CC BY
8
DOI: 10.17516/1997-1397-2020-13-3-350-359 УДК 517.518
LP-bound for the Fourier Transform of Surface-Carried Measures Supported on Hypersurfaces with DType Singularities
Nigina A. Soleeva*
Samarkand State University Samarkand, Uzbekistan
Received 02.02.2020, received in revised form 06.03.2020, accepted 06.04.2020 Abstract. Estimate for Fourier transform of surface-carried measures supported on non-convex surfaces of three-dimensional Euclidean space is considered in this paper.The exact convergence exponent was found wherein the Fourier transform of measures is integrable in tree-dimensional space. This result gives an answer to the question posed by Erdosh and Salmhofer. Keywords: Fourier transform, oscillatory integral, surface-carried measure.
Citation: N.A.Soleeva, LP-bound for the Fourier Transform of Surface Carried Measures Supported on Hypersurfaces with DType Singularities, J. Sib. Fed. Univ. Math. Phys., 2020, 13(3), 350-359. DOI: 10.17516/1997-1397-2020-13-3-350-359.
1. Introduction and preliminaries
Let S C R3 be a smooth surface and ^ € Cq°(S) be a smooth function with compact support on S. Consider the measure d^ = ^da, where da is the surface-carried measure. Fourier transform of the measure is defined by:
m := i
■Js
It is well-know that 1 is an analytic function.
In this paper the following problem is considered: find y := inf{p : ¡1 € LP(R3)}. This problem has a long history [1,2]. Recently L. Erdos and M. Salmhofer [2] considered the problem for partial class of non-convex surfaces in R3. The main class of such surfaces was level set of dispersion relation of discrete Schrodinger operator on the lattice Z3. It should be noted that the phase function of the corresponding oscillatory integrals has singularities of type A1; A2, A3 or D4. In particular, except the case D4 one of the principal curvatures does not vanish at every point. The case D4 type singularities was excluded in [2]. A more general class of hypersurfaces for which the Gaussian curvature has only simple roots was considered [3]. However, it was assumed that only one of the principal curvatures can vanish. The case when both principal curvatures vanish at a point of the surface in R3 is still one of the open problems.
We consider the problem for hypersurfaces in R3. More precisely it is assumed that the phase function (x, w)|S (where w € S2 is the unite sphere centred at the origin) is small perturbation of the so-called Dtype singularity (see [4] for definitions and basic properties of such singularities).
* niginasol@yahoo.com © Siberian Federal University. All rights reserved
It is shown that in this case y = 3. It can be shown that for any hypersurface S C R3, ( £ Lp(R3) for p < 3, whenever Supp(^) = 0. The main result is the following.
Theorem 1.1. Let S be an analytic hypersurface in R3. If S has Dtype singularities at the origin then there exists a neighborhood U of the origin such that for any 0 £ Cq°(U) the inclusion ( £ Lp (R3) holds for any p> 3.
Moreover, if S is any smooth surface in R3 and -0(0,0) = 0 then ( £ L3(R3). The paper is organized as follows. In Section 2 the problem for the model case is considered. In this case the result is obtained with the use of simple methods. The Section 3 is devoted to special function with Dtype singularity at the origin.
In Section 4 the general case is considered. Main theorem is proved in Section 5.
2. Model case D^
Let us consider a measure supported on hypersurface X3 — x^x^. The singularity of that function is called to be Dtype singularity at (0,0). The Fourier transform of the measure can be written as
((£) = / ei(«lXl+«2X2+53XlX2)01 (x)dx,
J R2
where 0i(xi, x^) = 0(xi, x2, xix^)//^/1 + x| + 4x1x|.
Following B.Randol [3], we define the following maximal function:
M (w) = sup r\((rw)\,
r>o
where r = |£| and w £ S2, S2 is the unite sphere centred at the origin.
Let us note that ((£) = 0(\£\-w) (as \£\ ^ x>) provided \£3\ < max{\^1\, \£2\} and 0 is a smooth function concentrated in a sufficiently small neighbourhood of the origin [5]. It is also assumed that \£3\ > max{\^1\, |£2|}. Let us consider the associated oscillatory integral
J(X, s) = f eiX®(x's)0i(x)dx,
R2
where $(x, s) = xix2 + sixi + S2x2, X = £3, Sj = j, j = 1, 2.
