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On the Convergence Exponent of the Special Integral of the Tarry Problem for a Quadratic Polynomial
Isroil A. Ikromov*
Institute of Mathematics named after V. I. Romanovsky Uzbekistan Academy of Sciences Samarkand, Uzbekistan Samarkand State University Samarkand, Uzbekistan
Akbar R. SafaroV
Uzbek-Finnish Pedagogical Institute Samarkand, Uzbekistan Samarkand State University Samarkand, Uzbekistan
Akmal T. Absalamov*
Samarkand State University Samarkand, Uzbekistan
Received 03.04.2023, received in revised form 05.05.2023, accepted 20.06.2023 Abstract. In this paper it is considered the summation problem for trigonometric integrals with quadratic phase. This problem was considered in the papers [7-9] in particular cases. Our results generalize the results of those papers to multidimensional trigonometrical integrals. Keywords: trigonometrical integral, exponent, sums, phase, polynomial.
Citation: I.A. Ikromov, A.R. Safarov, A.T. Absalamov, On the Convergence Exponent of the Special Integral of the Tarry Problem for a Quadratic Polynomial, J. Sib. Fed. Univ. Math. Phys., 2023, 16(4), 488-497. EDN: SKBBCK.
Introduction
Let P(x, s) £ R[x] (where x G Rk) be a polynomial with real coefficients s G RN. We consider the trigonometric integral given by
T (s) = exp(iP (x,s))dx, (1)
Jq
where Q C Rk is a compact set.
Problems related to such kind of integrals arise in mathematical physics (see [1]), harmonic analysis (see [2-5]), analytic number theory (see [6-11]) and so on. Surely, the given references are not complete.
* ikromov1@rambler.ru tsafarov-akbar@mail.ru ^ absalamov@gmail.com © Siberian Federal University. All rights reserved
One of the well known problems related to the trigonometric integrals is the issue on convergence of the special integral of the Tarry problem, which is given by the following:
The integral 9 arises as the coefficient of asymptotic representation for a number of integer solutions of a Diophantine system [2,6,7]. Therefore, it is important to find a minimal value of the parameter p, where the special integral is convergent, which is also essential in the Fourier restriction problem in harmonic analysis [3].
Definition. A real number y is called to be a convergence exponent of the special integral if for every p > y the integral (2) is convergent and for every p < y it is divergent. In other words Y = inf{p : T e Lp(RN)}.
It should be noted that the convergence exponent essentially depends on the form of the polynomials P(x, s). Thus the main problem can be formulated as:
Problem: Find the number y.
This problem was considered by I.M.Vinogradov [11] in connection with the problems of analytic number theory. He obtained an upper bound for the number y in the case k = 1. This bound was improved in [10].
The exact value of y was indicated in [6] for the case k =1. It is interesting to note that in one-dimensional case depending on form of the polynomial P(x, s) the exact value of y can be expressed by the sum of exponents of the non-trivial terms of the polynomial P(x, s). Moreover, it was proved un upper bound for the number y in multidimensional cases.
It should be noted that, in [12] a lower bound was found for the number y. Moreover, it was found the number y provided that the coefficients of the polynomial vary in some subspace of Rn. Similar problems were considered in the works [13,14,15].
In [7] a lower bound was obtained for y and also, it was investigated analogical problem for more subtle object trigonometric sums in the case k = 2. In [7] and [9] a similar problem was considered in the case k = 2. Moreover, in [7], it is shown that if P is a homogeneous quadratic polynomial and k = 2, then y = 4 in the case when Q = [0,1]2, more precisely, the special integral 9 is convergent if p > 4 and divergent if p ^ 4.
It was interesting to extend the results proved by L. G .Arkhipova, V. N. Chubarikov related to trigonometric integrals to multidimensional case.
In this paper we study the problem in the classical setting. In other words, P is a quadratic polynomial function and Q = [0,1]fc is the unit cube and also for the case when Q is a compact domain. Analogical problem was considered by J.Makenhaupt [2], who obtain the number y in the case when the polynomial P(x,s) satisfies some "non-degeneracy" condition.
