KUROKAWA, TAKAHIDE

Osaka City University Repository

Kurokawa, T.
Osaka J. Math.
27 (1990) 721-738
ON HYPERSINGULAR INTEGRALS
Dedicated to Professor Tatsuo FujiTe on the
occasion of his 60th birthday
TAKAHIDE KUROKAWA
(Received October 11, 1989)
l Introduction
The purposes of this paper are to generalize a result of E.M. Stein con-
cerning hypersingular integrals and to give its application. The same problems
for singular difference integrals are discussed in [2]. Let Rn be the n-
dimensional Euclidean space and for each point x=(xly •••,#„) we write \x\ =
(#i+ +#ίί)1/2. For a multi-index *y=(γ1, •••, γn) we denote |7 |=7iH h%»
7 ! = ^ ! ... γ
u
\9 χ
y
=χ{ι.... χl» and D v=Dϊi D2». The Riesz kernel κ
Λ
 of order
a>0 is defined as
j \x\* n, a<n or a^n, a—n not e
=
 I (8-..-log|*|)|*r--, a^n, ct-n
: even
i even
where δ^,, is the constant (see [1]).
For β>0, £>0 and a positive integer /, we consider the integral (hyper-
singular integral)
where (Rιtu) (x) is the remainder of order /, namely
(Rlu)(x) = u(x+t)-(
We denote μ,ι-*-\x) = H[*ιc
Λ
(x). E.M. Stein showed in [3; p. 162] that
/ ' • ' G L 1 for 0 < α < 2 . In §2 we generalize this result (Theorems 2.3 and
2.14). R.L. Wheeden [4; Theorem 1] established an inequality of the following
type for functions u in Bessel potential spaces:
where || 1^  is the L^-norm and || \\Ptβ is the norm in Bessel potential spaces.
722 T. KUROKAWA
In § 3 we give integral estimates of Hιt'βu for Beppo Levi functions a (Theorem
3.7).
Throughout this paper, the letter C is used for a generic positive constant
whose value may be different at each occurrence.
The author expresses his thanks to Prof. Fumi-Yuki Maeda who gave the
proof of Lemma 2.12.
2. Hypersingular integrals of Riesz kernels
For a positive integer /, Cι denote the space of all / times continuously
differentiable functions on Rn. For a point t in Rn we write t'=tj\t\ By
Taylor's formula we have
Lemma 2.1. // ψ e C\ then
(Rιtφ)(x) =
For x,ytER\ we put LXt,= {sx+(l—s)y; O^s^l} and Lx=LXt0. We
denote by d(y, L
x
) the distance between y and L
x
. By Lemma 2.1 we obtain
Corollary 2.2. (i) If φ e C \ then
\{R\φ){y)\^\t\ι Σ (1/7!) max \D>φ{z)\.
(ii) If l>a—n, then for d(y, L_t)^\t\βwe have
Proof, (i) By Lemma 2.1 we have
Σ ( / ! ) max \D*φ{»)\
IYI=/ ί e £ y > y + / Jo
(ii) We note that if d(y, L_t)^\t\/2, then
\y\β£\y+s?\
for O^ί^ 111. Further, if 171 =/ and l>a—n, then
HYPERSINGULAR INTEGRALS 723
Hence by Lemma 2.1, for d(y, L_t)^> \t\β we have
m=/Jo γ !
Σ
I7l=/Jθ
\y\ [
ιγι=/ Jo
= C\t\'\y\"-'-'.
For a>0, β>0 and a positive integer /, we set
and
We note that /Λ^  **^^) is finite for β>a—n, β>l—\ and y
2.1. Integrability of fJ **β
For K ί ^ o o , we denote the conjugate exponent to q by q\ namely, (1/?)+
')= 1. At first, we prove
Theorem 2.3. Let K ? ^ o o and β>a—(njq).
(i)
(ii) J/ / ίί an odd number, then for l—l<a—(nlq)<l+ί we have μ'
Proof, (i) By Minkowski's inequality for integrals we obtain
Further, we have
724 T . KUROKAWA
Since l>a—(njq)>a—ny it follows from Corollary 2.2 (ϋ) that
( \y\
l.yl^ l/t/2
Moreover it follows from a—(nlq)>l—1^0 that
\t\ *~(n/q) a—n is not a nonnegative even number
111 --(«/f>(l -}-1 log I ί I I), α—w is a nonnegative even number .
