RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL . 103 (1), 2009, pp. 17–48 An´alisis Matem´atico / Mathematical Analysis
On the structure of certain ultradistributions Manuel Valdivia Abstract. Let Ω be a nonempty open subset of the k-dimensional euclidean space Rk . In this paper we show that, if S is an ultradistribution in Ω, belonging to a class of Roumieu type stable under differential operators, is a family fα , α ∈ Nk0 , of elements of L∞ loc (Ω) such that S is represented in the P then there α form α∈Nk D fα . Some other results on the structure of certain ultradistributions of Roumieu type are 0 also given.
Sobre la estructura de ciertas ultradistribuciones Resumen. Sea Ω un subconjunto abierto no vac´ıo del espacio eucl´ıdeo k-dimensional Rk . En este trabajo demostramos que si S es una ultradistribuci´on en Ω, perteneciente a una clase de tipo Roumieu k ∞ estable bajo operadores diferenciales, P entoncesα existe una familia fα , α ∈ N0 , de elementos de Lloc (Ω) tal que S se representa en la forma α∈Nk D fα . Tambi´en se dan otros resultados sobre la estructura de 0 ciertas ultradistribuciones de tipo Roumieu.
1 Introduction and notation Throughout this paper all linear spaces are assumed to be defined over the field C of complex numbers. We write N for the set of positive integers and by N0 we mean the set of nonnegative integers. If E is a locally convex space, E will be its topological dual and · , · will denote the standard duality between E and E . σ(E , E) denotes the weak topology in E and β(E , E) is the strong topology in E . E stands for the topological dual of E [β(E , E)]. We identify in the usual manner E with a linear subspace of E . We represent by ρ(E, E ) the topology in E given by the uniform convergence on every compact absolutely convex subset of E [β(E , E)]. If A is a closed bounded absolutely convex subset of E, we write EA to denote the normed space given by the linear span of A in E, with A as closed unit ball.We say that a subset B of E is locally compact (weakly compact) whenever there is a closed bounded absolutely convex subset A of E such that B is contained in EA and it is a compact (weakly compact) subset in this space. Given a Banach space X, B(X) denotes its closed unit ball and X ∗ is the Banach space conjugate of X. Given a positive integer k, if α := (α1 , α2 , . . . , αk ) is a multiindex of order k, i.e., an element of Nk0 , we put |α| for its length, that is, |α| = α1 + α2 + · · · + αk , and α! := α1 !α2 ! · · · αk !. Given a complex function f , defined in the points x = (x1 , x2 , . . . , xk ) of an open subset O of the k-dimensional euclidean space, which is infinitely differentiable, we write Dαf (x) :=
∂ |α| f (x) , k . . . ∂xα k
α2 1 ∂xα 1 ∂x2
x ∈ O,
α ∈ Nk0 .
Presentado por / Submitted by Dar´ıo Maravall Casesnoves. Recibido / Received: 8 de Diciembre de 2008. Aceptado / Accepted: 14 de enero de 2009. Palabras clave / Keywords: Roumieu class, ultradifferentiable class, ultradistributions. Mathematics Subject Classifications: 46F05. c 2009 Real Academia de Ciencias, Espa˜na.
17
M. Valdivia
We take a sequence of positive numbers M0 , M1 , . . . , Mn , . . . satisfying the following conditions: 1. M0 = 1. 2. Logarithmic convexity: 3. Non quasi-analiticity:
Mn2 ≤ Mn−1 Mn+1 ,
n ∈ N.
∞ Mn−1 < ∞. Mn n=1
We consider an open subset Ω of Rk . We shall say that a complex function f defined and infinitely differentiable in Ω is ultradifferentiable of class {Mn } if, given an arbitrary compact subset K of Ω, there exist C > 0 and h > 0 such that |Dα f (x)| ≤ C h|α| M|α| ,
x ∈ K, α ∈ Nk0 .
We put E {Mn } (Ω) to denote the linear space over C of all the ultradifferentiable complex functions of class {Mn }. We write D{Mn } (Ω) to mean the linear subspace of E {Mn } (Ω) formed by those functions that have compact support. Given h > 0 and a compact subset K of Ω, by D(Mn ),h (K) we denote the linear space over C of the complex functions f , defined and infinitely differentiable in Ω, with support in K, such that |Dα f (x)| < ∞. |α| M x∈Ω h |α| α∈Nk 0
|f |h := sup sup
We assume that D(Mn ),h (K) is endowed with the norm | · |h . We take now a fundamental system of compact subsets of Ω: K1 ⊂ K2 ⊂ · · · ⊂ Km ⊂ · · · We have that D{Mn } (Ω) =
∞
D(Mn ),m (Km ).