It should be noted that the condition of J. Makenhaupt does not hold for the general case (see [2]). Actually, J.Mokenhaupt used an interesting approach. He computed the multidimensional trigonometric integral, for which the amplitude function is the gauss function. Then he be able to get the sharp value of the convergence exponent for some cases. It should be noted that using the gauss functions to investigate behavior of oscillatory integrals goes back to E. M. Stein [1]. We obtain the exact value of y, whenever P is a homogeneous polynomial of degree two.
We use the idea of J. Makenhaupt and then we able to investigate the obtained integrals. We observe that the integral over RN can be written as an iterated integral over the orbit of the orthogonal group and then over the corresponding fundamental domain. It is interesting that the
(2)
integrant in the trigonometric integrals with quadratic phase with special amplitude function, more precisely gauss functions, is invariant under action of the orthogonal group. Thus, our approach is natural in this case. Unfortunately, it seems such approach does not work for trigonometric integrals with more general polynomial phase functions.
The paper is organized as follows in the next Section 1 we formulate our main results. In the next Section 2 we give some auxiliary results on integrals. In particular, we obtain transformation of the volume form under the natural action of the orthogonal group. Then we give a proof of our main results in the next Section 3. Finally, we give some results related to two-dimensional integrals in the last Section 4.
1. Formulation of the main results
Let P be the polynomial given by
P(x, A, b) = (Ax, x) + (b, x),
where A = (alm)kl m=1 is a symmetric k x k matrix with real entries, b := (b1,b2,... ,bk) £ Rk and (■, ■) is the inner product of the corresponding vectors. Consider the trigonometric integral
T(A,b) = exp (iP(x, A, b))xQ(x)dx,
J Rk
where Q is a compact set and xq(x) is its characteristic function. Consider the integral
6 = / \T(A, b)\pdbda,
J RN
where db = dbidb2 ... dbk and da = H daim. The following is true:
Theorem 1.1. Let Q be a compact set, then the integral 6 converges, whenever p > 2k + 2 and if Q contains an interior point x0 and there exists a line l passing through point x0 such that the boundary of the set {l fl Q} contains only a finite number of points, then the integral diverges provided p ^ 2k + 2. In particular, if Q = [0,1]k, then y = 2k + 2.
1. The case when P is a homogeneous polynomial of the second order
Now suppose that P(x,A) = (Ax,x). In [9] it has been proved that if Q is a quadratic polynomial in R2, then for p > 4 the 6 integral converges and when p ^ 4 the 6 integral diverges. In this paper we extend those results to the case when Q is a polyhedron in Rk. By polyhedron we mean a finite union of nondegenerate simplexes [5].
Theorem 1.2. If P(x, A) = (Ax,x) and Q is a polyhedron, then for p > 2k the integral 6 converges. If Q = [0,1]k, then for p ^ 2k the integral 6 diverges.
Remark 1. In this case, we cannot apply the results of [3] as the corresponding set {xxj is not a smooth surface.
Remark 2. Depending on the set Q, the exponent p may be smaller than 2k. For example, if k = 2 and Q is a sufficiently small square centered at (1,1), then it can be proved that for p > 3 the integral 6 converges.
2. Preliminaries
Consider the following integral
To(A,b) = / exp (iP(x,A,b) — (x,x))dx.
J Rk
It is easy to check that this integral, whose calculation details are given in [2], is absolutely and uniformly converges with respect to the parameters A and b.
Lemma 2.1. The following equality holds
T.(A, b) = (2*)k (det(I — iA))-2exp (— ((I ~ A b ^ ),
where the square root is determined in the following way
(det(I — iA))-1 := (1 — iAi)-i • (1 — iA2)-i • ... • (1 — i\k)-1,
with A1,...,Ak being eigenvalues of A. The branch cut of the multiply-valued function z-i is taken on the complex plane by cutting the negative part of the real axis and 1-2 = 1.
