On the maximal operators generated by quasiradial Fourier multipliers and its applications to P.D.E.'s

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ON THE MAXIMAL OPERATORS GENERATED BY QUASIRADIAL FOURIER MULTIPLIERS

AND ITS APPLICATIONS TO P.D.E. 'S

YONG-CHEOL KIM

ABSTRACT. Let

e

EC=(JRn \ {O}) be a distance function which is homogeneous with respect to a dilation group{tPh>o. For

f

E 6(JRn), we consider the maximal operators generated by quasiradial Fourier multiplierm⁰

e

which is defined by

Mmoef(x)

=

sup

1;::-1 ^[m

⁰

^(e/t)1] ^(x)1

t>o

wheremis a function given onJR+- Suppose that the sphere Ee^~

{~EJRn

I

e(e)

=

1} satisfies a certain finite type condition and that m vanishes at infinity and satisfies

Jo=

scllm(cl+ll(S)lds ::; C for 8> (n - 1)11/p - 1/21, 1 ::; p ::;00. Then we prove that Mmoe is bounded on £F(JRn) for 1

<

p ::; 00; moreover, it is of weak type (1,1) ..

1.

Introduction

Let

]Rn

be the n-dimensional Euclidean space, denote the inner product in

^]Rn

by (x,

oE) = _L~=l

xioEi for x,

oE E

]Rn, and denote by

]RQ

=

]Rn \

{O}. Let

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(]Rn) be the space of all infinitely differentiable and rapidly decreasing functions on ]Rn. For f

^E

6(]Rn), we denote the Fourier transform of f by

F[f](x) = j(x) = r e-i(x,~) ^f(O ^doE.

JIRn

Received May 20, 1999.

2000 Mathematics Subject Classification: 42BI5, 42B25.

Key words and phrases: quasiradial Fourier multipliers, weak type (1, I)-estimate.

The author was supported by Dong-A University Research Grant and KOSEF.

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Then the inverse Fourier transform of f is given by

Let P be a real n x n matrix. whose eigenvalues ai satisfy Re(ai) > 0;

set a = minl<i<nRe(ai) and A = maxl<i<nRe(ad, and denote by v the trace of P.- Then we consider the dila.tion group {tPh>o in ~n generated by the infinitesimal generator P, where t

^P =

exp(Plogt) for t > O. We introduce quasi-homogeneous distance functions

g

associated with the dilation group {tPh>o; that

is,

g

is

a continl.l,ous function on

~n

with positive values satisfying g(

tP~)

=

tg(~)

for all

~ ^E ~n.

One can refer to [2] and [10] for its fundamental properties. In what follows we shall denote the adjoint of

P

by

P*

and we shall always assume that

g

E Coo (lR8).

IT m(s)

is

defined on (0,00) we consider its extension to

~(j

via

m0

g( e), which is called a quasiradial function, and its corresponding maximal operator defined by

Mmo(!f(x) == sup 1.r-1[m

⁰

(g/t)!](x) I' ^f

^E

6(~n).

t>O

In order to state our result on the maximal operator associated with quasiradial Fourier multipliers, we introduce the notion of fractional derivatives and certain surface condition on dilations of the unit sphere

For w > 0, 0 <

'Y

< 1, and a locally integrable function h on

~

vanishing identically on (-00,0), one can define ( see Gasper and Trebels [3]') the fractional integrals by

I2(h)(s) = r~)X(-oo,W)(s) l

^w

(T - sP-1h(T) dT

where

X(-oo,w)(s)

is the characteristic function on (-oo,w) and r(r)

is

the Gamma function, and define fractional derivatives of order 0 >

0,0= [0] +

'Y

with [0] being the largest integer less than or equal to 0, by

h(-Y)(s)

=

lim !£[-I1-'Y(h)(s)], 0 <

'Y

< 1,

w--.+oo ds

w

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and

whenever the right-hand sides exist; in fact, if /

E

N then h b ) means the classical derivative. In the following we assume that

I~^-"Y

(h)

is

lo- cally absolutely continuous for each w > 0 as well as

h b )

and

h(d).

