On the problem of mean periodic extension Текст научной статьи по специальности «Математика»

138 Probl. Anal. Issues Anal. Vol. 9 (27), No 2, 2020, pp. 138-151

DOI: 10.15393/j3.art.2020.8630

UDC 517.444

V. V. Volchkov, Vit. V. Volchkov

ON THE PROBLEM OF MEAN PERIODIC EXTENSION

Abstract. This paper is devoted to a study of the following version of the mean periodic extension problem:

(i) Suppose that T e S'(Rn), n > 2, and E is a non-empty subset of Rra. Let f e C(E). What conditions guarantee that there is an F e C(Rra) coinciding with f on E, such that F * T = 0 in Rra ?

(ii) If such an extension F exists, then estimate the growth of F at infinity.

In this paper, we present a solution of this problem for a broad class of distributions T in the case when E is a segment in Rra. Key words: convolution equation, mean periodicity, continuous extension, spherical transform

2010 Mathematical Subject Classification: 44A35, 45E10, 46F10

1. Introduction. Let Rra be the real Euclidean space of dimension n with Euclidean norm |-|. By V(O) (respectively, S(O)), we denote the space of distributions (respectively, the space of compactly supported distributions) on a domain O C Rra, and by V(O), the space of compactly supported infinitely differentiable functions on O. Given T e S'(Rra), we denote the support of T by supp T, and put

Ot = [x e Rra : x - y eO for each y e supp T}.

If Ot = 0, then, for any f e T>'(0), the convolution f * T is defined on Ot by the formula

(/ * 7» = {fy, (Tx,<p(x + y))), <p eV(0T)

(the subscripts of f and T mean the action with respect to the variable indicated). We set

VT(O) = [f e V(O) : f * T = 0 in Ot} © Petrozavodsk State University, 2020

in the case where Ot = 0, and let VT(O) = V(O) if Ot = 0.

Distributions from the class V'T(O) are said to be mean periodic in O with respect to T. If M(O) is a subset of V(O), then we denote the intersection VT(O)nM(O) by Mt(O). For example, Ct(O) = VT(O)n C(O).

The general mean periodic extension problem is stated as follows.

Problem 1.

(i) Suppose that f E MT(O) and a domain 0\ contains O. Under what conditions does there exist an F E MT(0\) coinciding with f on O?

(ii) If such an extension F exists, is it unique?

Even the one-dimensional case of Problem 1 is profound and deeply related to various branches of analysis. It was studied by J.-P. Kahan, V. D. Golovin, A. F. Leont'ev, A. M. Sedletskii, and other authors (see [3], [5]- [10] and the references therein). The answer to Problem 1 (ii) is positive for every T E £'(R) by Titchmarsh's support theorem (see [2, Theorem 4.3.3] as well as [3, Chapter 5, Section 1], [7, Part 3, Chapter 1, Theorem 1.1]. The answer to Problem 1 (i) depends essentially on properties of T related to the distribution of zeros of its Fourier transform, that is, the entire function

T(z) = (T, e-lzt) , z E C.

The strongest results in this direction for the classes VT and CT^ were obtained by the first author ( [7, Part 3]). In particular, the extension in Problem 1 (i) for these classes was shown to exist under the following conditions.

(a)

|ImA|

sup -—;-r-— < +oo,

XJT ln(2+ |A|) ,

where 2(T) = [z E C : T(z) = 0}.

(b) The sequence of multiplicities m\ of the zeros A E Z(T) is bounded.

(c) For every A E Z(T) we have

(3)

mx-l £

i=o

(z - \)mx \ T(\) )

z=A

< (2+ IAD"

where the constant 7 > 0 is independent of A.