X
One can define the Randol type maximal function [3] associated with the oscillatory integral J(X, s) as
M (s) = sup\X\\J (X,s)\. Now, the following statement is proved. Theorem 2.1. The inclusion M £ L3-0 (R2) holds true.
Taking into account that 0 has a compact support and using integration by parts, the integral
Ji(X,si,x2) = i eiXxi(x2+Sl)0(xi,x2)dxi
R
can be estimated by
\Tf\ m ^ c\\0\\c2
\Ji(X,si,x2)\ < 1 + \X\2\x2 + s i \ 2 .
Consider the following integral
dx2
J 1(A, si) = i J R
1 + |A|2|x2 + si|2 • First, we prove the auxiliary statement. Lemma 2.1. The following estimate holds true:
IJ 1(A, si)| < -^.
|A||Si| 2
Proof. First consider the case A|s11 < 1. If s1 =0 then there is nothing to prove. Let us assume that s1 = 0. In this case we use change of variables x2 = |s1|2y2 and obtain
J 1(A,S1) = |s1|2 f
R
dy2
1 + |As1|2|y2 +sgn(81)|2'
For the sake of definiteness we assume that sgn(s1) = —1, e.g. s1 < 0. Actually the case sgn(s1) = 1 or equivalently s1 > 0 is much more easy to prove. Thus, we have
J(A,s1) = |s112 f
R
dy2
1 + |As1|2|y2 — 1|2' It is easy to see that the following estimate
f dy2
J y2 — 1I
|ASi||y|-1|>1
< C|As112
holds. Indeed
dy2 = 2 f dy2
"2
y2-1l> ^ V2>^ 1+
oo
'2 — 1
Iy2 — 1I J y2 — 1
1 ' dy2 = ln y2 1
y — 1 y2 + 1J y2 + 1
V1+ Txin
1^4+5^ ) =l^|As1|f2+ ^ + 2^1 + 1
lAsi
— 1) V v |As11 v |As11
= 1n(1 + 2|As1| + VI As 112 + |As1|) < 2|As1| + V |As112 + |As1| = = + V1 + I As 11) < v1M](2 + 2V2) = ^vl^
for |As1| < 1. An analogical estimate holds true for |As1| < 2.
c
Also ¡{y2 : |As1||y2 — 1| < 1} < -1. Hence the inequality
(A|s1|) 2
IJ (A, s 1) | < ^
A|s1| 2
holds true provided A|s1| < 2.
Now, we consider the case \Xsi\ > 2. In this case, we have
f dy2 c
J |Asi|2|y2 - 1|2 |AS1|2'
\y22-1\>1
It is easy to see that the following estimate
f dy2
J |y2 - 1|2
1>\y2-1\>\Asi\-1
< ^As;!^1
holds. Indeed, using symmetry of arguments, the last integral can be estimated as [ dy2 < 2 f dy2 <
J \yl - 1\2 " J \y2 - 1\2\y2 + 1\2
i:?\^1-I\>\\SI\-1 \y2-i\>\Asl —1
4 2 / f^ < 4 j ^ = 4 |Xs,| .
|y2 i\>\Asl\-1 \y2 i\>\Asl \ — 1
On the other hand the inequality n{y2 : \ y2 — 1 \ < \ Xsi\ -i} 4 c \ Xsi\ -i holds true for the measure of the set {y2 : \ y2 — 1 \ < \ Xsi\ -i}. Hence we obtain
\ Ji (X, s) \ 4 —^r.
X \ si \ 2
Lemma is proved. □
It is easy to see that the oscillatory integral J(X, s) can be estimated as follows:
,-N
/N
| J1(A,s,X2) | dx2,
N
N
where the number N is
N = max{\x2\ : there exist xi, such that (xi,x2) £ Supp0}. (1)
Hence
\J (X, s)\ 4 c\0\c2 \J i(X, s)\. Consequently, it follows from the Lemma that
\tî\ m ^ dlwic1 J(a,s)| <
|A||s1|2
because 0 has a compact support. If |s| > m, where m is a big positive number depending on the support of 0, then the phase function has no critical point. Hence we can use integration by parts and obtain
Therefore we have
c
J(A's) < X\s\■
X{\S\>m}(s)M(s) < — G L~(R2\B(0, m)), (2)
where B(0, m) is the ball of radius m centred at the origin, and X{|s|>m} is the indicator function of the set {|s| > m}. Let us denote the indicator function of the set A by xa, e.g., xa(x) = 1 for x € A otherwise xa(x) = 0.
