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DOI: 10.17516/1997-1397-2022-15-4-459-466 УДК 517.518.5
Estimates for Mittag-Leffler Functions with Smooth Phase Depending on Two Variables
Akbar R. Safarov*
V. I. Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences Tashkent, Uzbekistan Samarkand State University Samarkand, Uzbekistan
Received 26.12.2021, received in revised form 21.03.2022, accepted 26.05.2022 Abstract. In this paper we consider the problem on estimates for Mittag-Leffler functions with the smooth phase functions of two variables having singularities of type DTO, D± and Ar. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory type integrals. We extend results of paper [1] and [2] to two-dimensional integrals with phase having some simple singularities.
Keywords: Mittag-Leffler functions, phase function, amplitude.
Citation: A.R. Safarov, Estimates for Mittag-Leffler Functions with Smooth Phase Depending on Two Variables, J. Sib. Fed. Univ. Math. Phys., 2022, 15(4), 459-466. DOI: 10.17516/1997-1397-2022-15-4-459-466.
Introduction
The Mittag-Leffler function Ea(z) is named after the great Swedish mathematician Gosta Magnus Mittag-Leffler (1846-1927) who defined it by a power series
^ k
Ea(z) = E r(aZk + 1)G C, Re(a) > 0, (1)
and studied its properties in 1902-1905 in five subsequent notes [3-6] in connection with his summation method for divergent series.
A classic generalization of the Mittag-Leffler function, namely the two-parametric Mittag-Leffler function
^ k
Ea,p (z) = J2r{oZk + ft , e C, Re (a) > 0, (2)
which was deeply investigated independently by Humbert and Agarval in 1953 [6-8] and by Dzherbashyan in 1954 [9-11], see also [12] and the references therein.
In this paper we also consider a special case the generalized Mittag-Leffler function defined as in (2) by
k xk
(x) = £ rakTW) ,a> G R-
* safarov-akbar@mail.ru © Siberian Federal University. All rights reserved
Obviously,
Ei,i(x) = ex. (3)
We consider the following integral with phase f and amplitude ^
Ia,l3 = Ea,p (i\f (x))^(x)dx, (4)
•J a
where 0 < a < 1, 3 > 0 and A > 0.
If a = 3 =1 in the integral (4) the integral Ii,i is called an oscillatory integral. In harmonic analysis estimates, the most important estimates for oscillatory integrals is van der Corput lemma [13]. Estimates for oscillatory integrals with polynomial phase can be bound, for instance, in papers [14-19]. In the this paper we replace the exponential function with the Mittag-Leffler-type function and study oscillatory type integrals (4). In the papers [1] and [2] analogues of the van der Corput lemmas involving Mittag-Leffler functions for one dimensional integrals have been considered. We extend results of [1] and [2] for two-dimensional integrals with phase having some simple singularities.
The main result of the paper is the following
Theorem 1. Let —m < a <b < m. Assume that the phase function is a homogenous polynomial of third degree in two variables and let ^ G Lp[a,b]2, 1 < p ^ m. Then for any a G (0,1),
3,A G (0, +m)
[a,bY f
[a,b]
Ea,ß (i\x'lx2)^(x)dx Ea,ß(i\(x1x2 ± x\))^(x)dx
< CW-, (5)
A 2 2p
< c(6) ^ > (6)
A 3 3p
Ea,ß (iAx1)^(x)dx
< Cfr, (7)
A 3 3p
[a,b]2
where the constant C depends only on p.
1. Some auxiliary statements
First we give auxiliary statements. Let us consider a homogeneous polynomial of third degree in two variables.
Proposition 1 ([20]). A homogeneous polynomial of third degree in two variables may be reduced by a R-linear transformation to one of the forms: 1) x2x2, 2) x2x2 ± x2, 3) xf, 4) 0.
Definition 1. Given f G (1, m], a critical point equivalent to the critical point of the function x2x2 ± 1 is said to be a critical point of type D±, where 1 = 0 for f = m.
Definition 2. A critical point equivalent to the critical point of the function xf+1, r ^ 1 is said to be a critical point of type Ar.
