Math. Nechr. 160 (1991) 161-161

Local and Global Descriptions of Periodic Pseudodifferential Operators

By WILLIAM MCLEAN of Sydney

(Received May 11, 1989)

Abstract. Periodic pseudodifferential operators can be defined either globally, via FOURIER series, or else locally, via partitions of unity and FOURIER integrals. Here, it is proved that the two defi- nitions are equivalent.

Introduction

Consider a function f : R* -+ C. We will say that f is periodic if it is unit periodic in each of its n variables, i.e., if

f ( x + k) = f (z) for x E R* and k E 2”.

A periodic operator is one which takes periodic functions to periodic functions. The aim of this paper is to establish the equivalence of two alternative descriptions of periodic pseudodifferential operators. The first of these may be called the global descrip- tion, and relies on the FOURIER series representation of a periodic function. The second is the local description, in which a partition of unity enables one to work with the FOURIER integral representation of a non-periodic function.

The global approach is much less cumbersome than the local approach, and has for this reason proved to be extremely useful in the study of pseudodifferential equations on simple, smooth, closed curves - via parametrization, these are equivalent to one- dimensional, periodic pseudodifferential equations. For example, the spectral properties of such operators were studied by AQRANOVICH [ 13, and the solvability of degenerate equations was investigated by ELSCHNER [3]. The global approach has also proved to be a valuable tool for analysing numerical methods for boundary integral equations; see, e.g., the papers by ARNOLD and WENDLAND [2], MCLEAN and WENDLAND [8], SARASEN and WENDLAKD [lo], SCHMIDT [12], and SLOAN [13].

The virtue of the local approach is that i t constitutes a special case of the theory of pseudodifferential operators on manifolds. Thus, in proving the equivalence of the

This rcsrarch WRS carried out while the author was a QE2 Fellow at the University of Tasmania.

162 Math. Nachr. 160 (1991)

local and global descriptions, we are showing that the global definition of a periodic pseudodifferential operator is fully consistent with the standard definition of a pseudo- differential operator on a manifold.

AQRANOVICH [ 13, [ 151 gave a proof - which he attributed to L. R. VOLEVICH (private communication) - that the local and global definitions of a classical pseudodifferential operator on a closed curve (or on a torus) are equivalent. SILRANEN and WENDLAND [ 111 published a detailed, elementary proof for the case of a onedimensional pseudo- differential operator possessing a homogeneous principal symbol - a result which suffices for most applications. In this paper, we deal with the general case of a periodic pseudodifferential operator of order m and type (e, 6).

The paper is divided into four sections. The first of these introduces the tools used for relating the local and global descriptions, namely, the periodization operator and a partition of unity. The role of 0 2 is to fix a consistent notation for dealing simul- taneously with FOURIER integrals and FOURIER series. Section 3 discusses the local and global symbols of a periodic operator. Finally, $ 4 contains the main result of the paper, which is stated as Theorem 4.4.

In this paper, we deal only with 0" functions. The technical report [7] contains a detailed treatment of some classes of periodic distributions which are important for the theory of pseudodifferential operators. In particular, the equivalence of the local and global definitions is proved for SOBOLEV, BESSEL potential and BESOV spaces. These results are useful for the study of boundary integral methods, because whereas trace theorems and a priori estimates for boundary value problems are given in terms of local norms, it is usually more convenient to use global norms when dealing with boundary integral operators.

1. The n-Torus

For an open set B 2 Rn, let

a(s) = cZD,,p(s)J is(B) =

and equip these vector spaces with their usual locally convex topologies; see, e.g., [5]. We use the standard multi-index notation for partial derivatives, writing

where = (ul, . . ., a,) is an n-tuple of non-negative integers. The set 2% is an additive subgroup of Rn, and any periodic function on R' induces

a unique, well-defined function on the quotient group

T" = R*/Zfl.

Conversely, every function defined on the n-torus T n gives rise to a unique periodic function on Rfl. To lessen the notational burden, two such functions will be identified, by writing f ( x ) = f (x + 2.) when f is periodic. In this way, we suppress the distinction between the point x E R" and the coset x + 2" E Tn.

McLean, Local and Global Descriptions 163

Let Q,, denote the closed subspace of I(Rn) conaisting of the C" periodic functions, then the relative topology of Qo coincides with the locally convex topology induced by the family of seminorms

f I+ max Iaaf(z)I. reR"

(1.1)

Define the perfcdization operator

then it is easily verified that

9: .B(Rn) + Qo

is a continuous linear map satisfying

(1.2) q e f ) = (se) 1, e E B ( R ~ ) , 1 E go.