The relation (2) suggests that it is sufficiently to consider the oscillatory integral and the associated maximal function on the set {|s| < m}. Let us assume that x = x0 € Supp(^) is a critical point, and s = s0 € B(0, m) is a fixed point. If x0 is not a critical point of the phase function $(x, s0) then one can use integration by parts and obtain better estimate than needed. Equations for critical points are
(x°)2 + s0 = 0, 2x0x2 + s0 = 0.
Let us assume that s° = 0. Then x0x2 = 0. Hence x0 = 0 and also x2 = 0, s0 = 0. Let us consider the integral
Jx(A, s) := / eiX®(x's)^(x)x(x)dx,
J R2
where x is a smooth cut-off function defined in a sufficiently small neighbourhood of x0 and s is close to s0. One can use stationary phase method in two variables because
Hess$(x°, s0) = -4(x0)2 =0.
Therefore for | s — s01 < e we have the estimate
c
JX(A,s^ < -
provided x is a smooth function defined in a sufficiently small neighbourhood of x0. If x0 is not a critical point then one can use integration by parts and obtain the same type of estimate (even better estimate than needed). Hence M(s) is a bounded function in V(s0), where V(s0) is a sufficiently small neighbourhood of s0 = 0. Let us consider the case when s0 = 0, e.g., when s belongs to a sufficiently small neighbourhood of the origin. This case will be considered in the next section.
3. Case {|s1| 2 ^ |§2|}
Then trivial estimate for J (A, s) is
J (A, s)| <-C—^ <-^^-—
|A||si|2 |A||si|3 |s213
and the estimate is obtained because-j1-r € L3-0 (V), where V is a bounded neighbourhood
|si| 3 | s2 | 3
of the origin.
Let us assume that |s2| > |s1|2.
Let us consider the one-dimensional integral
J2(A,s2,xi) = [ eix(xjx2+s2x2)^(xi,x,2)dx2.
R
If |Ax1| < 1 then we have the trivial estimate
J | J2(A, s2,xi)|dxi < c|A|-1.
[0,A-i]
Hence we may assume |Ax1| > 1. If |Ax1| > 1 and |x1| < |s21 then the phase function has no critical point on the support of 0 provided N < 2, where N is defined by relation (1). Then one can use double integration by parts and obtain
rn N, ^ cll0llc2
|J2(A,s2,x1)| < | 2 .
Therefore
J \J2(X,s2,xi)\dxi 4 x^ ■
[0, \ S2 \ ]
Finally, let us suppose that \xi\ > \s2\. Then we use stationary phase method in x2 and obtain
2
c s2 \ ( s2 \
J2(X,s2,xi) = ---y e-4X1 mxi, — -— + R(X,xi,s2).
\Xxi \ 2 V 2xi/
c
For the remainder term R(X,xi,s) we have \R(X,xi,s2)\ 4 -r. Then
1 + \Xxi \2
c
f \R(X, xi)\dxi 4 ttt. Thus, it is sufficiently to consider the integral \ X\
r iXs2(-X+S2xl) e l s2 s2
Ji(X,s)= -:—-1-0lxl, dxi.
JR \xi\2 V 2xi/
c
If \Xs2\ < 1 then we have \Ji\ 4 —1-. Hence we assume \Xs2\ > 1. Let us estimate the
|X| 2 \ s2 \
integral
i^K-¿T + Si xi) 0(x1, - 2X1)
t—I— / \ \ 4xi + s2 xl! ' \ 2x1/ 7
Ji (X,s)= e 1 s2 -1-— dxi.
J R+ xf
Using the change of variables xi = y2, we obtain
J—+(X,s) = 2 f eiX4(-412 +ffly2)0(y2, — )dyi,
JR+ v 2yi'
where ai := s2.
s2
The phase function has no critical points provided 0 is a smooth function defined in a sufficiently small neighbourhood of the origin so one can use integration by parts. Thus, we obtain
\Ji(X,s)\ 4 .