Proposition 2 ( [21]). If 0 < a < 22, 3 is an arbitrary real number and f is such that
na/2 < f < min{n, na}, then there is C > 0 such that
C
\Ea,p (z)\ , z G C, fi < | arg(z)| < n. (8)
1 + |z|
Proposition 3 ([1]). Let a, 3 > 0 and f : [a, b] ^ C. Then for all A G C
Ea,p (iAf (x)) = E2a,i (-A2f 2(x)) + iAf (x)E2a,p+a(-A2f 2(x)). (9)
2. Proof of the main results
Proof of Theorem 1. Since for small A the integral (4) is clearly bounded, we prove Theorem 1 only for A > 1. Without loss of generality, we can consider the integral on [0,1]2, otherwise we reduce to this case using a linear transformation. Since we are given a homogeneous polynomial of third degree in two variables, by Proposition 1 we can represent it as one of the following: 1) x'x2; 2) x'2x2 ± x2; 3) xf 4) 0. If the phase function is f (x) = 0 it is clear that integral will be identically zero. So we will consider the other three cases separately. Using the inequalities (8) and (9) we obtain:
I Ea,p (iAf (x)) \ < I E2a,p (-A2f 2(x)) | + A I f (x) | | E2a,p+a(-A2f2(x)) | <
< C + CA|f (x) | < C(1 + A | f (x) | ) < C (10)
< 1 + A2f2(x) + 1 + A2f2(x) < 1 + A2f2(x) < 1 + A |f (x)|. (0)
Case I. First we assume that the phase function has a critical point of type Dso that f (x) =
2
= x^x2 .
We consider the integral (4) of the form:
Ia,p = / Ea,p (iAx"[x2)^(x)dx. (11)
J[ 0,1]2
We use the inequality (10) in the integral (11) and we obtain:
\Ia,ß \ =
^ \Ea,ß(i^x1x2)\\^(x)\dx ^
J[ 0,1]2
/ Ea,p (iAx1x2)4'(x)dx '[0,1]2
<^jfdijf ^ «12»
Let q be such that 1 + 1 = 1. Assume first that p = x>, so that q > 1. Then using the Holder
p q
inequality in the inner integral we get
j = i 1|5(x)|dx2 < (i'mx)^' d'1 dx2 1
ln ' J0 1 + Ax\x2 ^ \J0 J \J0 |1 + Ax2x21q
Thus,
| <C /00 H'(1 ^^d' dxi.
Then using again the Holder inequality in this integral we obtain
i ia,f i < c( / ¡0| «*> r "c^'(/1 -q-^ p «*)' <
< Ct fo 1 '¿'J+I^"' dxi
q
Let
K f1 1 - (1 + Ax2)1-q
K =Jo (q - 1)Ax2 dx1.
Since (1 + Ax1 )1-q = 1 + O(Ax2) near x1 =0 and q > 1, the integral K is convergent. To estimate K, first we use the change of variables t = ^/~Ax1 to get
K = 1 i^ 1 - (1+ t2)1-qdt = 1 i^ (1+ t2)q-1 - 1 dt = (q - 1 )VAJo t2 (q - 1)VAJo t2(1+ t2)q-1
1 f1 (1 + t2)q-1 -1 1 r^ (1 + t2)q-1 -1,
=-^-—rr,—r-dt +--= / ^-—— dt =: K1 + K2.
(q - 1)VAJo t2(1+ t2)q-1 (q - 1 )yAJi t2(1+ t2)q-1 ^ 2
Since q - 1 ^ [q], where [q] ^ 1 is the integer part of q > 1, by the Newton binomial formula
(1 + t2)q-1 < (1 + t2)q = 1 + [q]t2 + [q]([q2-1)t4 +... + t2[q\
and hence
where
Kl (q -i )Vai W +t2)— dt ^ CA'
_ i f 1 [q] + mq-11t2 +...+t2[q^-2
Cq :_ q—i Jo (TTi^ dt-
( i + t2)q-1 -i i
Moreover, since ^( 1+ ^i < ^
K _ i r^ ( i + t2)q-1 - id i r^ i
K2 (q - i)CAj1 t2(i+ t2)q-1 dt< (q - i)CaJ1 T2dt
i ^i-iu 1
Hence,
and
(q - i )y/A\ VAJ (q - i )VA'
C' i
K < -¡L, cq :_ Cq +
vA' q' q q -i'
, . C'q\W\\Lv ^a\ ^ -X-,
A 2q
where Cq is some coefficient depending only on q, and hence only on p.
Now we consider the case q = 1. Notice that the coefficient C' ^ +m as q ^ 1 and therefore we cannot directly conclude the required estimate from the one for q > 1. As q = 1, we have p = m and 0 G In view of (12), first we estimate the inner integral as
f1 ^^^ . , .. f1 dx2
Jin^ = 1, ^ 2 < SUP ^^ T^T^ ^ Jo 1 + Ax(x2 x26[0,1] JO 1 + Ax2x2
sup |0(x)| 1 sup |0(x)| ln(1 + Ax2)
X2E[0,1]
< -'t-^-ln(i + Ax2x2)
Ax1
Ax21
o
Thus
1 sup \ ^(x) \ln(1 + Ax1)
\ Ia,ß\ < f -r-2-dxi < CM\l-
0 Ax1 0
We use the change of variables Ax1 = y in the last integral and get
ln(1 + Ax1)
Ax\
dx1
,r C u\\l- rx in(i+y) , cu\\l- r
\Ia,ß \ <-71- -1-dy <--r-
lo y2 A2 Jo
ln(1 + y)
dy.
