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Regularity Properties of Distributions and Ultradistributions PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 12, Pages 3531{3537 S 0002-9939(01)06013-0 Article electronically published on June 28, 2001 REGULARITY PROPERTIES OF DISTRIBUTIONS AND ULTRADISTRIBUTIONS S. PILIPOVIC´ AND D. SCARPALEZOS (Communicated by Jonathan M. Borwein) Abstract. We give necessary and sufficient conditions for a regularized net of a distribution in an open set Ω which imply that it is a smooth function or Ck function in Ω. We also give necessary and sufficient conditions for an ultradistribution to be an ultradifferentiable function of corresponding class. Introduction Algebras of generalized functions are usually defined as factor algebras of cer- tain algebras of sequences (nets) of smooth functions (cf. [1], [2], [3], [8], [11]), and the classical spaces of functions, distributions and ultradistributions are em- bedded into the appropriate algebra through their regularizations; they are also equivalence classes of sequences (nets) of smooth functions. Solutions in algebras of generalized functions of PDE are also presented by such sequences (nets) and even differential operators (for example, with singular coefficients) are sometimes replaced by sequences (nets) of differential operators with smooth coefficients. So the natural question arises: Under what conditions is a given generalized function (a generalized solution of PDE) actually a classical function of appropriate class, distribution, or ultradistribution? The partial answers to such questions are given in Propositions 1, 2 and more gen- erally in Theorems 1, 2, respectively, in terms of asymptotic behaviour of sequences of functions provided that these sequences are evaluated on the ultraproduct ΩN: Denote by (θn) (respectively, (φn)) a δ-sequence, also called a sequence of molli- fiers, of smooth functions (respectively, of appropriate ultradifferentiable functions). Precise definitions will be given in sections 2 and 5, respectively. We will prove: 0 Proposition 1. (i) Let (Tn) be a regularized sequence of T 2E(Ω); i.e. Tn = N (α) m T ∗ θn;n 2 N: If (9m 2 R)(8(xn) 2 Ω )(8α 2 N0)(Tn (xn)=O(n )),then 2 1 O T C0 (Ω).( is the Landau symbol.) 0k (ii) Let (Tn) be a regularized sequence of T 2E (Ω);k2 N0.If(8(xn) 2 N 8 ≤ (α) O 2 k Ω )( α k)(Tn (xn)= (1)),thenT C0 (Ω): ∗ Let (Mp) be a sequence of positive numbers satisfying conditions (M.1) ,(M.2) and (M.3)'. These conditions and the definitions of corresponding ultradistribution spaces will be given in section 1. Received by the editors February 21, 2000. 2000 Mathematics Subject Classification. Primary 46F05, 46F30, 03C20. c 2001 American Mathematical Society 3531 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3532 S. PILIPOVIC´ AND D. SCARPALEZOS 0(Mp) Proposition 2. Let (Tn) be a regularized sequence of T 2E (Ω); respectively, 0fMpg T 2E (Ω); of the form Tn = T ∗ φn;n 2 N; where (φn) is a sequence of mollifiers of Mp-type such that: N N (i) (9h>0)(9m 2 R)(8(xn) 2 Ω ) respectively, (8h>0)(9m 2 R)(8(xn) 2 Ω ) α (α) h jT (x )j ∗ sup n n = O(eM (mn)): α2N0 Mα f g (ii) (9b>0)(8φ 2E Mp (Ω)) respectively, (9b>0)(8φ 2E(Mp)(Ω)) −M ∗(bn) hT − Tn,φi = O(e ): f g Then T 2D Mp (Ω), respectively, T 2D(Mp)(Ω): Note that in Proposition 1 we can use any regularized sequence, while in Propo- sition 2 we need a special regularized sequence. This is discussed in section 5. In relation to distribution theory, our results can also be considered in the frame- work of asymptotic analysis; cf. [4] and references therein. 