Lemma 2.1 is proved by reducing A to the diagonal form. Consequently, the calculation of the integral is reduced to a one-dimensional integral and it is explicitly calculated (see. [1]). Obviously, the following equations are satisfied:
( ((I - iA)-1 b, b). exP (--4-)
((I + A2) 1b,b)p) = exp(----),
/exp (— ((I + A2)-1 ^)db = ^2(det(! + A2))2 .
J 4 p 2
Rk
Let us introduce the following notation:
9.. = / \T..(A, b)\pdbda,
J RN
k(k + 2) where N =-2-.
Proposition 1. The integral 9o converges when p > 2k + 2 and diverges when p ^ 2k + 2. Due to Lemma 2.1, the proof of the Proposition 1 comes by studying the following integral
9. = c(p) [ , , (T d'\,-2 , (3)
J (det(I + A2)) —
RN-kV V "
where c(p) is some positive number.
As the determinant is an invariant of the orthogonal group, it is convenient to integrate it first by the orbits of the orthogonal group and then by the quotient space, e.g. over fundamental domain with respect to action of the orthogonal group.
Let M be the set of symmetric matrices with real entries and G = SOk be a special subgroup of orthogonal matrices. This group naturally acts in the space M as g(A) = gAg, where g€ SOk and A M.
p
It is known that for any real symmetric matrix A, there exists g£ G such that g(A) = diag(X1 ,...,Xk), where diag(X1,..., Xk) is a diagonal matrix with diagonal elements X1,...,Xk. In other words for any matrix A there exists g£ G such that A = glAg, where A = diag(X1,..., Xk). Hence it is possible to define a surjective smooth map
$ : RkxG^M
which is defined by the formula $(A, g) = glAg.
Let da = da11 A... Adakk be the standard volume form in the space M. We can define the image of this form under the map denoted by $*da£AN-k(RkxSOk).
Lemma 2.2. The following equality holds
$*da = (Xm — Xl)dX1A ... AdXk Au,
where u is the volume form on the orthogonal group SOk.
Lemma 2.2 can be proved by using the zero sets of the Jacobian of the map Note that the equality (Xm — Xl)2 = pA(X) holds, where pA(X) is the discriminant of the characteristic
polynomial of the matrix A.
By Lemma 2.1 the integral (3) can be rewritten as
, , n \ Xm — Xl\
da / 1^l<m^k A , ,, /
dX1A ... AdXk u.
J (det(I + A2))— J n (1 + Xl2)— ^
From the last equality, it follows that the convergence of the integral (3) comes from the investigation of the convergence of the following integral
. n \Xm — Xl\
I 1^l<m^k ,, ,,
—=—-dX1A ... AdXk.
Jh n (1+Xl2)-
Note that this integral converges when p > 2k + 2 and diverges when p < 2k + 2 and this proves the Proposition 1. □
3. Proofs of the main results
Proof of the Theorem 1.1. The upper bound for y follows from the main Theorem 1.1 of paper [3]. Consider the following subset 0(a11) in RN-1:
\a12\ + \a13\ +-----+ \a1k\ < ^an, —1 < — < —1, \aj — a1la1jC2, \bl — 2blC2,
2 a11 4 a11 a11
where l = 2,... ,n and c1, c2 are sufficiently small fixed positive numbers and a11 > 1. Lemma 3.1. There is a positive number c such that, the following equality holds:
p(il(an)) = c ■ a^1, for the Lebesgue measure of p of the set Q(an).
Proof. Consider the following maps:
Cii(A,bi, ...,bk) = an,
e1(A,61 ,...,bk ) = bi,
Zl(A, bx,...,bk) = bi - ^, a11
£ij (A, bi,..., bk) = aij - a'11 aij ,
a11
j < l, j,l = 2, 3,..., k. Jacobian of this map is equal to ±1.
