/21k21Fn12k(W(s)s^12b/22n/22k)(t) 4c_kt^n/21k21Fn12k(W(s)s^2g)(t) , tD0 .

First assume 22n22kEbG1 . ThengG1 /21n/21k21En12k andgD12n/22k1n/21k2140 . It follows thatFn12k(W(s)s^2g)4 4c_k(Fn12kW)˜s^2n22k1g, where the convolution is taken inR^n12k. Since Fn12kWS^(Rn12k) we obtain

NFn12k(W(s)s^2g)(t)N GC( 11t)^2n22k1g, tD0 . (9)

In the remaining case b422n22k we have g40 and it is clear that (9) holds also in this case.

We have n12k2g4n12k2b/22n/22k114n/21k2b/211 , and hence

NI(t)N GCt^n/21k21( 11t)^2n/22k1b/221, tD0 .

ThusNI(t)N GCt^n/21k21for 0EtG1 , and NI(t)N GCt^b/222 for tD1 . Invoking (8) we then get

r^kNF(r)N GCr^12n/22b/2

0 Q

NF(t)Nc(t)dt, rD0 , (10)

where c(t)4t^n12k21 for 0EtG1 , and c(t)4t^n/21k1b/222 for tD1 . Using the Schwarz inequality we obtain

0 Q

NFNcdtG

u

0 Q

NF(t)N²t^2k1a21dt

v

(11)

u

0 Q

c(t)²t^22k2a11dt

v

We have

0 1

c(t)²t^22k2a11dt4

0 1

t^{2n14k2222k2a11}dt

4

0 1

t^2n12k2a21dtE Q

since 2n12k2aFn. On the other hand we also have

1 Q

c(t)²t^22k2a11dt4

1 Q

t^{n12k1b2422k2a11}dt

4

1 Q

t^n1b2a23dt, which is finite if n1b2a23E 21 i.e. bE21a2n.

Invoking (10) and (11) we conclude that r^2kNF(r)N²GCr^22n2b

0 Q

NF(t)N²t^2k1a21dt, rD0 ,

if 22n22kGbG1 and bE21a2n. Setting M4 ]b; 22n22kG GbG1 and bE21a2n( we obtain

b(a)Fsup

bM

(b1n22 )4n221supM. (12)

Then assume 0EaGn21 . We have 22n22kE21a2nG1 , and it follows that supM421a2n and thus b(a)Fn22121a2n4a in this case.

Then assumen21EaGn. In this case 21a2nD1 and it follows that supM41 . Invoking (12) we obtain b(a)Fn21 .

Thus we have obtained lower bounds for b(a). We shall now obtain upper bounds, and we first assume 0EaGn21 . Also assume that (6) holds for allF given by (5) with fR. We shall prove that then bGa. First choosefR^withfg0 . Then there existsbD0 such thatF(b)c0 . Also set fa(s)4f(as), aD1 , and

F_a(r)4c_kr^12n/22k

0 Q

f(as)J_n/21k21(rs)s^n/21kds, rD0 .

Performing a change of variable as4t we obtain F_a(r)4c_kr^12n/22k

0 Q

f(t)J_n/21k21(rt/a)t^n/21kdt a^2n/22k21

4a^2n22kF(r/a), and (6) yields

r^2ka^22n24kNF(r/a)N²GCr^2b

0 Q

NF(r/a)N²r^2k1a21dr a^22n24k. Performing a change of variable we then get

r^2kNF(r/a)N²GCr^2b

0 Q

NF(s)N²s^2k1a21ds a^2k1a (13)

4Cr^2ba^2k1a

for allaD1 andrD1 , whereCdepends on fbut not onaorr. We now choose a4r/b, where r is large, and it follows from (13) that

r^2kNF(b)N²GCr^2br^2k1ab^22k2a.

We conclude thatr^bGCr^aand it follows thatbGa. Henceb(a)Ga for 0EaGn21 and we have proved that b(a)4a in this case.

It remains to study the casen21EaGn. Assume as above that (6) holds for all fR^.

Let L₁denote the class of all fL¹[ 0 ,Q) withfF0 and satisfying f(r)40 for rF7 /8 and f(r)40 for 0GrGe for some eD0 . It is then easy to see that (6) holds also for allfL₁. In fact, this follows from ap- proximation of fL1 with f˜W_e, where the convolution is taken in R^n12k and W_e is an approximate identity in R^n12k.