The notation Q C Rn is used to indicate that 9 is an open subset of Rn, whose closure 0 is compact and satisfies

(1.3) a r ! ( k + 8 ) = 0 for O + k k E * .

In this case, for any set E G 9, we define

9 E = U (k + B ) , kcZn

80 that if the support of the function f satisfies supp f c 9, then 9f is periodic with

We shall view T" as an n-dimensional manifold, with a preferred atlas of local co- ordinate systems, given by the restrictions of the canonical map x H x + Zn t~ those open sets 9 satisfying 9 C Rn. The manifold TI admits partitions of unity, 80 if Q1, . . ., QN are open sets satisfying

Qj C R* for 1 5 j 5 N

SUPP 9f = WUPP f ) and (9f ) la = 1.

N R" = U 9 Q j ,

j=l

then (since the sets 52, + 2" form an open cover of Tn) there exist functions O,, . . ., 0, whose periodizations form a partition of unity, i.e.,

supp 0, c Qi for 1 5 j 5 N, N

j= I bei = 1 on Rn.

If follows at once from (1.2) that every periodic function f can be decomposed as fol- lows:

N

j=1 f = z Jv,f) -

154 Math. Nachr. 160 (1991)

Also, since the periodization operator commutes with a/ax, (1 I I 5 n), the partial derivatives of a periodic function f satisfy

N

j-1 aaf = c s[ayejf)]. (1.4)

When T n is viewed as a manifold, the vector space B(Tn) = Cm(Tn) is defined as the set of all periodic functions f satisfying (O f ) l a E B(0) whenever B C R* and 0 E B(0). The compactness of T" means that B(T") is a F R E C H ~ T space, with the defining family of seminorms

indexed by u and j , Obviously,

8 0 = B(Tn),

and one can see from (1.4) that the topology of B(Tn) can also be described using the global seminorms (1.1).

2. Fourier Analysis

Being a compact abelian group, the n-torus possesses a unique, normalized translation- invariant integral, which we write simply as

j f ( 4 ax, T"

for every f E Ll(Tn). Of course, this is just the ordinary LEBESOUE integral of the periodic function f taken over one whole period, i.e., over any translate of the cube [O, lln.

For E E R*, define the group character x E : Rm + { z E C : IzI = 1) by

xt(x) = ei2n€.z , xERn,

where E . x = Elxl + . + ERxR is the EucuDean inner product; notice that x E is periodic if and only if E E 2". The FOURIER transform for the group R" is defined by

(s,./) ( E ) = 1 e-iZn?.z/(x) ax, 5 E ~ n , R"

when f E Ll(R"), and the FOURIER transform for Tn is defined by

( S T n f ) (6) = 1 e-iZnt,z/(x) ax,

f = Sf,

5 E ~ n , T"

when f E L,(Tn). For brevity, wc will write

provided the context makes clear which FousmR transform is intended. With these definitions, the inversion formulae are, for the non-periodic case,

f (z) - 1 f (6) e""nE.z at, x E Rn, R"

MoLeen, Local and Global Descriptions 155

and for the periodic case,

(Of course, this is just the complex FOURIER series expansion.)

multiplication operator by a ( X ) , 80 that Given a function x H a(x), it will be convenient to denote the associated pointwise

(2.1) [ a m 4 (4 = 44 44 *

xa = ( ~ p . . . (xn)=n,

Also, we write

D" = (24- l4 a=, then

(2.2a) ( x ~ D f ) ( E l = E " / ( E ) , E E R", (2.2b) 3 n 4 - X ) " f = Oaf,

for any f E .B(Rn). In order to state the corresponding formulae in the periodic setting, we define

1)*1 . . . (e-ianzn - 1)*", (e-i2nz - 1)" = (e-i2nz1 - and introduce the partial forward difference operator on Z",

Aa = A:;, AT;, , where, for 1 5 j 5 TZ and ?I: Zn --f C,

A , ~ ) ~ ( E D * * ., 6j, - * - 3 E n ) z= ~((1, . . ., E j + 1, * * - 3 E n ) - v(E1, . . - 9 E j , * * 6 n ) . It is easy to see that Acb commutes with A,,, for all j , 1 E (1,2, . . .) n}, and that

J e-i2nn".z(e-iZnzj - 1) = A (1) . f (0, 5 E 2".

T" Thus,

(2.34 (J-,&"f) (6) = E"f(6) 9 E E zfl 7

(2.3 b)

for any f E B(T").

operator will be denoted by a(D) , so that

(2.4) ;f,*[a(D) U ] ( E l = a ( [ ) a(5), 5' E R".