Note that a heuristic calculation gives F[hCd)](O")

=

(-l)ld ^l (-iO")dh(u),

^0"E ]R.

Let

~

be a smooth convex hypersurface" of

]Rn,

n

~

2, with which every tangent line makes finite order of contact; that is,

~is

called to be of finite type. Let

£(~)

be the set of points of

~

at which the Gaussian curvature

^fi,

vanishes, and let

_N(~) =

{n(e)le

E £(~)}

where nee) denotes the outer unit normal to

~

at e

^E~.

^For ^x

^E ^]Rn,

^{denote by}

B(e(x), r) the spherical cap near e(x)

E ~

cut off from

~

by a plane parallel to

T~(x)(~)

( the affine tangent plane to

~

at e(x) ) at distance r > 0 from it; that is,

where e(x)

is

the point of

~

whose outer unit normal is in the direction x. For t > 0, let

~~

= {e

E ]Rn

I gee) = t} and let Bt(et(x),r) = {e

E

~~ld(e,T~t(x)(~~))

< r}, where et(x)

is

the point of

_~~

whose outer unit normal

is

in the direction x. We say that

~ is

of finite type k, if

k _~

2 is the maximal order of contact on

~.

DEFINITION 1.1.

Let

~e

be a smooth convex hypersurface of finite type and {tPh>o be a dilation group as in the above. Then we say that

(~e'

t

^P) is

uniformly spherically integrable near t

=

1 if there exists some

^f

> 0 such that nE

^E

^Ll(sn-l) ^where

(1.1)

dO"t

is

the surface measure on

~~,

and the polar coordinates on

]Rn

is

denoted by x = r() for r = Ixl and () = x/lxl

^E

^sn-l.

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REMARK.

We shall now give several examples that exemplify the above definition.

(i) Let t

^P

be a general dilation matrix as in the above and suppose that I;u has nonvanishing Gaussian curvature. Then it is obvious that (I;u' t

^P)

is uniformly spherically integrable near t

=

1. (ii) Let us consider a general dilation matrix t

^f

as in the above, Et =

E(I;~),

and M =

N(I;~),

and for

^E

> 0 set NE =

U19~l+EM.

Let I;u be a smooth convex hypersutface offinite type k

~

2 and suppose that there is an

EO

> 0 such that NEo is a m-dimensional submanifold of JRn which is on sn-l, where m < [ken - 1)]j[2(k - 1)]. Then we see ( refer to [7] ) that (I;u, t

^P)

is uniformly spherically integrable near t

= 1.

(iii) Let

t^P =

diag(t

¹/>'1,t 1/>'2, ...

,tl/>'~) where each Ai is even inte- ger, and consider its associated quasi-homogeneous distance functions defined by

e(~)

=

L~=l I~il>'i.

Then it is known in [8] that (I;u' t

^P)

is also uniformly spherically integrable near t

= 1.

THEOREM

1.2. Let (I;u,

^t^P⁾

be uniformly spherically integrable near

t = 1.

Let m be a measurable function on JR+ vanishing at infinity and satisfying

(1.2)

for 8 > (n - 1)11/p - 1/21, 1 ::; p ::;

00.

Then M

mou

is bounded on V(JRn) for 1 < p::;

00;

moreover, it

is

of weak type (1,1).

REMARK.

It is well-known in Dappa and Trebels [2] that under less smoothness condition on e, Le., e E

c[n/2+l]

(JRH), if m satisfies (1.2) for 8> (n-1)(1/2-1/p) then M mou is bounded on V (JRn) for 2 ::; p ::;

00.

Thus we concentrate on the proof of the above theorem on the range 1 ::;

^p

75: 2.

Next we write out a routine result for almost everywhere convergence

of averages F-1[m

⁰

(e/t)j] given by quasiradial Fourier multiplier

mo e, which is controlled by the weak type (1, l)-estimate and V(JRn)-

estimate of

_~.