Denote by the image of the map. Since the Jacobian is ±1, then we have
Mft(aii)) = )).
It is easy to verify that for the set with
I Ci2 | + | Ci3 I + ••• + I Cik \< ci • aii,
1 C1 1 --^ — <--,
2 < Cii < 4'
we have
Hence,
I I < C2, I Sij | < C2, j < l, j,l = 2, 3,...,k
M(^(en) = c • eii = c • akn.
^(O(an)) = c • aki.
Lemma 3.2. There exists a positive number L such that when a11 > L and (A,b)e 0(a11) for the integral T(A, b) the following asymptotic equality holds
T(A, b) = + o(as
ii
Moreover, there exists a positive number 6 such that for any (A,b)e Q(a11), the following inequality holds:
\c(A,b)\ >6.
Proof. Lemma 3.2 is proved by the method of stationary phases. Note that for the sufficiently small c1, c2 and for the sufficiently large L, the phase has oscillation only in the x1 direction on the set (A, b) e Q(a11). Consequently, for fixed values of x2,..., xne [0,1], the non-degenerated critical point x1(A, b,x2,..., xn) lies in (0,1). □
Finally, for integral 0 we have the following lower bound:
0 >f f \T (A,b)\pdbda > 6c [ ak- p da11.
jL Jn(a 11) jL
Thus, when p ^ 2k + 2 the last integral diverges, which proves the Theorem 1.1. □
2
Proof of the Theorem 1.2. We use the classical Young inequality.
p
Let f £ Lp(Rk) and g£ Lr(Rk) be arbitrary functions. The following inequality holds:
\\f*ghq < \\f \\lp\\ghr,
where f *g is a convolution of the functions f and g. Moreover, constants 1<p, q, r^x are related by
1,11
- + 1 = - + -. q p r
Let Q be a compact polyhedron in Rk and
h(b)= I eIx^ XQ(x)e-2ni(b'x)dx.
J Rk
Lemma 3.3. The following relation h£ L1+0(Rk) holds true, where L1+0(Rk) := ^^ Lp(Rk).
Proof. Note that, for any e > 0, Xq £ L1+£(Rk) (see. [4]). Then the statement of Lemma 2.1 easily follows from the Young's inequality.
Now let us return to the proof of Theorem 1.2. According to the Plancherel theorem we have:
TA)= f ^(AxXdx = f SAx*XQ(*)d* = i ¿•.Axx-x?e|xI 2x^d* = f famM
JQ J Rk J Rk J Rk
where f(A,b)= J ei(Ax'x)-1 x 12-2ni(x'b)dx and g(b) = J e1 xx2e-2ni(x'b)dx.
Rk Rk
Let q > 1 be a fixed number. Then, using the Holder inequality, we have:
\T (A)\ < \\f(A, )\l, (Rk )\g\Lq
where —|—7 = 1. qq
According to Lemma 2.1 , we have
\T(A)\ < c
P__
4
(det(I + A2))4 2q
' p 1 k p 1 k
Thus, if p > 2k, then we can choose q > 1 such that - — —T ^ —. It follows that if - — —T ^ —,
4 2q 2 4 2q 2
then T £ Lp(Rk).
It remains to prove the sharpness of the result. Consider the following subset Q+(a11) in
TON-1 „.1____AT _ k(k + 1)
where N
2
auaij
an > 0, |ai21 + |ai3| +-----+ |aifc| < cian,
aij
aii
< C2,au < 0
where l < j = 2,n, l = 2,... ,n and c1, c2 are sufficiently small fixed positive numbers.
According to the Lemma 3.1 there exist positive numbers c1 and c2 such that the following equality holds for the Lebesgue measure of Q+(a11):
p(Q+(a11)) = c ■ ak-1. □
2,
Lemma 3.4. There exists a positive number L such that when a11 > L and (A,b)e 0(an) for the integral T(A) the following asymptotic equality holds
T (A) = ^ + o(—) as
Moreover, there exists a positive number 6 such that for any (A,b)e 0+(a11) the inequality
\c(A)\ >6> 0
holds true.
