Kučera, Jan

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Institute of Mathematics AS CR, v. v. i.

Czechoslovak Mathematical JournalJan KučeraOn representation of temperate distributionsCzechoslovak Mathematical Journal, Vol. 22 (1972), No. 2, 191–194Persistent URL: http://dml.cz/dmlcz/101089Terms of use:© Institute of Mathematics AS CR, 1972Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.czCzechoslovak Mathematical Journal, 22 (97) 1972, Praha ON REPRESENTATION OF TEMPERATE DISTRIBUTIONS JAN KUCERA, Pullman (Received December 8, 1969) Every temperate distribution can be represented as a linear combination of deriva­tives of square integrable on R" functions with polynomials as coefficients (see e.g. [3]). In this paper we substitute differentiation and multiplication by a family of bounded linear operators and show that then the problem of representation is equiva­lent to the problem of existence of solutions of some operational equations in the dual space. Let Я, X, be Hubert spaces with inner products (,, .)fj, (., .)̂ .̂, respectively. For a bounded linear operator T : H ~> К we denote by T* its adjoint, i.e. (Гх, y)j^ = = (x, Г*у)я for all X e Я, yeK, and by T its dual, i.e. y{Tx) = (Ту) (x) for all xe H, у G К\ where К' is the strong dual space of K. The space of all infinitely differentiable complex functions on i^" is denoted by ^^{R") and CO{R'') is its subspace of all functions with compact support. ^ = = {fEC'^{R"); sup Ix^DVWI < +00 for all multiindices a, Д}, 6^' denotes its dual. For Ä cz H, where Я is a Hilbert space, l.h. Л means the linear hull of A, Due to Fréchet-Riesz Theorem for each Hilbert space L it exists a semi-linear bijective map Ij^ : L^ L such that /^(j^) x — (x, д)^ for all x, у e L. Evidently T = IjjT^I^^, As an immediate consequence of this identity we get Theorem 1. Let Tp j e J, be a family of bounded linear operators Tj : H -> К, where Я, К, are Hilbert spaces. Then l.h. и T;K' = H' iff l.h. и TfK = я . JeJ jeJ Let us have a sequence (1) H"" :D H' ZD H^ ZD . . . of Hilbert spaces with an inner product (., \ and a norm||. ||̂  respectively. Let they satisfy the following hypothesis: 191 (H) For each integer к ^ 0 the space m+^ is dense in H^ and the identity operator / : H^^^ -> H^ is continuous. The intersection H^ = f) H^ is a locally convex space with a topology defined by the norms ||. ||̂ ,̂ к = 0, 1, 2, ,.. It was shown in [4] that if (H) holds then the dual 00 space {Н'^'У equals to the space of restrictions on Я°° of all elements of \J (H^y. If we moreover assume that Я^' = ^ (including the topology) then 6^' is a union of Hubert spaces (Я^)', A: = 0, 1, 2, . . . In this particular case Theorem 1 implies Theorem 2. Be given a sequence (1) satisfying (H) and a family of linear bounded operators Tj : H^ -^ H^, j e J, such that for each integer к ^ 0 the restriction of Tj on H^^^ (which we denote again by Tj) is a bounded operator from H^^^ into H^ for each j e J. Then, for each integer m ^ 0, elements of (Я"")' can be represented as linear combinations of Т^Л^^^ ... Tj^g, where g e (Н^'У iff ].h.