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COROLLARY

1.3. Suppose that (El?' t

^P) is

uniformly spherically

in-

tegrable near t = 1. Let

m

be

a

measurable function on lR+ vanishing at infinity and satisfying (1.2) for

~

> (n - 1)11/p - 1/21, 1 :s ^p :s

^00.

If f

E LP(Il~.n)

for 1 :s ^p :s

^00,

then we have that

lim ;:-1 [m

⁰

(e/t)j] (x) = ( rX> ^s6 m(H1)(s) dS) f(x) a.e..

t~oo lo

2. Proof of the main theorem

In this section, we prove L

²

(lR

ⁿ

)-bOlmdedness and weak type (1,1)- estimate for the maximal operator mt

moe

generated by quasiradial Fourier multiplier m

⁰

e. ^An application of the complex interpolation theorem for analytic family of operators along the lines of Stein and Weiss [11] leads us to complete the proof of the main theorem. We shall see that there is an interrelation between the maximal operator

Mmol?

and quasiradial Bochner-Riesz means. For f

^E

^6(lR

ⁿ^),

we con- sider quasiradial Bochner-Riesz means

R~,d

of index

~

> 0 defined by

1ifJ(f,)

=

(1 - e(f,)/t)~j(f,)

and the associated maximal operator

~f(x) =supIR~d(x)l·

« t>O «,

Let

J.L

2: 0 and

~

> o. Then it follows from Gasper and Trebels [3] that

for 0 :s

^J.L

<

~.

For f

^E

6(lR

ⁿ^),

by simple calculation we now have that

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and so we have that

(2.1) _Mmoef(x) _~ r(5+ ^l[m]I.s,l

l)'Jl^~

lJ(x).

Thus the proof of the main theorem relies heavily upon the mapping properties of the maximal quasiradial Bochner-Riesz operator

_9Jt~.

In order to obtain weak type (1, l)-estimate for

~,

we recall the lemma [6] about asymptotics of quasiradial Bochner-Riesz kernel.

We now introduce polar coordinates with respect to a

^t^P

-homoge- neous distance function {l

E

Coo

(]Rn),

which is given by the following

diffeomorphism .

~+

x Ee

^~

lRQ, ({l, ()

^I---T

t

^P( = ~,{l

> 0, (

E

E _e.

It follows from this that the transformation rule of the Lebesgue mea- sure df.is given by

where dCF denotes the surface measure on

Ee

and n(() is the outer unit normal vector to E e at (.

Fix

some (0

E

_e•

Then the unit sphere

E

e can be parametrized near

(0

by a map

W I---T ~(w),W E ]Rn-I,

Iwl <

1

such that

~(O) = (0.

Then there is a neighborhood U o ^of

⁽⁰

with compact closure, a neighborhood Vo of the origin in

]Rn-I,

and an interval 1

₀ =

(1 -

fO,

1 +

^fO)

so that the map

Q :

1 0 x Vo

~

Uo, ({l, w)

^I---T

Q({l, w) =

(lP~(w)

is a diffeomorphism with Q(l,O) =

(0.

The Jacobian of Q is given by J({l, w)

= (l';-l(p~(w), n(~(w)))

9l(w),

where 9l(w) is positive and

[9l(W)]2

=

det ([r []) .

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LEMMA

2.1. Let (o,Uo, and

~

be as above. Then there is co> 0, a neighborhood U1 of(0 whose closure is contained in Uo, a neighborhood V1 ofthe origin

in~n-l,

and an interval 1

₁

=

(1-f1l1+f1)

with

f1

«

¹

such that

Q

maps 1

₁x

V1 toU!, and such that for

^all

X

E

Ccgo(U1) and for all t

E

1

₁

and

N-l

]="-1[(1_ e/t)~x](x)⁼

L

^{P£(x, t)}

+ O(lxl-

^N)

for I{x/lxl,

n«(o)}

I

_~1-eO

£=0

where

and ql E

Ccgo(~(Vo)

x 1I x sn-1) for.e

=

0,1, ... , N -

1. In

particu- lar, qo((,t,x/lxl) = C(o)X(tp()tV{P(,n(()>[{PtP 1~1,()rO-1 ^where C(O)

=

r(o +

1)

e-

^i1r(0+1)/2.