A 2 J0 y 2 A 2 J0 y 2
Note that the last integral converges. Now, using integration by parts we obtain
(13)
ln(1 + y)
dy = -
lim
2ln(1+ y)
N2
+
2dy
N1 Jo (1+ y)y2
Thus from (13) we get
4dy 2 = lo l + y =
\ Ia,ß \ <
4 arctan y \ = 2^.
C Ml-Ai .
Case II. Assume that the phase function has a critical point of type D± so that f (x) = x'x2 ±xf. We estimate the integral (4) when the phase function has a critical point of type D+ and the case D- can be done similarly. We consider the integral
Ia,ß
'[o,i]2
Ea,ß (iA(x2x2 + x\ ))^(x)dx.
(14)
Using the inequality (10) for the integral (14) we get
\ Ia,ß \
i[o,i]2
Ea,ß (iA(x\x2 + x|))^(x)dx
^ \Ea,ß(iA(x^x2 + x2))\ \^(x)\ dx ^
J[o,i]2
, 1 ,1 ^ dxi /
oo
\^(x) \dxi
1 + A(xix2 + x2)
ii
= dxi
\ -0(x) \ dxi
1 + Ax2 + Axix2
We use the Holder inequality for the last inner integral and obtain
i
\ Jin2 \ :
\ -0(x) \ dxi
<
"i \ p \^(x) \pdx2
i
dx
J0 | 1 + Ax2 + Ax2x'21 ^ \J0 J \J0 | 1 + Axf + Ax2x' |q
Then using again the Holder inequality for this integral we establish
\ Ia,ß \ <
ii
oo
\ ^(x) \ pdx2dxi
1 + Ax2 \ 2
ii
dx
oo
\ 1 + Ax\ + Ax2xi \'
dx2
Changing the variables x' = ( ———- ) t we get
\ Ia,ß \ < MLp
Ax2
o lo \ 1 + Ax22 + Ax2xi \'
dxi dx2
j r ii±M)t!d„ r
dt
o
(Ax2)2
o
(1+12)'
CXJ
CXJ
y
2
y
2
o
o
o
q
p
q
q
q
where A
Ax2 \ 2 A dt
ï+ÂXl) and 0 (T+t2jq
< C as A ^ œ. Thus,
\Ia,ß \ < C11^ ( 0 (1 ' dx^ '
Replacing x2 by A 3 t and using 1 = 1 — 1 we get
q p
\Ia,ß \ <
C \\v\\
LP
3
dr
2__I I 1 / Q -l\/7_ 1
A 3 3p \ Jo t 2 (T3 + 1)q 2
<
c\m
LP
dr
A 3 3p \Jo t 2 (T3 + 1)q 2
Since the last integral is covergent,
\Ia,ß \ <
c\m
LP
2__1
A 3 3p
Case III. Assume that the phase function has a critical point of type A2 so that f (x) = x3. We estimate the integral (4) with the phase function f (x) = xf
,■1 r 1
T-jr 3::::ix)
lo Jo
First, we use the inequality (10) for the last inner integral to obtain
"1 \'^>(x)\dxi
\Ia,ß\ ^ / \Ea,ß(iAx31)\\^(x)\dx1dx2-■Jo Jo
\Jin3 \ ■ — /
o
/0 1 + Axf
Then we use the Holder inequality for the last integral Iatwice and we get:
^ r1 dx 1
io jo J \Jo Jo \1 + Axf\q
Replacing A _3 xf by t in the above inequality, we obtain
, i . i 3
\Ia,ß \ <
1 r1 \ P
\^(x) \pdxidx2
dx2
\Ia,ß \ <
C \\V\\
LP
dt
A 3q
o
\1+ t3\
q
<
C\m
LP
A 3q
dt
o
\1+ t3\
Since 1 + 1 = 1 and the last integral converges, pq
\Ia,ß \ <
c\m
LP
, i _ A.
A 3 3p
The proof is complete.
Remark. If a = 3 =1 in the integral (4), it is called an oscillatory integral and the theorem holds for it.
Declaration of competing interest
This work does not have any conflicts of interest.