Lemma 3.4 is proved by the method of stationary phases. Note that if 62 > 0 and 61 < 0 are fixed numbers then the following relation holds true
s^VA
J cos y2dy = c(6i,62, A)
SiVx
and there exist Ao, e > 0 such that the inequality c(61; 62, A) > e > 0 holds for all A > A0. Indeed, we have the following relation
s^Vx
lim cos y2dy
A^+TO J 2
SiVx
Note that, for sufficiently small c1, c2 at A e 0+(a11) and for sufficiently large L, the phase has oscillations only in the x1 direction. Also, for fixed values x2,... ,xne [0,1], the nondegenerate critical point x1(A, b,x2,... ,xn) lies inside (0,1).
Finally, for the integral 0, we have the following lower bound:
0 \T(A)\pda > 6c ak- 2da11.
JL Jn(a11) JL
Thus, the last integral diverges, whenever p ^ 2k. The Theorem 1.2 is proved. □
4. Two-dimensional case
Note that in the homogeneous case the results of [3] are not applicable. The proof of Theorem 1.2 essentially uses the property xq e L1+0(Rk).
In Lebedev's paper, it is given an example of the domain dD e C, where w is the continuity module of the gradient y that locally defines dD, such that xq e L1+0(Rk). Therefore, we can assume that D is a compact domain with sufficiently smooth boundary. The following is true
Theorem 4.1. Let D be a compact domain such that xd e Lq(R2) and T(A) = J ei(Ax'x)dx.
D
2
Then T e Lp(R3) for p > 6--. Moreover, if xD e L1+0(R2), then for any p > 4, the inclusion
T e Lp (R3) is valid.
Remark 3. From the results given in [4] it follows that there exists a domain D other than a polygon such that xq e L1+0(R2).
Corollary 1. If D c R2 is a compact set such that dD c C1, then for p > 4.5 the relation T e Lp(R3) holds.
References
[1] E.M.Stein, Harmonic Analysis: real-valued methods, orthogonality and Oscillatory Integrals, Princeton, 1993.
[2] G.Makenhaupt, Bounds in Lebesgue Spaces of Oscillatory Integral Operators, Habilitation-sschift zur Erlangung der Lehrbefugnis im Fach Matematik der Gesamthochschule, Siegen, 1996.
[3] J.-G.Bak, S.Lee, Restriction of the Fourier transform to a quadratic surface in Rn, Mathematische Zeitschrift, (2004), no. 247, 409-422. DOI: 10.1007/s00209-003-0626-8
[4] V.V.Lebedev, On the Fourier transform of the characteristic functions of domains with C1 boundary, Functional Analysis and Its Applications, 47(2013), no. 1, 27-37.
DOI: 10.1007/s10688-013-0004-1
[5] V.V.Lebedev, The dissertation on competition of a scientific degree of physical and mathematical sciences, Superposition operators in some spaces of the harmonic analyzer the translator, Moscow, 2013.
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О показателях сходимости особого интеграла проблемы Терри для квадратичного многочлена
Исроил А. Икромов
Институт математики имени В. И.Романовского Академия наук Республики Узбекистан Самарканд, Узбекистан Самаркандский государственный университет Самарканд, Узбекистан
Акбар Р. Сафаров
Узбекско-финский педагогический институт Самарканд, Узбекистан Самаркандский государственный университет Самарканд, Узбекистан
Акмал Т. Абсаламов
Самаркандский государственный университет Самарканд, Узбекистан
Аннотация. В статье рассматривается проблема суммируемости для тригонометрических интегралов с квадратичной фазой. Аналогичная задача рассмотрена в работах [7-9] в частных случаях. Наши результаты обобщают результаты этих работ на кратные тригонометрические интегралы.
Ключевые слова: тригонометрический интеграл, экспонент, сумма, фаза, многочлен.