We point out that the set of distributions T e S' (Rra) satisfying conditions (a)-(c) is quite large and contains many of the distributions used in applications ( [7, Part 3]). On the other hand, there are examples of distributions T e S'(R) for which any two of the conditions (a) - (c) hold but the third fails. The question of the necessity and unimprovability of these conditions in extension theorems was studied in [10]. First, the counterexamples showing that condition (a) is unimprovable and (c) is necessary, were constructed (see [10, Theorems 3, 5]). Second, condition (b) can be weakened as follows:

SUV -—:-r-TV < +CO,

xeJcr) 1^(2+ IM) ,

and this estimate is already unimprovable even if the extension in not required to be mean periodic (see [10, Theorems 1, 2, 4]). Finally, the investigation of the one-dimensional case in [10] contains a result on non-existence of a continuous extension of a continuous mean periodic function for which m\ grows faster than |ImA| (see [10, Theorem 6]).

Some results on the possibility of a mean periodic extension of functions from the classes C and Lp for distributions T of the form

(T,p) = J (p(t) da(t), p eV(R),

—r

where a is a function of bounded variation on [-r, r] with a jump at one of the points ±r, were obtained in [3, Chapter 5], [5]. The presence of jumps of a at the points ±r imposes a number of constraints on the zeros of T. Among them are the following conditions (see [3, Lemma 5.1.1]):

(a) All zeros A of the function T are contained in some horizontal strip.

(b) The sequence of multiplicities [m\} of the zeros A is bounded.

(c) For any 8 > 0, there exists a constant cs > 0, such that

IT(z)I> c5er|Imzl

outside disks of radius 8 centered at A.

The multidimensional case of Problem 1 is more complicated and more specific. For example, the continuation is not generally unique. This can be seen from the example of linear hyperbolic differential equations with

r

constant coefficients. Moreover, when speaking of the existence of an extension, one must impose additional conditions on f (see [10, Proposition 10]). The main results in this direction in dimensions n > 2 were obtained by the authors (see [7] - [10]). This is done using the technique of transmutation operators related to expansions in eigenfunctions of the Laplacian (see [8]).

This paper is devoted to a study of the following version of the mean periodic extension problem. Problem 2.

(i) Suppose that T E £'(Rn), n > 2 and E is a non-empty subset of Rn. Let f E C(E). What conditions guarantee that there exists an F E CT(Rn) coinciding with f on E?

(ii) If such an extension F does exist, then estimate the growth of F at infinity.

In this paper, we present a solution of Problem 2 for a broad class of distributions T in the case when E is a segment in Rn (see Theorem 1 below). The precise statement of the main result is given in the next section. In Section 3, we prove some auxiliary lemmas. The proof of Theorem 1 is contained in Section 4.

2. The main results. Throughout, we assume that n > 2 and

E = [x = (xi,... ,xn) E Rn : -L < xi < L,x2 = ... = xn = 0}

for some L > 0. Let SO(n) be the group of rotations of Rn. A distribution T E V (Rn) is said to be radial if

{T,p(x)) = {T,<p(rx)), ip EV(Rn)

for all r E SO(n). We write £'(Rn) for the set of all radial distributions from the class £'(Rn). Given T E £^(Rn), we denote the spherical transform of T by T, that is,

~ n 1 / n \ / J--\(z M)\

T(Z) = 2,-) {T,,-^) , * E C

(here, r is the gamma-function and J™-i is the Bessel function of the first kind). We set

2(T) = [z E C : T(z) = 0}.

Let a > 0. We say that a distribution T from belongs to Za if

there is a sequence {ATO}^=1 of points in Z(T) such that

Xm = am + £m, m = 1, 2,..., (1)

where

^ l£m\2< (2)

m=1

We emphasize that for each a > 0 the class Za is fairly large. It is often quite easy to verify whether a given distribution T belongs to Za, using asymptotic expansions of T, which are known under very general assumptions on T (see [4, Chapter 2, Theorem 10.2]). We note also that if T e Za, then T * U eZa for each U e ^'(Rra).

For f e C(E), T e , let CT,f (Rra) denote the set of all F E CT(Rra) coinciding with f on E.

Theorem 1. The following assertions are valid.