This paper was supported by "El-yurt umidi" Foundation of Uzbekistan and partially supported in parts by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). The author thanks Prof. M.Ruzhansky for proposing the problem and constant attention to the work and also thanks the referees for numerous suggestions which greatly helped to improve the exposition.
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References
[1] M.Ruzhansky, B.Torebek Van der Corput lemmas for Mittag-Leffler functions, arXiv:2002.07492.
[2] M.Ruzhansky, B.Torebek, Van der Corput lemmas for Mittag-Leffler functions. II. a-directions, Bull. Sci. Math., 171(2021), 103016.
[3] M.G.Mittag-Leffler, Sur l'integrale de Laplace-Abel., C. R. Acad. Sci. Paris, 135(1902), 937-939.
[4] M.G.Mittag-Leffler, Sur la nouvelle fonction Ea(x). Comp. Rend. Acad. Sci. Paris, 137(1903) 554-558.
[5] M.G.Mittag-Leffler, Sopra la funzione Ea(x)., Rend. R. Acc. Lincei, 5(1904), no. 13, 3-5.
[6] P.Humbert, Quelques résultats relatifs à la fonction de Mittag-Leffler. C. R. Acad. Sci. Paris 236, (1953), 1467-1468 .
[7] R.P.Agarwal, A propos d'une note de M.Pierre Humbert, C. R. Acad. Sci. Paris, 236(1953), 2031-2032.
[8] P.Humbert, R.P.Agarwal, Sur la fonction de Mittag-Leffler et quelquenes de ses generalisations, Bull. Sci. Math., Ser. II, 77(1953), 180-185.
[9] M.M.Dzherbashyan, On the asymtotic expansion of a function of Mittag-Leffler type, Akad. Nauk Armjan. SSR Doklady, 19(1954), 65-72 (in Russian).
10] M.M.Dzherbashyan, On integral representation of functions continuous on given rays (generalization of the Fourier integrals), Izvestija Akad. Nauk SSSR. Ser. Mat., 18(1954), 427-448 (in Russian).
11] M.M.Dzherbashyan, On Abelian summation of the eneralized integral transform, Akad. Nauk Armjan. SSR Izvestija, fiz-mat. estest. techn. nauki, 7(1954), no. 6, 1-26 (in Russian).
12] R.Gorenflo, A. Kilbas, F.Mainardi, S.Rogosin, Mittag-Leffler functions, related topics and applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2014.
13] K.G.van der Corput, Zur Methode der stationaren phase, Compositio Math., 1(1934), 15-38.
14] G.I.Arkhipov, A.A.Karatsuba, V.N.Chubarikov, Theory of multiple trigonometric sums., Moscow, Nauka, 1987.
15] A.N.Varchenko, Newton polyhedra and estimation of oscillating integrals Functional Analysis and Its Applications, 10(1976), 175-196.
16] J.Duistermaat, Oscillatory integrals Lagrange immersions and unifoldings of singularities, Comm. Pure. Appl. Math., 27(1974), no.2, 207-281.
17] V.N.Karpushkin, Uniform estimates for oscillatory integrals with parabolic or hyperbolic phase, Tr. Seminara im. I. G. Petrovskogo, vol. 9, Izd-vo MGU, Moscow, 1983, 3-39 (in Russian).
[18] A.Safarov, Invariant estimates of two-dimensional oscillatory integrals, Math. Notes., 104(2018), 293-302.
[19] A.Safarov, On invariant estimates for oscillatory integrals with polynomial phase, J. Sib. Fed. Univ. Math. Phys., 9(2016), 102-107. DOI: 10.17516/1997-1397-2016-9-1-102-107
[20] V.I.Arnold, S.M.Gusein-Zade, A.N.Varchenko Singularities of Differentiable Maps, Birkhauser, Boston Basel, Stuttgart, 1985.
[21] I.Podlubny, Fractional Differensial Equations, Academic Press, New York, 1999.
Оценка для функции Миттаг-Леффлера с гладкой фазой, зависящей от двух переменных
Акбар Р. Сафаров
Институт математики имени В. И. Романовского Академия наук Республики Узбекистан Ташкент, Узбекистан Самаркандский государственный университет Самарканд, Узбекистан
Аннотация. В статье рассматривается задача об оценках функций Миттаг-Леффлера с гладкими фазовыми функциями двух переменных, имеющими особенности типа и Аг. Мы обоб-
щаем результаты статей [1] и [2] на двумерные интегралы с фазой, имеющей некоторые простые особенности. Обобщение состоит в том, что мы заменяем экспоненциальную функцию функцией типа Миттаг-Леффлера для изучения типа осцилляторного интеграла.
Ключевые слова: функция Миттаг-Леффлера, фаза функция, амплитуда.