Thus
j
—' — at
 y a—n is not a nonnegative even number
S ^ J v T I °51 1 1 ^ α — n is a nonnegative even number.\t\n+β
By the condition β>(X—(nlq)> we obtain 7<oo.
(ii) Let / be a n odd number. We have
Q' \l/ί'
Since / is an odd number, we see that
Hence we have
dt
Note that | y \ ^ 3 1 / 1 β implies rf(y, L . , ) ^ 111/2. Since /+ l>a-(n/q)>a-n
and β>a—(n/q)> we obtain from Corollary 2.2 (ii)
HYPERSINGULAR INTEGRALS 725
$
In the same way as in (i) we obtain J2<°° since a—{njq)>l—1^0 and β>
a—(njq). The proof of the theorem is complete.
2.2. Integrability of /*'•*•*
By Taylor's formula we have
Lemma 2.4. If P is a polynomial of order m and l>m, then (RltP)(x)=0.
The symbol 3) denotes the LF-space of all O°°-functions with compact
support and 3)' stands for the topological dual of 3). For/eiZ), the Riesz
potential of order a of / is defined as
Uί(x)=\κ
ct(x-y)f(y)dy.
We easily see
Lemma 2.5. Let f^W. Then
(i)
(ii) For \y\<a
(iii) If I γ | = / is odd, then
exists for all x and
τtix) = lim
If a—n is not a nonnegative even number, then the Riesz kernel κ
Λ
(x) is
homogeneous of degree α—n. Hence
(2.1) D-tfc^Xx) = x'-w-'Dt^x), λ>0.
When a—n is a nonnegative even number, we have
Lemma 2.6. Let a—n be a nonnegative even number. Then
726 T. KUROKAWA
withX
Λ
{x)=\x\*-*.
Proof. By Leibniz's formula we see that
withyC8=7!/δ!(γ-δ)!. For δ>0, we have
where P^x) is a homogeneous polynomial of degree 181. Hence we have
= ( -
Thus we obtain the lemma.
Lemma 2.7. Lei / fo α positive integer, and moreover we assume that />
a—nin case a—nis a nonnegative even number. Then
(i) (B\κ.){y)=\t\m-\R\,Km){yl\t\),
(ϋ) (#«.)(?)= IyI—(fl/wOOO-
Proof, (i) If α—w is not a nonnegative even number, then by (2.1) we
have
If a—n is a nonnegative even number, then by Lemma 2.6 we see that
HYPERSINGULAR INTEGRALS 727
iγ Σ ^
log 111 )D1X
a
{yl 111 )+D>κJίyl | ί | ))(ί')γ }
= 1*1 " " " ( - l o g 111 ) ( Λ { ' 3 C ) ( y / 1 < I ) + 1 * I β - " ( - R ί ' « < . ) 0 ' / 1 * 1 ) .
Since l>a—n, by Lemma 2.4 we obtain the required equality.
Using (2.1) and Lemma 2.6 we can prove (ii) in the same way as in (i).
We complete the proof of the lemma.
Lemma 2.8.
(i) Let l<a-\-\, and moreover we assume that l>a—nin case a—nis a nonnega-
tive even number. Then
(R\ Uί)(x) = I f Γ j (Rl>*
m
)(x)f(x-111 z)dz.
(ii) If I is an odd number, then
(Rιt+ι W){x) = 111' Hm
Proof, (i) Since / — l < α , by Lemma 2.5 (ii) we have
(R'tUί)(x) =
= j M*+t-y)-
n
Σ_
ι
(D'*Jί*-y)lΎi)Wy)dy
Furthermore it follows from Lemma 2.7 (i) that
{R\Ui){x) = 111-- J (#,«
(ii) By Lemma 2.5 (ii) and (iii) we have
728 T. KUROKAWA
{x) = Uf,(x+t)- ^ (OT{
= J <c,(x+t-y)f(y)dy-,yΣJ
By Lemma 2.7 (i) we see that
(R>t+1U',)(x) = lim 111'- j ( ^ e i ( i {R'S\)(yl 11 \ )f{x-y)dy
= Um 1*1* \^{R>t<"κ,){z)f{x-\t\z)dz.
Thus we obtain the lemma.