Then choose WC₀^Q( 0 , Q) with suppW%( 1 /2 , 7 /8 ), WF0 , and W( 3 /4 )41 . Also set

f(s)4fR(s)4e^2iRsW(s) , sD0 , where R is large. Then

f4f₁2f₂1if₃2if₄,

wheref_jL₁^andfjG NfNforj41 , 2 , 3 , 4 . LetF_jcorrespond tof_jin the same way asF corresponds tofin (5). ThenF_j4c_kFn12kf_j and since (6)

holds for F_j we obtain R^2kNF(R)N²GCR^2k

!

j NF_jN²GCR^2b

!

j

0 Q

NF_j(r)N²r^2k1a21dr

4CR^2b

!

j

R^n12k

NF_j(r)N²NjN^a2ndj.

In the case n21EaEn we invoke Lemma 12.12 in Mattila [5], p.

162, and then get

R^2kNF(R)N²GCR^2b

!

j

R^n12k

R^n12k

Nx2yN^2a22kf_j(x)f_j(y)dx dy

GCR^2b

R^n12k

R^n12k

Nx2yN^2a22kNf(x)N Nf(y)Ndx dy. Hence

R^2kNF(R)N²GCR^2b, (14)

whereCdepends on Wbut not onR. In the casea4n(14) follows from an application of the Plancherel theorem.

We have

R^kF(R)4cR^12n/2

0 Q

e^2iRsW(s)J_n/21k21(Rs)s^n/21kds and we shall use the asymptotic formula

J_n/21k21(t)4c₁e^itt^{21 /2}1c₂e^2itt^{21 /2}1O^{(t23 /2) , t}_{K Q} (see [8], p. 158). We obtain

R^kF(R)4cR^12n/2

0

1

y

(Rs)^{1 /2} 1O((Rs)^{23 /2})

z

Qs^n/21ke^2iRsW(s)ds

4cc₁R^{1 /22n/2}

0 1

s^{n/21k21 /2}W(s)ds

1cc₂R^{1 /22n/2}

0 1

e^22iRss^{n/21k21 /2}W(s)ds

1O^{(R21 /22n/2)}_F^{cR1 /22n/2} and hence

R^2kNF(R)N²FcR¹²ⁿ for large values of R.

The formula (14) then yields

R¹²ⁿGCR^2b i.e.

R^bGCRⁿ²¹

and we conclude that bGn21 . Hence b(a)Gn21 for n21EaGn and it follows thatb(a)4n21 in this case. The proof of the theorem is complete.

We shall finally discuss how results for radial functions (i.e. the case k40) can be used to study the casekF1 . Therefore assumenF2 ,kF1 and 0EaGn. If fR ^{and F} is given by (5), then the estimate

NF(R)N²GCR^2b

0 Q

NF(r)N²r^a12k21dr

is equivalent to the estimate

R^2kNF(R)N²GCR^2(b22k)

0 Q

NF(r)N²r^a12k21dr. It follows that

bn,k(a)4bn12k, 0(a12k)22k. (15)

Assume then that we know that bn, 0(a)4min (a,n21 ) for all nF2 and 0EaGn. For kF1 , nF2 and 0EaGn (15) then yields

bn,k(a)4min (a12k,n12k21 )22k4min (a,n21 ) , which is the desired formula.

R E F E R E N C E S

[1] J. BOURGAIN, Hausdorff dimension and distance sets, Israel J. Math., 87 (1994), pp. 193-201.

[2] P. MATTILA,Spherical averages of Fourier transforms of measures with fi- nite energy; dimension of intersections and distance sets, Mathematika,34 (1987), pp. 207-228.

[3] P. MATTILA,Geometry of sets and measures in Euclidean spaces, Cambridge studies in advanced mathematics,44, Cambridge University Press, 1995.

[4] P. SJO¨LIN,Estimates of spherical averages of Fourier transforms and dimen- sions of sets, Mathematika,40 (1993), pp. 322-330.

[5] P. SJO¨LIN,Estimates of averages of Fourier transforms of measures with fi- nite energy, Ann. Acad. Sci. Fenn. Math., 22 (1997), pp. 227-236.

[6] P. SJO¨LIN- F. SORIA,Some remarks on restriction of the Fourier transform for general measures, Publicacions Matemàtiques, 43 (1999), pp. 655-664.

[7] P. SJO¨LIN- F. SORIA, Estimates of averages of Fourier transforms with re- spect to general measures, Manuscript 2000.

[8] E.M. STEIN - G. WEISS, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, 1971.

[9] T. WOLFF,Decay of circular means of Fourier transforms of measures, In- ternat. Math. Res. Notices (1999), pp. 547-567.

Manoscritto pervenuto in redazione il 2 febbraio 2002.