In this way, if u ( t ) = p, then a(D) = D". Sindarly, any suitable function a defined on 2" gives rise to a periodic FOURIER multiplier operator, which we denote by a(D,), in order to avoid confusion. Thus,

3T,(e- i2nX - 1)" f = Pf ,

If a is a suitable function defined on Rn, then the corresponding FOURIER multiplier

3Tn[a(Do) U ] (6) = ~((6) a(E), t E Zn,

where u is periodic.

156 Math. Nachr. 160 (1991)

3. Symbols and Operators

Suppose 8 is an open subset of R*. A function a : 8 x R" --f C is called a symbol on 8 if a E Cm(S x R") and if, for every multi-index ,!l and every compact set K c Q, there exist constants c = c(@, K ) and r = r(@, K ) such that

(3.1) l@(z, E ) l 5 c(1 + z E K , E E R".

The set of all symbols on 8 is denoted by sm(8). For a E Sm(S) and u E a(8), we define

(3.2)

this notation generalizes (2.1) and (2.4), because if a(%, 5 ) = al(s) a2(E), then a(X, D )

[u(x, D ) u] (x) = J a(2, 6) a(() ei2nt*zdt, x E A?; R"

= a m a2m.

The estimates (3.1) imply

a ( X , D ) : a(0) --f S(8) is a continuous linear map, and it is a remarkable fact that every continuous linear map from a(8) to S(8) is locally of this form. Indeed, fix a cutoff function t3 E CZmp(8), then by the FOURIER inversion formula,

eu = J w) eXE a, u E am, R"

(3.3)

with the integral converging in a@). Hence, if the map

(3.4) A : a(Q) --f S(8)

is continuous and linear, then

A ( W = J W) 4 e X b ) dE, R"

with the integral converging in g(8); cf. [S, p. 1681. Define

(3.5) a@(%, 5) = e-i*n).z [A(BX~)I (XI, z E 8, E E R", then a, E Sm(8) and

(3.6) A(t3u) = a,(X, D ) u, u E a(8).

We say that a, is a local symbol of the operator A . Turning now to the periodic setting, let Cm(T" x 2") denote the set of all functions

a : T n x Zn + C having the property that for each 5 E Z", the map x H a(x, t ) belongs to Cm(Tn). A function a E Cm(T* x 2") is called a periodic symbol if, for every multi- index 8, there exist constants c = c(@) and r = r(8) such that

(3.7) la!&(x, E)1 5 c(1 + z E T", E E 2".

The set of all periodic symbols is denoted by Sm(Tn).

McLeen, Local and Global Descriptions 167

By analogy with (3.2), we write

[ a ( X , Do) u] (2) = a(x, E ) d(E) eiZnE.z, x E T", EEZ"

for a E S"(Tn) and u E 2)(Tfl), then the estimates (3.7) imply that

a ( X , Do): . B ( T n ) --f .B(Tn)

is a continuous linear map. Conversely, if

(3.8)

is continuous and linear, and if u E B(Tn), then

Au = A Z 4 6 ) X E = a(E) A X E ,

A : . B ( T n ) --f .B(Tn)

[&a. 1 EEZ"

with the sum converging in B(Tn). Hence,

A = a@, Do) Y

where n E S"(Tn) is defined by

(3.9) a(2, t ) = e-i2nE.z(AXE) (2) , z E 2'" , E E 2".

We call a the (global) periodic symbol of the operator A .

Indeed, if l2 Q Rn and 8 E C&,(SZ), then by analogy with (3.5) we define It is also possible to describe the periodic operator (3.8) in terms of local symbols.

a,(z, t ) = e-i2nE.z (3.10) ~ [ q o x o i (x), 2, E E ~ n .

Notice that a&, E ) is periodic in x when E E Z", so a, is well-defined on Tn x Zn. In fact, one finds that a, E Sm(R*) and a, E Sm(Tn). Applying the periodization operator to the FOURIER integral representation (3.3) gives

q e u ) = J a ( ~ ) w X e ) at , E BW), R"

with the integral converging in B(T*), 80

A[9(8u)] = a,(X, D) u, u E B(R*); cf. the analogous result (3.6) for the non-periodic operator (3.4).

4. The Equivalence Theorem

The following symbol class was introduced by HORENLNDER [4], and plays a central role in the theory of pseudodifferential operators.