1. Regularized sequences and regularity properties We assume that the reader is familiar with Schwartz's distribution theory and its traditional notation. Here, we will recall some basic facts concerning ultradis- tribution spaces (cf. [6]). Denote by (Mp) a sequence of positive numbers with M0 = 1 satisfying the assumptions ∗ ∗ 2 ≤ ∗ ∗ 2 N (M.1) (Mp ) Mp−1Mp+1;p : p (M.2) Mp ≤ AH MqMp−q;p2 N;q ≤ p; for some A>0andH>0: P1 (M.3)' Mp−1=Mp ≤∞: p=1 ∗ ∗ ∗ ∗ ∗ 2 N Recall, M0 =1;Mp = Mp=p!;mp = Mp=Mp−1;mp = Mp =Mp−1;p : (M.1)∗ implies the well-known condition (M.1) of [6]. We refer the reader to [6], [7] and [9] for the analysis of these conditions. ∗ The associated function M and the growth function M related to (Mp)p are defined by p p t ∗ t M(t)= sup ln ;M(t)= sup ln ∗ ;t>0: p2N0 Mp p2N0 Mp We use the convention M(x)=M(jxj);M∗(x)=M ∗(jxj);x2 R (cf. [6]). Let Ω be an open set in R.ThenK ⊂⊂ ΩmeansthatK (or its closure) is a compact subset of Ω. Recall, for ' 2 C1(Ω); α j (α) j k k h ' (x) ⊂⊂ ' K;h;Mp =sup ;h>0;K Ω: x2K,α2N0 Mα Denote by EMp;h(K) the space of smooth functions on Ω for which the above seminorm is finite and by DMp;h(K) its subspace consisting of smooth functions supported by K. Define E(Mp)(Ω) = proj lim proj lim EMp;h(K) = proj lim E(Mp)(K); K!Ω h!1 K!Ω f g f g E Mp (Ω) = proj lim ind lim EMp;h(K) = proj lim E Mp (K); K!Ω h!0 K!Ω License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use REGULARITY PROPERTIES 3533 D(Mp)(Ω) = ind lim proj lim DMp;h(K) = ind lim D(Mp)(K); K!Ω h!1 K!Ω f g f g D Mp (Ω) = ind lim ind lim DMp;h(K) = ind lim indD Mp (K): K!Ω h!0 K!Ω (K ! ΩmeansthatK runs over all compact sets exhausting Ω.) Their strong duals are spaces of compactly supported and general Beurling and Roumieu ultradistributions, respectively (cf. [6]). 2. R 0 Let T 2D(Ω); suppT = K ⊂⊂ Ω,θ2D; θ(x)dx =1andθn(x)= nθ(nx);n2 N: Let Tn(x)=T ∗ θn(x);n2 N: For any 2E we have −1 hTn − T; i = hT; ∗ θˇn − i = O(n ): This is clear because, by Taylor's formula, one has (for some C>0; some compact set K1;K1 ⊃ K and k 2 N) jhT ∗ θ − T; ij ≤ Cj ∗ θˇ − j n Z n K1;k ≤ j − − ˇ j ≤ −1 C ( (x y) (x))θn(ny)ndy K1;k Cn : Thus, any regularized sequence for a distribution T satisfies condition b) of (i) and (ii) in Theorem 1 given below. 3. In this section we prove more general assertions than the ones in Proposition 1. 1 Theorem 1. (i) Let (fn) be a sequence of C functions on Ω supported by a compact set K ⊂⊂ Ω and T 2D0(Ω); suppT ⊂ K: Assume: N (α) m a) (9m 2 R)(8(xn) 2 Ω )(8α 2 N)(fn (xn)=O(n )): −b b) (9b>0)(8φ 2E(Ω))(hT − fn,φi = O(n )): Then T 2 C1(Ω). k (ii) Let (fn) be a sequence of C functions on Ω (k 2 N is fixed) supported by K ⊂⊂ Ω and T 2D0(Ω); suppT ⊂ K: Assume: N (α) a) (8(xn) 2 Ω )(8α ≤ k)(fn (xn)=O(1)): k −b b) (9b>0)(8φ 2E (Ω))(hT − fn,φi = O(n )): Then T 2 Ck(Ω). 2 1 2 k Remark 1. (i) If T C0 (Ω), respectively, T C0 (Ω); then one can prove that conditions in (i), respectively, (ii), hold for T and its regularized sequence (Tn). (ii) Proposition 1 follows from Theorem 1. Proof. (i)Hypothesisa)isequivalentto j (α) j O m 2 N sup fn (x) = (n ); for every α : x2Ω In fact, if it is not so, then there would exist some α 2 N such that 8 9 2 N 9 2 j (α) j m ( C>0)( nC )( xC Ω)( fnC (xC ) >CnC ): By choosing C = N 2 N,wehaveasequence(xN ) such that j (α) j m 2 N fnN (xN ) >NnN ;n : License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3534 S. PILIPOVIC´ AND D. SCARPALEZOS Putting yn = xN ;nN ≤ n<nN+1 (assuming that (nN )N is increasing) we see that there does not exist C>0 such that j (α) |≤ m 2 N fn (yn) Cn ;n : By taking N>C;this inequality does not hold for yn;n>N;which contradicts the hypothesis. Let a = b=2. By assumption b), we have a −a hn (T − fn),φi = O(n ),φ2E(Ω);n2 N: a 0 Thus, (n (T − fn)) is a sequence in E (Ω)convergingto0asn !1: This implies that the convergence of this sequence takes place in some Ck(¯!); where ! is an a open bounded set such that K ⊂⊂ ! ⊂⊂ Ωandthat(n (T − fn)) is a bounded sequence in the dual space (Ck(¯!))0: By the Banach-Steinhaus theorem, we have a (9C>0)(8 2E(Ω))(8n 2 N)(jn hT − fn; ij ≤ Cjj jj!;k¯ ) j j j (p) j ( !;k¯ =supx2!;p¯ ≤k (x) ). −iy· k If y = e ; we have jj yjj!;k¯ ≤ C(1 + jyj) : For Fourier transforms T^ and f^n; this implies a k jn (T^ − f^n)(ξ)|≤C(1 + jξj) ,ξ2 R;n2 N: (C denotes positive constants which can be different.) Note that f^n 2S(R);n2 N: We have the following estimates coming from the hypothesis on (fn): For every r>0; there exists Cr > 0 such that m −r jf^n(ξ)|≤Crn (1 + jξj) ,ξ2 R;n2 N: Thus, for ξ 2 R;n2 N; −a k m −r jT^(ξ)|≤n C(1 + jξj) + n Cr(1 + jξj) : Now, for given ξ 2 R we choose n such that − − −1 p k n =[(1+jξj) a ]: By putting this in the previous inequality, we obtain −p p+k −r jT^(ξ)|≤C((1 + jξj) +(1+jξj) a ),ξ2 R: p+k − − ^ Choose r such that a r< p.
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