As

you can observe in the above lemma, the decay estimate for the Bochner-Riesz kernel

is

closely related with the Fourier transform of the unit sphere El? So it is very natural to compare the relative sizes of the spherical caps on dilations of El? with respect to a dilation group {tPh>o which is introduced in section 1. We now recall the following lemma [7] for this natural need.

LEMMA

2.2. There is an

fO

> ° ^{such that}

(i)

_Bt(~t(rP*

x),r/2)

C

t P [B1(6(x),r)]

^C Bt(~t(rP*

x),2r), (ii) a1 [B1(6(x),r)]/2 :S at [tP(B1(6(x),r))] :S 2a1 [B1(6(x),r)], and

(iii) at

_[Bt(~t(x),

r)]/2 :S a1

[rP(Bt(~t(x),

r))] :S 2 at

_[Bt(~t(x),

r)]

for any t

E [1-fO,1

+

^fO]

and for any x

E ~o

We also see that the proof of the doubling property in [1] leads us

to obtain the stronger estimate for the surface measure da on El? as in

the following lemma.

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LEMMA

2.3. Let Ee be a smooth convex hypersurface of Enite type. Then there is a constant C > 0 so that

(T [B(~(x),

Ar)] ::;

C

A(n-1)/2(T[B(~(x),

r)] for any x

E

IRQ, for any r > 0, and for

A?:

1. We learned from (1.2) and (2.1) that the study of the maximal operator M

moe

generated by quasiradial Fourier multipliers

mog

relies heavily upon that of the maximal quasiradial Bochner-Riesz operator

9J1~.

So we prove weak type (1, I)-estimate and V"(IRn)-boundedness of

~

for 1 <

p ::;

2

in

the following lemma.

LEMMA 2.4.

Let Ee be a smooth convex hypersurface of Enite type and {tPh>o be a dilation group as in the above. If(Ee,

^t

P ) is uniformly spherica11y integrable near t

= 1,

then

9J1~

is bounded on LP(IRn) for d> (n - 1)(I/p - 1/2),

1

< p::; 2; moreover, it

is

of weak type (1,1) for d > (n - 1)/2.

Proof. Let K(x) = .1"-1[(1 - g)t](x) and Kt(x) = F-

¹

[(1 - g/t)t](x) for t > o. Then we have Kt(x) = tVK(t

^P-

x). Choose

10

=

^lOO

> 0 with

10

« a satisfying (1.1) and such that for any t

E

[1 -

lOO,

1 +

^lOO]

(2.2) inf It

^P

1 ^-I

xl ?: 1/2 and sup It

^P^/

^-I ^XI ^::; ^2.

xERO' X xERO' X

Since fC

E

Loo(lRn) is compactly supported,

in

order to get weak type (1, I)-estimate for

9J1~,

it suffices by the results in [7] to show up that

{ sup /Kt(x)1 dx <

⁰⁰

lRn

^19::;l+€

and

:r _~ sup L 1 ^sup ^{IKt(x)1 dx} ^<

^00.

yERO' {hl(l+€)h r(y»l} r(x)2':(l+€)h r(y) 199+€

By Lemma 2.1 and Theorem A [1], we obtain that

IK(x)/ ::; (1 + C /xl)8+1

^(Tl

[B

¹

(6 (x), I/Ixl)]·

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So we have that for any t

E

[1- €, 1 + €],

C P* P*

(2.3) IKt(x)1 ~ (1 + _Ixl)8+1 udB1(6(t x), lilt ^xl)].

It also follows from (i) of Lemma 2.2 that

B1(6(t Px), lilt?" ^xl) c CP[Bt((t(x), 2/1tPxl)].