(i) Let a > 0, T e Za, and f e C(E). Then, for each £ > 0 there is F e CT,f (Rra), such that

F(x) = O (e£,x2|) , x e Rra. (3)

(ii) For each a > 0, there is a T e Za with the following property: if f e C(E), f = 0, then for each F e CTJ (Rra) there is an £ > 0, such that

m IF(x)le-£lxl > 0. (4)

Several remarks are in order here. The proof of assertion (i) shows (see Section 4) that the function F in (i) depends only on x1 and x2. It is unclear whether the condition for T in (i) can be relaxed, but it is easy to see that this condition cannot be removed. For example, if T e ), supp T = {0}, and Z(T) = 0, then the equation F * T = 0 is an elliptic differential equation with constant coefficients. Hence, each solution F must be real-analytic in Rra, and we see that CT,f (Rra) = 0 if f e C(E) is not real-analytic in E. If, in addition, T(X) = T'(X) = 0 for some A e C and

f (x) = X1 eiXxi, x e E, then the assumption F e CTj (Rra) implies that

i\x

F (x1,0,..., 0) = x1 e 1, x1 e R

In this case, Ctj(Rra) = 0, but condition (3) is not met.

Assertion (ii) shows, in particular, that condition (3) cannot be strengthen in the general case. However, if all the numbers \m in (1) are real, then (3) can be replaced by

F(x) = 0(1+ elx2\), x e R

(see the proof of (i) in Section 4). This means, in particular, that assertion (ii) is not generally true for each T e Za.

3. Auxiliary results. The following lemmas are needed in the proof of Theorem 1.

Lemma 1. Let z\,...,zm be pairwise different complex numbers. Then, for all a e R, b > a, Cj e C (j = 1,... ,m), there is a function g e CX(R) with a support on [a,b], such that

X

9(zj )= j g(t)e-^ *dt = Cj, j = 1,... ,m. (5)

— X

Proof. For brevity, we set

. a + b b - a

Z = —, v = —. (6)

Suppose that u e CX(R), u = 0 and suppu C [—rq,'q}. Then, for each N e Z+, there exists a positive constant ^fN, such that

e^|Im z\

\U(Z)I< >yN + ^|)W for all Z e C.

Consider the function

(z) = u(z) - zi), z £ C. i=i

Let {kj}1J=1 be the sequence of multiplicities of the zeros [zj}1J=1 of v. Then the function

^ i^p kj! v(z)

w(z) = ^ ^ eiz *-3

i=1 (z - zi)k]v{k) (zi)

is entire and

w(Zj) = cjelZj?, j = 1,... ,m. Assume now that I z — Zj|> 1 for all j. Then

( )

(z — Zi)'

g^|Im z\ m

<Iv (z)I<lN (Y+^n k — Zj I for all N e Z+,

j = 1,... ,m. Together with (7), this shows that for each N e Z+ there exists a constant > 0 such that

e^|Im z\

|w(^)|< l'N (Y + forall ¿e C.

By the classical Paley-Wiener theorem, there exists a function h e C^(R), such that supp h C [—rj, rj] and h(z) = w(z). Putting g(t) = h(t — £), we conclude from (8) and (6) that g satisfies the required conditions. □

To continue, for z = (z1,..., zn) e Cn, ( = ((1,..., (n) e Cn, we set

n

(z, o = .

3=1

Also, let

S = {ze Cn :(z, z) = 1}, Sn-1 = S n Rra. (9)

Lemma 2. Let T e ^'(Rra), A e C, f(A) = 0, £e S. Then the function

hc (x) = elX(x'°, x e Rra

is in the class C~(Rra). Proof. For each y e Rra, one has

(h *T)(y) = {T,hs(—x)) h^(y). (10)

Since T is radial, we obtain

{T,hc (—x)) = {T,hc (—rx)) (11)

for all r e SO(n). Let dr be the Haar measure on SO(n) normalized by J dr =1. Formula (11) ensures that

SO(n)

{T,hc(—x)) = J {T,hc(—rx))dT=(r, J hc(—rx)dr(12)

SO(n) SO(n)

Using now [1, Introducion, Section 3.1, formula (9)], we see that

[ ht(-TX) dr [ e-lX(lxlv'du(ri),

J ^n-1 J

SO(n) S"-1

where du is the area measure on Sn-1 and

w-

dw(rj) = n-Kt/r (l + ^ .