Corollary 2.9. (i) If a<l«x+\, then {R\κ
Λ
){y) is integrable as a func-
tion of y and for all
(ii) // / is an odd number, then (R^Wy) is integrable on {\y\^S} (£>0)
and for all
(2.2)
Proof, (i) We easily see that (R!tκΛ)(y) is locally integrable for / < α + l .
It follows from Corollary 2.2 (ii) that
\*-l-n
for \y\ ^ \t|/2. Hence for α < / < α + l , (R^^iy) is integrable with respect to
y. Next we shall show
Since
by Lemma 2.7 (i), it suffices to show
HYPERSINGULAR INTEGRLAS 729
for all | | | = 1 . Let \ξ\ = l. We take a function /ejg) such that / (0)=l .
By Lemma 2.8 (i) we have
(R'
rί
Uί)(0) = r- j (R>tKcί)(z)f(-rz)d2.
Since l>a, it follows from Corollary 2.2 (i) that
(R'rtum _»o
 ( r _ o ) .
On the other hand, since (ϋ^ t f)(^) *s integrable we see that
j (i«(s)/(-πO<fe - J (*{«.)(*)<** (r - 0).
Hence we obtain
(ii) We easily see that (i2{+V7)(j) is integrable on { | j | ;>£}. For (2.2) it
suffices to show
for | f | = l . We take a function/e5) which satisfies the condition/(#)=1 on
{I x I < 1}. By Lemma 2.8 (ii) we have
(Λί«+ιtf{)(0) = limr' J (i?^/)(^)/(-^)^
It follows from Corollary 2.2 (i) that
lim lim ( {Rl^xMzifi-rzjdz = 0.
We easily see that the left hand side is equal to
lim
Hence we obtain the required property.
Lemma 2.10. Let β>a—n,β<l—l, and moreover we assume that l>a—n
in case a—nis a nonnegative even number. Then
730 T. KUROKAWA
μ Ky>
 \y\«^-a ) 1,121/1,1 \z\»+β
Proof. By Lemma 2.7 (ii) and the change of variables tj\y\ =z, we have
[
1(121 \t\n+β
=
 f
Jllίlδl
= (
\y\n+β-a Ju
REMARK 2.11. Let
dt
Then
L e m m a 2.12. Let K
m
(x, y)= Σ (Dyκ*(x)lΎ Oj 7 * Then for \χ\ = \y\ we
K
m
(x, y)=K
m
(yy x).
Proof. Let | * | = | y | . We set FXfy(s)=κΛ(x+sy) for -ί<s<ί. We
easily see that F
s
,Js) = F,tS(s). Hence (d'nlds'n)FXty(0) = (dmldsm)Fy>x(0). Since
we obtain the required property.
Lemma 2.13. Let β>a—n) β>l—\, and moreover we assume that
—n in case a—n is a nortnegative even number. Then
Proof. By Lemma 2.10 we have
μ>.«.»{y) = i f &(yld».
μ yy)
 \y\'+*- Juiai/ivi | β | +ί
Let a—n be not a nonnegative even number. By the inversion u=vl\v\2, we
see that
HYPERSINGULAR INTEGRALS 731
- Σ
By Lemma 2.12 we have
Σ
Σ
Σ
/ - I
Further, since | | v \ yr-\-(vl \ v |) | = | v~\-y' | by an easy argument, we have
= _J (
lyl +ί- Juisι,ι
Next, let α — n be a nonnegative even number. By the inversion u=vj\v\2, we
see that
X
By Lemmas 2.12 and 2.6 we have
Σ (Dy
=
m
Σ _ i
Therefore, by Lemma 2.4 we obtain
γ !
M) Σ
Σ r»- Σ
732 T. KUROKAWA
)
 i v
\y\u+β-*
The proof of the lemma is completed.
Now we prove
Theorem 2.14. Let l<q^oo. If l-.\<
a
—{nlq)<i, then μ!
Moreover, when I is an odd number, μ!'"**^Lq' still holds for /— l<α—(rt
Proof. In case l<g r<oo> the conclusion follows from Theorem 2.3 since
a>a—(nlq). Hence it is sufficient to show that if /— l < α < / , then μι**'
and that if / is odd and / — I < α < / + 1 , then μ ' ^ e Z Λ
First let l—\<a<L By Lemma 2.13 and a>l— 1, we see
( \(Rιy<«Λ)(v)\dvdy<oo.