By (2.2), we get that for any t

E

[1 - €, 1 + €],

1 ~ ² _{It xl} ^P* ⁼ ^Ao ^< _- ^{2 sup}

_xElRlJ

^I _t ~I _x ^I ⁼

_.nf

² _{It xl} ^P* ~ ^4,

sup - -

¹ ^{- -}

xElRC'

Ixl

^xElRlJ

Ixl

where Ao

=

2

infXElRlJ

Itl:lxl. Combining this with Lemma 2.3,

P*

=.!.

Ut [Bt((t(x), 2/1t xl)]

_~

2C.AQ

²

Ut [Bt ((t (x), I/lxl)].

Thus by (i) and (iii) of Lemma 2.2, we have that (2.4)

U1 [B1(6 (tPx), I/lt* Pxl)]* _~ U1 [CP(Bt((t(x), 2/1tP* xl))]

~ 2Ut [Bt((t(x), 2/1tP

*

xl)]

n - l

~

4CAo-r Ut [Bt((t(x), I/lxl)].

From (2.3) and (2.4), we get that for all t

E

[1 - €, 1 + €], IKt(x)1 ~ ⁽¹ + c Ixl)8+1 Ut [Bt((t(x), I/lxl)].

Since

(I;(»

t

^P⁾

is uniformly spherically integrable near t = 1, using OE(O) defined in (1,1) we have that

sup IKt(x)1

l::;t::;l+E

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If

0> (n -1)/2, then since ne E L1(sn-l) we have that

<

^00.

Finally if 0> (n - 1)/2, then we have that

Thus we conclude that

~ is

of weak type (1,1) for 0 > (n -1)/2.

Finally we utilize the standard linearization technique of the

maxi-

mal operator in order to employ the Marcinkiewicz interpolation theo-·

rem. It now suffices to consider the analytic family of linear operators given by

z ^1---+ 'R~,t(x)f(x)

where x

^1---+

t(x)

is

an arbitrary measurable function on lRn with values in lR+. Combining this with the above weak type (1, I)-estimate for

_~,

we get V(lRn)- boubdedness of

_9J1~

near

p =

1. By the complex interpolation method, we interpolate this estimate with L

²

(lRn)-boundedness for

~

on 0 > O. Therefore we

complete the proof. 0

3. Applications to P.D.E. 's

In this section, we discuss about applications of Theorem 1.2 to

several partial differential equations.

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(a) It is well-known that the solution U of the Schrodinger initial value problem

t::.U =

ⁱ

~U,

^t

^> ^0, ^U(x,O) ⁼ ^{f(x), f}

^E

^6(IR

ⁿ⁾

is given by U(x, t)

=

(21r)-n fJRn

ei(x'~)eitl~121(~)~,

and can be ex- tended to all of V(IRn) only

ifp

= 2 ( see [5] ). Sjostrand [9] studied V(IRn)-boundedness of the a-th Riesz means of U(x, t) defined by

Rf(x)

=

~Q~11 ^lot (t - sy'U(x, s) ds.

Then we have that

So the associated Fourier multiplier is

mQ(tl~12)

⁼

~Q~11 ^lot ^{(t -} s)Qeisl~12 ^ds + 1

_rtl~12

=

~Q+l 10 (tl~12 ^- ^{S)QeiS ds.}

Now we consider the generalized Schrodinger operator defined by

for t > 0 and x

E

IRn, where (Ee, t

^P)

is uniformly spherically integrable near t = 1, and a-th Riesz means of Se,t[f](x) defined by

R~,t(x) ⁼ ~Q~11 Lt (t - s)QSe.tff](x) ds.

....

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Similarly to the above, the associated Fourier multiplier is

a + 1 taW .

mQ(tl!(~»

= _tQH lo

(tl!(~)

- S)Q e

^2S

ds.

Note that ma(t) = ~t~ f~(t - s)aeis ds = (a + 1) f0

¹

(1 - s)aeist ds.