\s2\\X\1
Let us show that
" 1 -\^ £ L3-0(V).
s2
Indeed for p < 3 we have
n1 ds2 fS2 , f1 ds■
ds1 = I , ""P 2 <
Jo \s2\p Jo Jo Mp-2
Combining the obtained estimates for the Rendol maximal function for oscillatory integral, we obtain
M(S) 4 c(X{ 1 sl\^s?}(s) + ^ff^
V \si\2 \s2 \
Since M £ L3oC°(R2) our consideration is completed.
4. The general case
The following proposition holds true.
Proposition. Let us assume that $(xi,x2) has DC type singularity at the origin
$(xi, x2) = xix2 + R(xi, x2),
where R(xi,x2) = 0(|x|4).
Then there exist analytic functions ^ and b such that function $ can be written as
$(xi,x2) = b(xi,x2)(xi — p(x2))(x2 — ^(xi))2,
where ^(0) = y>'(0) = 0, ^(0) = ^'(0) = 0, b(0,0) = 1 (see [2] and [6]).
Let us assume that ^(xi) = x™1 ip(xi), ^6(0) = 0 and y(x2) = x^2fi(x2), ^(0) = 0. Then
$(x, s) = b(xi,x2)(xi — ^(x2))(x2 — ^(xi))2 + sixi + s2x2.
Using the change of variables
xi — ^(x2) —> xi, x2 — -0(xi) —> x2,
we obtain
$(x, s) = b(xi,x2)xix2 + s i (xi + <£>(x2)) + s2 (x2 + ^(xi)). Let D be the annulus D = {i < |x| < 2} and Supp x C D with x € CC(D) satisfying
- 2
c
x)
ft=«0
y^ x(23x) = 1 for x = 0, x << 1. Then we have
/CO ~
a(xi,x2)eiA*l(x's)dx = ^ a(xi,x2)x(2f x)eiA*l(x's)dx.
Let
_ J. = / a<xi ,x=)x(2 3 x)e««l(x->dx
Let us use scaling 2 3 x —> x and obtain
Jk = 2-* J a(2-3x)x(x)eiA2^J (x's)dx,
V(x, s) = b(2-3x)xix2 + 2^si(xi + x^12-3(m2-i)^(2-3x2))+ + 2 231 s2(x2 + xm12-3 (mi-i)^,(2-3 xi)). Note that x € D. If |2331si| >> 1 or |2^s2| >> 1 then using integration by parts, we obtain
2-2K
J ^ C ¥ si| +2 ¥ | s 21).
Let us take the integral
f 23pds
J (|2^si | + 2^ |s2 |)p
2"3 |s|>i
After the change variable 2 f s = a we have
f 2kp-4f da=2K(p-4) f da=2f(p-4)cP.
J \a\p J \a\p p
\ " \ >i \ " \ >i
~ 2 r x\2 ¥ s\ 2k
Thus, if p < 4 then the series J2 —2K-2f- converges in Lp. Let 2 3 s = a and \a\ 4 1.
K=K0 \2tsi\ + 2"r \s2\ Now, we use compactness arguments.
Let us assume that a = a0 = 0 and (xi,x2) is a critical point of the phase function. Then $K(x,a) can be considered as a small perturbation of the function
$ = 6(0, 0)xix2 + a0xi + a0x2,
where (xi,x2) £ D. If (a0,a0) = (0,0) then x02 = 0. Hence
Hess$
0 2x2
2x2 2x0
62(0,0) = —4(x0)262(0, 0) = 0.
Then we can use stationary phase method in two variables and obtain
|JX| 4 X
in a neighbourhood of a0.
Finally, let us consider the case when (a0, a0) = (0,0). Since (xi,x2) £ D, then x2 =0 and xi = 0. Thus x0 - 1.
6(2 f x)xix2 + ai(x™12-f (m2-i)((2-f x2)) + a2x2.
= — 2xi^(2-fxi, 2-f(m2-i)ai)
2
a2
x2
Using stationary phase method in x2, we obtain oscillatory integral with phase g(0,0) = 0. 2
$K(a,xi) := 4xiG2-fxi, 2-f(m2-i)ai) + aixi + a2xf12-f(m2-i)0(2-fxi)
xi - 1, a\ - a22-f(m2-i)2f (m2-i)a2 - 1.
Let us consider the following one-dimensional oscillatory integral
JK(X,a) = i eiX2—f*f(a'xl)a(xi)dxi X 2 J
R
where \X2-K\ > 1.