Since α < / < α + l , by Lemma 2.13 and Corollary 2.9 (i) we have
Hence by Corollary 2.2 (ii) we obtain
J U ' β - ( j ) | < f y = J - l , IJ (ΛpΛ<l)(β)Λ;
Ji>i>i/2 J I y I Ji»ι>ι^ιi>i>i/2 I y I
( ~^-,( \v\*-ι-*dviy
y\>ll2 \y\ )\v\>\y\
( \y\*-ι~*dy<oo
\y\>\/2
since /<«. Tnus ^ ' e L 1 for / — l < α < / .
Next let / be an odd number and / < α < / + l . Since a>l>l— 1, we see
by the same reason as above. Since / is an odd number and l<a, for | γ | = /
we have
vlέl l
Hence by Lemma 2.13 we have
( Diκ
a
{v)dv = 0.
Jlvlέl l
HYPERSINGULAR INTEGRALS 733
If If
Iy \n JM^IJI y * \y |Λ Jι»ι^ !vi
Furthermore, because of a<l+l<a+l by Corollary 2.9 (i) we obtain
Therefore by Corollary 2.2 (ϋ) we have
\y\>\!2 \y\Λ )\υ\>\y\
l>l>l/2
since a<l-\-l.
Finally let / be an odd number and a=L We easily see
Since / is an odd number, for | y \ ~l we obtain
( D*φ)dυ = 0, £ > 0 .
Hence by Lemma 2.13 and Corollary 2.9 (ii) we have
Therefore by Corollary 2.2 (ii) we obtain
t \μUJ(y)\dy^c\ J--A \υ\-ι-*dυdy
The proof of the theorem is completed.
2.3. /^'"^potentials
For a locally integrable function/we set
734 T. KUROKAWA
Theorem 2.15. Let f be a nonnegative Lp-function, β>a—{nlp) and β>
/—I. // a—(nlp)<l<a+\y then Vfιtΰitβ(x)<oo for almost every x. In parti-
cular, if a—{nlp)<l<a—{nlρ)+\, then VfιtΛtβ(x)<°o for all x.
Proof. We have
L,
It follows from Corollary 2.2 (ii), Holder's inequality, a—(njp)<l and
(n/p) that
For V2(x) we see that
Let α—w be not a nonnegative even number. Since
\t\n+β
we obtain
V
Ά
{x)<ίC j (1+1y I)»-*-»f{x-y)dy<
because of β>a—{njp). By β>l— 1 we have
HYPERSINGULAR INTEGRALS 735
^_i\\y\
Λ
-
m
-
nf^-y)dy\
m>idhL t)<UhA^
dt
11y\*-\i\-n(\-\- \y\)~β+mf(x—y)dy
c
It follows from β>a—(njp) that V222(x)<oo for all x. Moreover, by /— ί<a
and Young's inequality we obtain V22l{x)<°° for almost every x. In particular,
if l—l<a—(njp)y then by Holder's inequality we have V22l(x)<oo for all x.
In case a—n is a nonnegative even number, the proof is similar. Therefore we
obtain the theorem.
3. Hypersingular integrals of Beppo Levi functions
For an integer k<a, we set
Further, for a locally integrable function /we set
We denote by [r] the integral part of a real number r and by / 1 E the restric-
tion of a function / to a set E.
Lemma 3.1. ([1]) Lei k=[a—(njp)] and f^Lp. If a—(n\ρ) is not a
nonnegative integer, then Uζtk exists, and if a—{njp) is a nonnegative integer, then
Uί\k_λ and U
f
a
% exist where}•=/1Bχ9 f2=f-f andB,= { | x | < 1}.
Lemma 3.2. Let k=[a—{njρ)] andf^LK
(i) 7/ a—(njp) is not a nonnegative integer, then for a—(njp)<l<a-\-\ we have
(R'tUik)(x)=$R'tKa)(y)f(x-y)dy.
(ii) If cί—(njp) is a nonnegative integer, then for a—(njp)<l<aJ[-l we have
where f and f2 are as in Lemma 3.1.