We see that the major contribution of the asymptotic decay for the function m~H)(t) comes from some small intervals near t

=

° ^and

t

=

1. Using a function ifJ

E

coo(lR) such that ifJ(s)

=

1 for s 2:: 3/4 and ifJ(s) = ° ^{for s::;} ^1/2, we split the function m~+1)(t) as follows;

m~H)(t)

⁼

^C 1 ¹ ^(1- ^{s)a s}

^8H

^{eist ds}

=

va(t) + wa(t),

whereva(t) = C f:(1-s)a s

^H1

eistifJ(s) and wa(t) = C f~ (1 - s)a sH1 eist (l-ifJ(s» ds. It then follows from an usual asymptotic formula

in

A. Erdelyi ( Asymptotic Expansion, Dover, 1956 ) that va(t)

^rv

1/(1 + t)a+1 and wa(t)

^rv

1/(1 + t)H2. So we have that for a > 8 >

(n - 1)11/p - 1/21,

fa1 SO

Im~+1)

^(s)1 ^ds

~ c 10

⁰⁰^SO ^[(1

⁺

^s)-(a+1)

⁺

⁽¹

⁺

^s)-(0+2)] ^ds

^<

^CXJ.

Thus by Theorem 1.2 we conclude that if a> (n -1)11/p -1/21, then M

_maoa

is bounded on LP(lR

ⁿ)

for 1 < p ::;

00;

moreover, it is of weak type (1,1) for a > (n - 1)/2.

(b) We now consider the heat initial value problem

(~ ^- ~) ^U

⁼

^O,t ^> ^0, ^U(x,O)

⁼

^f(x),f

^E

^6(lR

ⁿ^).

Then it is well-known that the solution U is given by U(x, t)

= _1_ [

ei(x,{)

e-tl{l^z

j(~) d~.

(27r)n lIR

ⁿ

Suppose that (Ea' t

^P⁾ is

uniformly spherically integrable near t

=

1. Then we define the generalized heat operator by

lla,t[f](x)

⁼

(2~)n Ln ei(x,{) e-ta({) j(~) ~

(13)

for t > ° ^and ^x

^E ^]Rn.

We easily see that the associated Fourier multi- plier is m(te(e)) where m(s)

=

e-

^s.

Since m(s) trivially satisfies (1.2) for all 8 > 0, by Theorem 1.2 we conclude that the associated maximal operator M

_moe

is bounded on

LP(]Rn)

for 1 <

p ::; 00;

moreover, it is of weak type (1,1).

(c) Suppose that (Ee, t

^P)

is uniformly spherically integrable near

t

= 1. Since m( s) = (1 - e-

^S^{) /}

s obviously satisfies (1.2) for all 8 > 0, by Theorem 1.2 we have that

and

which may be useful to prove almost everywhere convergence theorems of Voronovskaya type for the generalized Weierstrass means (see [4]).

(d) Suppose that (Ee, t

^P)

is uniformly spherically integrable near t

=

1 and let m6(s) = (1- s)t. Then we get that m~>")(s)

⁼

C(l- s)~->") for). < 8 + 1. By Theorem 1.2, we have that

_~

is bounded on V

(]Rn )

for 8>

(n

-1)11/p-1/21, 1 <

p _~ 00,

and also it is of weak type (1,1) for 8 >

(n -

1)/2. This result coincides with that of Lemma 2.4.

References

[1] J. Bruna, A. Nagel and S. Wainger, Convex hypersurfaces and Fourier trans- forms, Annals of Math. 127 (1988), 333-365.

[2] H. Dappa and W. Trebels, On the maximal junctions generated by Fourier multipliers, Ark. Mat. 23(1985), 241-259.

[3] G. Gasper and W. Trebels, A characterization of localized Bessel potential spaces and applications to Jacobi and Hankel multipliers, Studia Math. 65 (1979), 243-278.

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On the maximal operators generated by quasiradial Fourier multipliers and its applications to P.D.E.'s