\J T^H^ = H^''' for к = 0, 1,2, ..., Ш - Ï. " '^-^ Example 1. For any integer /c ^ 0 let (2) H' = {/• : R" ^ C; V f x'f^\Dj{x)\' dx < + o ) ! , where derivatives DJ are generalized. Define operators T^ = h Tj = Xp Tj+^^ = = iidjôxj), j = 1, 2, ..., Л, T = YJ '^P ^^^^^ ^^ ^̂  ^ Hubert space with an inner product (/, g\ = U. (TV ) (TVj'dx. ^ 00 00 The spaces (2) fulfil (l) and (H). Moreover OH^ = 6^. Hence (J (H^y = ^ ^ To simplify some formulae we will further write (Н^У = H~^. It was shown in [3] that for e a c h / 6 H~^ and for each multiindex a, |a| ^ /<, there exists/^ e H^ = L?{R") and a polynomial P.^ of degree ^/c — |a| such that (3) / = I P.D% , W\uk where the derivative D ^ is in the ditributional sense. For each integer к '^ 0 and each j = 0, 1, ..., 2n, the operator Tj is a linear bounded operator from H"'^^ into Я^. Further Tj = Tj, where the bar denotes the complex-conjugate operator. Hence, according to Theorem 2 we have л (4) Lh. и1;*Я^ = Я^+1 f^j, ic = 0 , 1 , 2 , . . . J = 0 192 Fix j = 0, 1, ..., 2n, and /< = 0, 1, . . . By definition ( r / , g\ = (/, T^'g),^, for feH'-"', деНК This is equivalent to {f^){T'Tjf) = (f^^^ffg)(T''-''f) and {T/F^J)f = {T^^^Yfg)f. Hence TjT^^g = T^^^^Tfg as elements in H-^-\ By (4) we have (5) j2k+2^k+ 1 _ 11̂  у Т.Т^̂ ^Я'̂  , /с ^ о . Prove that (5) implies (6) j2k^2f^k+l ^ j^j-k-i f̂ .̂ /c ==0,1,2, . . . Evidently 72^ + 2^/0+1 ^ я - ' ^ ' Ч To prove the opposite inclusion let first к = 0. 2n Then by (5) T^H' = l.h. (J TjH"^ = H'K For the induction assume that (6) holds for an integer к - i ^ 0. Then by (5) T^^-^^H^^^ = Lh. IJ TjT^^H^ = 2n ' j = 0 = l.h. и Tß-^ = н-^'К j = o Now for any integer/с ^ 0 we can write Я "'^'^ = 7-2̂  + 2^^+1 ^ j^j2k+ij^k+i^^ ^ c: ГЯ . As the opposite inclusion is trivial we have Theorem 3. For any integer /c ^ 0 and any g e H ^ the equation с has a solution f e H~^. Evidently, the representation (З) follows immediately from Theorem 3. Example 2. Let ^ : i^" -> R be measurable. Assume that there are constants A, B, n n such that A{i + ^ ^j) й % ) è В{1 + Y. ^]) ^^^ ail x e R'\ If we define an inner product in spaces (2) by (8) { L 9 \ - l W\âk J .9^"i"'DyD^^dx, /c = 0, 1,2, ... then the representation (З) remains the same. Each operator Г,, i = 0, 1, ...,2/7, from H^^^ into Я^', /с ^ 0, remains bounded on Я^^Ч Similarly as in Example 1 we get an identity (in fj"^"^) (9) TjiY. D%^'-\^^D^g))= Y D^(S^^^-\ß\DPTf'g), \а\^к \ß\ak+l where g e Я ^ /с - 0, 1, 2, ..., j = 0, 1, ..., 2п. Then from Theorem 2 it follows 193 Theorem 4. For any he H ^ the equation (10) Y. i>%9'"'"'i)y) = h has a solution fe H^, к = 1, 2, ... P roo f by the mathematical induction. References [1] L. Schwartz: Théorie des distributions. Nouvelle édition, Hermann, Paris 1966. [2] L. Bers, F. John, M. Schechter: Partial Differential Equations, Interscience 1964. [3] J. Kucera: Fourier L2~transform of Distributions, Czech. Math. J., 19 {94), Praha 1969, pp. 143-153. [4] Ю. M. Березинский: Пространства с негативной нормой, Успехи матем. наук, т. 18, вып. 1(109), январь—фебраль 1963, стр. 63—96. Author's address: Washington State University, Pullman, Washington 99163, U.S.A. 194 

On representation of temperate distributions

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On representation of temperate distributions

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Institute of Mathematics AS CR, v. v. i.