Definition 4.1. Suppose m, Q and B are real numbers satisfying

(4.1) - m < m m m , O < e S l , O S S < l ,

and let Q be an open subset of Rn. A function a E Cm(Q x Rn) is called a symbol of order m and type (p , S) if, for all multi-indices (Y, ,4 and every compact set K c l2,

168 Math. Nechr. 160 (1991)

there exists a constant c = c(u, 8, K) such that

lQ;a(x , t ) ] 5 c(1 + Itl)m-plul+61bI, x E K , 5 E R". The set of all such symbols is denoted by Sr'(l2).

class as follows. Comparing (2.2) and (2.3), i t is natural to define the corresponding periodic symbol

Definition 4.2. A function a E C"(T* x 2.) is called a periodic symbol of order m and type (e, S) if, for all Iqulti-indices OL and 8, there exists a constant c = c(01, /?) such that

18: A";(x, E) l 5 c(l + Itl)m--Plul+d161, x E T", t E 2".

The set of all such symbols is denoted by SF,(T**).

Also, it is customary to define S-" = n S m , then The most important case occuw when p = 1 and IS = 0, so on0 writes S m = 8Eo.

meR s-" c SF6 c 8"

for all m, e and 6 satisfying (4.1).

The connection between the periodic and non-periodic symbol classes is given in the next theorem, which is the main result of this paper. The proof makes use of the follow- ing simple lemma.

Lemma 4.3. If f E Cr(Rn) and la1 = r, then

for every s E R. Proof . Let E E 2". There exists vl such that E l 5 5 E l + a, and

Successively taking differences with respect to the variables f P , . . ., 5, we eventually arrive at

1 A"f(E) == 7 a"f(q),

a.

for some r] satisfying Ej 5 qi 5 lj + ai (1 5 j 5 n) . PEETRE'S inequality in the form

(1 + I E l r S 5 2lSV + IE - r]l)lal (1 + and the fact that 16 - 71 5 C lajI2 5 1611, now imply the result.

(j:1

Theorom 4.4. Suppose that A : B(T") --+ B(T") is a continuow linear map. The global periodic symbol a, defined by (3.9), belong8 to SE,(Tn) if and only if the local symbol ag, defined by (3.10), belongs to Sr,(ln) whenever ln C R" and 8 E C~o,,,(sZ).

McLean, Local and Global Descriptions 169

Proof . Assume first that a E SZ,(T"), then

end so

e(z, E ) = z a(z, 7) J e - ~ * ~ ( q - ~ ) . ( u - z ) ~ ( Y)dY. a

(4.2) qEZ"

Note that this sum converges because the integral over 9 is rapidly decreaaing aa a function of 7.

For every multi-index u, define

then 0. E Cm(Q x Q) because Q satisfies (1.3). Let

b.(z, 7) = J e-i2*v(u--+)@ a (5 , Y ) dy 9 zEQ, vER", a

then integration by parts shows that

(4.3) b. E S-=yQ),

because @,,(z, y) = 0 for y E Q \ supp 0. Recalling (2.2b) and (2.3b), we find

a; J e-i2n(tl-~)4u-z),3( Y) dy = (- 1)'"' A;h(z, 7 - t ) , a

so, using (4.2) and applying summation by part,s, i t follows that

Therefore, since a E Senfd(T"), PEETRE'S inequa.lity implies

(4.4) pp;aO(z, E)1 5 cN(2, E ) (1 + IEl )m-e la l+@I , 2 E Q, E E R",

where c = c(a, ,!?, 0 ) and

M ( s , E ) = z Z laZb.(z, 7 - E)I (1 + ( r ] - E l ) l m l + Q l = l + V VEZ" B'SB

Since b,, satisfies (4.3), and since M ( z , 6) is periodic in 5, it is clear that M is bounded on K x R" for any compact set K c S, and hence the estimate (4.4) implies E SE,(SZ).

Now assume ae E S:,(Q) whenever S C R" and 0 E C~,,,,,(Q). If we choose a parti- tion of unity as in 0 1, then

160 Math. Nachr. 160 (1991)

Thus, in order to prove that the global symbol satisfies a E SZ,(Tn), it suffices to show that any local symbol satisfies E SE,,(T").

Given 9 C R" and 0 E C~,,(B), choose a compact set K c 9 such that

(4.5) dist (supp 8, 9 \ K ) > 0.

Taking 10) = ~~$us (x , E ) in Lemma 4.2, we see that

(4.6) Ia$A&(X, E)I 5 ~ ( l + ~ E ~ ) m - t ' ~ a l + d l ~ ~ , x E K , 5 E 21,

because E S,?&2). In fact, this estimate holds for x E 9 K , since ae(x, E ) is periodic in x for each E E 2". The case x E R* \ 9 K can be treated using the pseudo-local property of the operator A , as follows.