Proof. By / — l < α , l>a—(njp) and Lemma 2.4 we have
736 T. KUROKAWA
CRljTf \(γ\ JJf (γJL.f\ V1 (ΏITJf
^\{φ+t-y)- Σ ((x+t)sl8$D'Ka(-y))f(y)Jy
J [O|<Λ
n \ (*A*-y)- Σ (x'β\)D<Ktl(-y))f(y)dy
ι8Σ j ({(χ+t)ψ\)-
- Σ
Since the proof of (ii) is similar, we obtain the lemma.
REMARK 3.3. Let/eZΛ Then for α—(n/p)<l<a+l we have
JI (ΛίO(y)/(*-y) I ιfy< oo
for almost evrey x. In particular, if a—(nlp)<l<a—(nlp)+l, then
for all x.
For a positive integer I and β > 0 , we set
Recall that in Remark 2.11 we defined μlt" β as
„/.•.*/VN _ f (R't*.)(y) dt
Proposition 3.4. Lei α—(»/p)</<α+l, β>a—(nlp), β>l—ί, k =
[a-(nlρ)] andf el/.
(i) If a—(nip) is not a nonnegative integer, then
(ii) If cί—{njp) is a nonnegative integer, then
HYPERSINGULAR INTEGRALS 737
f
a
\k.^ U'ά)(x) = j μ['"-p(y)f(χ-y)dy.
Proof, (i) By Lemma 3.2, Theorem 2.15 and Fibini's theorem we have
The proof of (ii) is similar. Hence the proposition is proved.
Lemma 3.5. Let β>a—ny β>l—ί, and moreover we assume that l>a—n
in case a—n is a nonnegative even number. Then
Proof. This lemma follows from Remark 2.11.
Let Rj (j=lf •••, n) be the Riesz transforms, namely
*,/(*) = lime, j , *'-y* f(y)dy
with c
n
=T((n+ l)/2)τr"(w+1)/2 ϊoxf^LK For a multi-index y=(γl9 •••, γΛ) we set
By = κ{i... RI» .
For a positive integer τ« and />>1, the Beppo Levi space J2p
m
 is defined as fol-
lows :
Lemma 3.6. ([1]) Le* k=[m-(nlp)] and
(i) 7/* m—(nlp) is not a nonnegative integer, then u can be represented as
u(x)= Σ aBx8+Uitk(x)| δ | ^ « l
where f=c
mn
 Σ (m\\η\)WDΊu and c
mn
 is the constant.
' \y\=m
(ii) If m—{njp) is a nonnegative integer, then u can be represented as
u(χ)= Σ a^+Uίn^W+Uίn^x)
738 T. KUROKAWA
where f
x
=f \
 BvU=f-fx andj is as in (i).
Theorem 3.7. Let u(=Xp
m
.
(i) Iftl<q<oo,p^q9 β>m-{nlq) and (l/r)=(l/ί)-(l/ff), then
(ii) Ifβ^m,then
Proof. Let m—{njp) be not a nonnegative integer. By Lemma 3.6 (i) we
have
By Lemma 2.4 we [see that H?<βu=H?'βUίntk. Therefore by Proposition 3.4
we obtain
H?>βu(x) = J μ?-m>β(y)f(χ-y)dy .
Hence by Young's inequality we have
Consequently, (i) and (ii) follows from Theorems 2.3, 2.14, Lemma 3.5 and
| | / I ! ^ C \u\
m>p. In case m—(n\p) is a nonnegative integer, the proof is similar.
Thus we obtain the theorem.
References
[1] T. Kurokawa: Riesz potentials, higher Riesz transforms and Beppo Levi spaces,
Hiroshima Math. J. 18 (1988), 541-597
[2] T. Kurokawa: On singular difference integrals> Preprint.
[3] E.M. Stein: Singular integrals and differentiability properties of functions,
Princeton Univ. Press. Princeton, 1970.
[4] R.L. Wheeden: On hypersingular integrals and Lebesgue spaces of differentiable
functions II, Trans. Amer. Math. Soc. 139 (1969), 37-53.
Department of Mathematics
College of Liberal Arts
Kagoshima University
Kagoshima 890, Japan


On hypersingular integrals (Dedicated to Professor Tatsuo FujiTe on the occasion of his 60th birthday)

http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/111F0000002-02703-13.pdf

On hypersingular integrals (Dedicated to Professor Tatsuo FujiTe on the occasion of his 60th birthday)

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