Wn-1 . w, ■-■■ ! ' y • 2

Sn-1

Together with [1, Introducion, Section 3.2, Lemma 3.6], relation (12) brings us to the formula

{T,ht (-x)) = T(\) = 0. Now, the assertion of Lemma 2 follows from (10). □

4. Proof of the main result. Let us prove assertion (i) of Theorem 1. We start by noting that there is no loss of generality in assuming that

f(-L, 0,..., 0) = f(L, 0,..., 0) = 0 (13)

(otherwise, it is enough to enlarge the number L and consider an appropriate continuous extension of f). By the assumption on T, there is a sequence {ATO}^=1 of points in Z(T) satisfying (1) and (2).

According to what has been said above, there exist d > 0, M > 0, such that

L 1

d > max] — , M , (14)

[ n a j

and

m < d\Xm\ (15)

for all m e N, m> M. Using (13) and (14), define the function f1 e C(R) by the formula

i } (t, 0,..., 0) if t € \-L,L], /l(i) = \o if t € \-L,Ll (16)

Lemma 1 ensures us that there is a function g e C(R), such that supp g C [L,nd] and

Trd, Trd

J g(t)e-i * dt = -J fi(t)e-i * dt (17)

—Trd —Trd

for all k e Z, m< 2M. Now define

Î2(t) = h(t)+g(t), te R.

Since suppf1 C [—L,L] and suppg C [L,nd], we obtain f2 = f1 on [—L,L]. Owing to (17), the Fourier series of the function f2 on [—nd,nd] has the form

/2(t) = £ cmeit«, (18)

\m\>M

where the sequence cm e C satisfies the condition

y^ |cml2<

\m\>M

Relation (2) shows that

£m ^ 0 as m ^ (19)

For m > M, define

m

v

Amd

For z e C, z = 0, we set -Jz = \f\z\et(argz)/2, where —t < argz < t. Bearing (19) in mind, we see from (15) and (14) that

Hence,

where and

W1 - C = ^^ + ^ m>M, (20)

7

= ^(ad)2 - 1, Sm e C

= 0(em) as m ^ (21)

Next, for m e Z, \m\> M define the function hm e C~(Rra) by the formula

{

exp(iAm(xi(m + X2^J 1 - C)) if m > M,

exp (-iA-m(xi(-m + X2^1 - C-m)) if m < -M.

Since

(u V1-&, 0,..., 0)

for each m > M (see (9)), we infer, by the definition of the sequence (Am}™=l and Lemma 2, that

hm € C^(Rn) for all lml> M. (23)

For m € Z, lml> M, we set

Vm = S\rn\ (24)

(see (20)). In view of (15), (20) and (24), relation (22) yields

hm(x) = el ^(Xl+7X2) eiVmX2 (25)

for all m € Z, lml> M.

Let p € C^(R) be a non-negative function, such that

suppp c [-1,1] and J<p(t)dt=1. (26)

R

Setting pk(t) = kip(kt), k = 1, 2,... we conclude from (26) that

Wk(u)l < 1 for all u € R, (27)

(pk(u) = O (|u|-2) as u ^ <, (28)

and

lim (pk (u) = 1 for each u € R. (29)

k

Consider now the sequence of functions

m

Cm'Pk ^

\m\>M

Fk (x) = ^ cm(pk (m) hm(x), x € Rn

(30)

Relations (25), (24), (21) and (28) show that the series in (30) converges locally uniformly on Rn. Bearing (25) in mind, we have

Fk(x) = Fk, i(x) + Fk>2(x),

where

Fk, i(x) = ^ cmLpk{^ m)e"d

m\ J1 (XI +7X2) d

\m\> M

Fk, 2(x)= £ c^ *1 {X1+1X2) [eiVmX2 — 1) .