We prove the following Lemma.
Lemma 4.1. Let x° = 0 be a fixed point. Then there exist a cut-off function x supported in a neighborhood of xi, k0, c0, c such that for any k > k0 the following estimate holds true:
\JX\ 4 — f 1 , x \ Vl \ <c*2 (ai, a2 )
X2 V \a\3 \a2 \3 \a2 \2 \ai — c0a|\4
Proof of the Lemma follows from the results presented in [7]. It is easy to see that for any
p< 3 e Lfoc (R2), where
T , , 1 X\a1\^ca22
) = ~—ixi—a + 2
3M3 M2K - co4
Corollary. There exists ko such that for any k > ko the following estimate holds true:
I T (A * ^(g1,g2)23
*-—-J-,
|A| 2
where ^ e L^-c° (R). The following theorem holds true.
Theorem 4.1. Let s be an analytic hypersurface such that it has Dtype of singularity at the origin. Then there exists a neighbourhood U C R3 such that for any ^ e C^(U), M e L3-0(S2).
5. Summation of the Fourier transform of measures
Let S be an analytic hypersurface and
d/ = ^(x)dS.
We prove the following Theorem.
Theorem 5.1. Let S be an analytic hypersurface. If S has Dtype of singularity at the origin then there exists a neighbourhood U of the origin such that for any ^ e C0^(U) the inclusion ( e Lp(R3) holds for any p > 3.
Proof. It is well known that there exists a neighbourhood U of the origin such that for any ^ e C§°(U) the following estimate holds true (see [8])
~ C
|/(0| * -T. (3)
According to Theorem 4.1, there exists a function ^(w) e L3-0(s2) such that
|/(rw)| * . (4)
(1 + r)
Let p > 3 be a fixed number. Let us take q < 3. We interpolate estimates (3) and (4) and obtain
C
(1 + r) '
If p > 3 one can choose a and 3 such that p{0, + > 3 and p3 < 3.
3 s
For instance, we take a sufficiently small positive number S > 0 and set 3 = - and
p
p-3 + S ,
a =-. Then it is easy to see that
p
f f^ r2dr i |d/(e)|pe * c/ ^ dw< +*>.
JR3 Jo (1 + z)( 2 +p)p Js2
Theorem 5.1 is proved. □
References
[1] G.I.Arkhipov, A.A.Karatsuba, V.N.Chubarilov, Tringonometric integrals, Izv. AN SSSR, Ser. Mat., 43(1979), no. 5, 971-1003 (in Russian).
[2] L.Erdos, M.Salmhofer, Math. Z, 257(2007), 261-294. DOI: 10.1007/s00209-007-0125-4
[3] B.Randol, On the asymptotic behavior of the Fourier transform of the indicator function of a convex set, Trans. AMS, 139(1969), 278-285.
[4] V.I.Arnold, A.N.Varchenko, S.M.Huseyn-zade, Features of differentiable mappings, Part 1. Classification of critical points of caustics and wave fronts, Moscow, Nauka, 1982 (in Russian).
[5] E.M.Stein, Harmonic analysis, Real-valued methods and oscillatory integrals, Princeton University Press, Princeton, 1993.
[6] I.A.Ikromov, D.Mwller, Fourier restriction for hypersurfaces in three dimensions and Newton polyhedra, Annals of Mathematics Studies, Number 194, Princeton and Oxford, 2016.
[7] I.A.Ikromov, Functional Anal. and Its Appl., 29(1995), no. 3, 161-168. DOI: 10.1007/BF01077049
[8] V.N.Karpushkin, J. Math. Sci, 35(1986), 2809-2826. DOI: 10.1007/BF01106076
^-оценки преобразования Фурье поверхностных мер, сосредоточенных на гиперповерхностях с особенностью типа Бг
'ОО
Нигина А. Солеева
Самаркандский государственный университет Самарканд, Узбекистан
Аннотация. В этой статье рассматриваются оценки преобразования Фурье мер, сосредоточенных на невыпуклых поверхностях трехмерного евклидова пространства. Мы найдем точный покозатель, для которого преобразование Фурье мер с этой степенью интегрируемо по трехмерному пространству. Этот результат дает ответ на вопрос, поставленный Эрдошем и Салмхофером.
Ключевые слова: преобразование Фурье, осцилляторный интеграл, поверхностная мера.