Let G be the SCHWARZ kernel of A, so that formally

( A 4 (4 = J Y) U(Y) dy, 2 E T", T"

for any u E .B(T"), then

a&, 5) = e-iZn6.z J a(x , y) S[0;,] (y) dy = 1 e-i2nl.(v-z)G ( x ,Y)O(Y)dY* T" R

It can be shown that a, which is a distribution on T n x T", has its singular support along the diagonal - in other words, B is a C" function outside the set ((x, y) E R" x R" : x - y E 2"). (The proof of this result relies on the assumption that e > 0, and uses the fact that R" x R" can be covered by sets of the form 99 x 99 with 9 E R"; see [9, pp. 187, 2851 or [la, p. 391.) Consequently, the condition (4.5) implies that G is C" on a neighbourhood of the set (R" \ 9 K ) x supp 0. Integration by parts then showa that, for all multi-indices u, p and for every r > 0, there is a constant c = c(m, p, r, K ) such that

18; Aiae(5, E ) l 5 ~ ( 1 + lEl)-r, Combining this estimate with (4.6), we conclude that

x E R" \ B K , 5 E 2".

E S,?'(T"), as required. 0 If the indices e and 6 satisfy, in addition to (4.1), the inequalities

(4.7) then the symbol class Srd(f2) is invariant under smooth changes of coordinates. I n this caw, given a compact C" manifold r, it is possible to define the set Yzb(I') of pseudodifferential operators on r of order m and type (e , 6); see [6, pp. 95- 1061, [9, pp. 204-2081 or [14, pp. 47-49].

Assume now that I' is diffeomorphic to T I , then there exists a C" periodic para- metrization

1 - e 5 6 < e ,

y : T" -+ r, which is one-one modulo Z", i.e.,

y ( r ) = y ( y ) w 2 - Y E Z",

~ ( Y I , .-., Yn)

a(z,, . . .)xn)

and whose JAcoBIan satisfies

+ O for all%.

McLean, Local and Global Descriptions 161

Associated with any continuous linear operator

A : a(r) +- a(r) is a continuous linear operator

A,,: a(Tn) + .B(Tn), defined by A,u = [ A ( u o y- l ) ] o y. An immediate consequence of Theorem 4.4 is the following result.

Corollary 4.6. Assume that m, e and S satisfy (4.1) and (4.7). The operator A belmtgs to YJe,>(T) if and only if the global periodic symbol of the operator A , belongs to Sr’(T*).

References

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121 1). N. ARNOLD and W. L. WENDLAND, The convergence of spline collocation for strongly elliptic equations on curves, Numer. Math. 47 (1985) 317-341

[3] J. ELSCHNER, “Singular Ordinary Differential Operators and Pseudodifferential Equations”, Lecture Notes in Math. 1128, Springer, Berlin, New York 1985

[d] L. HdRM.4NDER, Pseudo-differential operators m d hypoelliptic equations, in “Singular Inte- grals”, Proc. Sympos. Pure Math. 10, 1967, pp. 138-183

1.51 .J. H O R V ~ T R , Topological Vector Spaces and Distributions, Volume 1, Addison-Wesley, Reading, Massachusetts 1966

[ G I K. KUMANO-GO, Pseudo-Differential Operators, The MIT Press, Cambridge, Massachusetts, and London, England 1981

[7] W. MCLEAN, Periodic pseudodifferential operators and periodic function spaces, Applied Mathematics Preprint, University of New South Wales

[S] W. MCLEAN and W. L. WENDLAND, Trigonometric approximation of solutions of periodic pseudodifferential equations, Operator Theory: Advances and Applications 41, Birkhiluser, Besel, 1989

[9] B. E. PETERSEN,! Introduction t o the Fourier Transform and Pseudodifferential Operators, Pitman, Boston, London 1983

[lo] J. SARANEN and W. L. WENDLAND, On the asymptotic convergence of collocation methods with spline functions of even degree, Math. Comp. 46 (1985) 91-108

[I13 -, -, The Fourier series representation of pseudodifferential operators on closed curves, Complex Variables 8 (1987) 55--64

[12] G. SCHMIDT, The convergence of Galerkin and collocation methoda with splines for pseudo- differential equations on closed curves, Z. Anal. Anwendungen 8 (1984) 371-384

[13] I. H. SLOAN, A quadrature-based approach t o improving the collocation method, Numer. Math. 54 (1988) 41-56

[14] M. E. TAYLOR, “Pseudodifferential Operators”, Princeton University Press, Princeton, New Jersey 1981

[15] M. S. AQRANOVICH, On elliptic pseudodifferential operators on a closed curve, Trans. Moscow Math. SOC. 47 (1985) 23-74

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