\m\>M

Let $ e C(R) be 2nd - periodic and assume that

$(t) = /2(t) for te [—nd,nd]. (31)

It is easy to verify the relation

Fk, 1 (x) = ($ * (k)(x1 + 7x2) = J $(t)(fik(^1 + 7^2 — t) dt

R

(see (18)). This implies that the sequence Fk> 1 converges locally uniformly on Rra and

lim Fk, 1(x) = $(x1 + 7x2). Next, using (27) and the estimate

< \VmX2\e^Im^^

etVmX2 _ 1| =

VmX2

j elzdz 0

we now get

^ KM7) (etVmX2 — < ^ I E I\e^X2lmn

|m|>M |m|>M

Hence,

IFk,2(x)I<Ix2Ie^ ^ (IcJ2+IVmI2) , (32)

|m|>M

where

u = max |Imrnm\.

|m|>M

In addition, the sequence Fk, 2 converges locally uniformly on Rra and

^(x) = lim Fk 2(x) = V cme%(X1+^X2) (eirimX2 — 1)

\m\>M

(see (29)). Estimate (32) yields

№)I=0 (\x2\eu|x2|) , x e Rra. (33)

We now define the function F e C(Rra) by the formula

F(x) = $(x1 + 7x2) + ^(x), x e Rra.

Together with (30), condition (23) shows that Ffc e CT(Rra) for each k, whence F e CT(Rra). Since ^(x) = 0 for x e E, by the definition of f2 and (31), we obtain F = f on E. Using now (33), (21), and (19), we see that for a sufficiently large M, the function F is the required extension of .

Now we prove (ii). Let {£m}'^=1 be a sequence of positive numbers satisfying (2). Consider the function

u(z)

fi 0- i), -C

(34)

where Am = am + iem. Setting am = Am/a, from expansion of function in infinite product, we have

sin KZ

u(az) =-v(z), z E Z,

■

where

*)=n (i -H1 -

m=1

Let us estimate lv(z)| for the case when

(35)

lz — ll> - for each I E Z.

One has

^ o

*) = Urn 2(

m=1

m\ y m—a^\ / m — am am) V m — z )\ m + z

so that

œ / 2 / lnlv(z)l< ^ (ln(—) +ln(

m=A am V

m — ar,

m

Using the inequality ln(1 +1) <t for t > 0, we infer that

m=1 x 1 1

ii

+

laml im — zl im + zl

)

<

(36)

)+ln(i+m^^ )) ) \ m + z J J

1

< -

<x * m=1 N

1

2 F2 +

, . . 1 I |2

TO=i \ ii

+

1

+

1

|m — z\2 |m + z\

o.

Hence, from (36) it follows that

where the constant 71 > 0 is independent of z. Then, owing to (35) and the maximum modulus principle,

Hz)|< 72e^|Im^, z G C,

where 72 > 0 does not depend on z. Due to the Paley-Wiener theorem for the spherical transform (see [7, Part 1, Theorem 6.5]), there exists T G £'(Rn), such that T = u. Moreover, from the definition of {ATO}^=1, we see that T g We now estimate |T"(Aq)| for sufficiently large q G N. Assume that £q < a/8 and |z — Aq |= a/8. This yields

and

i + i

N

a

> 2 — ^ > 1

> 1 + £« < 1.

* 8 a 4

(37)

(38)

Then, from (34) and (37) it follows that

u(z)

z - Xn

\ * + Ag \

\A*\2 m=i

m = q

n

1 —

n

! An !

n \ m=1

m = q

1

(am)2

Together with (38), this shows that there exist 73,74 > 0, such that

u(z)

z- An

>

Ta

I sin - Izl

I a 11

\ An \2

1

M2

(an )2

>

74

\ An \2.

Thus,

\ T' (An ) \ > min |*-A, | = I

u(z)

z- An

>

74

\ An\2.

(39)

(40)

Assume now that / G CT, /(Rra) for some / G C(E), / = 0. Using (40) and the definition of |ATO}X=1, we see from [7, Part 3, Theorem 3.2] that condition (4) is true for some e > 0. Thus, Theorem 1 is completely proved.

y

2

z

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Received March 14, 2020.

In revised form, May 23, 2020.

Accepted May 23, 2020.

Published online June 9, 2020.

Donetsk National University 24 Universitetskaya str., Donetsk 283001, Russia E-mail: [email protected]

Donetsk National University 24 Universitetskaya str., Donetsk 283001, Russia E-mail: [email protected]