m=1 {Mn }
(Ω) as the inductive limit of the sequence (D(Mn ),m (Km )) of Banach spaces. The We consider D elements of the topological dual D{Mn } (Ω) of D{Mn } (Ω) are called ultradistributions in Ω of the Roumieu type. We assume that D{Mn } (Ω) is endowed with the strong topology. By K(Ω) we represent the linear space over C of the complex functions defined continuous and with compact support in Ω. If K is a compact subset of Ω, we use K(K) to denote the linear subspace of K(Ω) formed by those functions with support contained in K. For f in K(K), we put |f |∞ := sup |f (x)|. x∈Ω
We assume K(K) is provided with the norm | · |∞ . We shall consider K(Ω) as the inductive limit of the sequence of Banach spaces (K(Km )). A Radon measure u in Ω is an element of the topological dual K (Ω) of K(Ω). Given a Radon measure u in Ω and a compact subset K of Ω, we write u(K) to indicate the norm of the restriction of u to the Banach space K(K). We use L∞ loc (Ω) to denote the linear space over C formed by the complex functions defined and Lebesgue measurable in Ω which are essentially bounded in each compact subset of Ω. Given a compact subset K of Ω and f ∈ L∞ loc (Ω), we put |f |K,∞ to denote the essential supremum of f in K, that is, |f |K,∞ := supess{ |f (x)| : x ∈ K }. 18
On the structure of certain ultradistributions
In the usual way, we shall consider the elements of L∞ loc (Ω) as distributions in Ω. In [4] and [5], a theorem on the structure of ultradistributions is given which we can state in the following way: Result a) Let uα , α ∈ Nk0 , be a family of Radon measures in Rk such that, for each h > 0 and each compact subset K of Rk , we have that sup h|α| M|α| uα (K) < ∞.
(1)
α∈Nk 0
Then the formula S :=
D α uα
(2)
α∈Nk 0
defines an element of D{Mn } (Rk ). Conversely, each ultradistribution S of D{Mn } (Rk ) may be written in the form of (2) for a family uα , α ∈ Nk0 , of Radon measures in Rk satisfying condition (1). In [3], certain objections to the method of proof used by Roumieu to obtain result a) are indicated and therefore the result is left as an open question. This led Komatsu to obtain result a) for ultradistributions in Ω under the additional assumption for the converse part that the class {Mn } of ultradifferentiable functions in Ω be stable for differential operators, that is, there exist A > 0 and L > 0 such that Mn+1 ≤ A Ln Mn ,
n ∈ N0 .
(3)
In this paper, in the first place, we give proof of the Roumieu-Komatsu result with no need of condition (3), i.e., we recover result a) . On the other hand, the method used to achieve this result will be used later to obtain the main result of this paper, which has as a particular case the following: Result b) If Mn , n ∈ N0 , satisfies condition (3), then, given S in D{Mn } (Ω), there is a family fα , α ∈ Nk0 , of elements of L∞ loc (Ω) such that, for each compact subset K of Ω and h > 0, we have that sup h|α| M|α| |fα |K,∞ < ∞
α∈Nk 0
and S=
D α fα ,
α∈Nk 0
where the series is absolutely and uniformly convergent on every bounded subset of D{Mn } (Ω).
2
Basic constructions
Let X be a Banach space. We use · to denote its norm and also for the norm of X ∗ . Given r ∈ N and α ∈ Nk0 , we put, for each x ∈ X, x |x|r,α := |α| . r M|α| ∗ for the conjugate of Xr,α . The norm We write Xr,α for the linear space X with the norm | · |r,α , and Xr,α ∗ ∗ of Xr,α will be denoted by | · |r,α . Clearly, if u belongs to X , then
|u|r,α = r|α| M|α| u. We represent by Yr the linear space over C formed by the families ( xα : α ∈ Nk0 ) of elements of X, which we shall simply denote by (xα ), such that (xα )r := sup
α∈Nk 0
xα |α| r M|α|
< ∞.
19
M. Valdivia
We endow Yr with the norm of · r . It follows that Yr ⊂ Yr+1 and that the canonical injection from Yr into Yr+1 is continuous. We write Y for the inductive limit of the sequence of Banach spaces (Yr ). Let (xα ) be an element of Y and let β be in Nk0 , we define xβ , if α = β, xβα := 0, if α = β Clearly, if (xα ) is in Yr , then (xβα ) belongs to Yr and (xβα )r ≤ (xα )r . For fixed r ∈ N and β ∈ Nk0 , we put Yrβ to denote the subspace of Yr formed by those elements (xα ) which satisfy that xα = 0 when α = β. Then (xα )r = sup
α∈Nk 0
xβ xα = |β| = |x|r,β , r|α| M|α| r M|β|
and thus Yrβ is isometric to Xr,β . On the other hand, if we denote by Y β the subspace of Y whose elements (xα ) satisfy that xα = 0 when α = β, then Y β is topologically isomorphic to X. We assume that Y is provided with the strong topology. If Yr∗ is the Banach space given by the conjugate of Yr , then the projective limit of the sequence of Banach spaces (Yr∗ ) is a Fr´echet space which coincides with Y . Setting Ur to be the polar set of rB(Yr ) in Y , it follows that Ur , r ∈ N, is a fundamental system of zero neighborhoods in Y . If we consider a bounded subset B of Y , its polar set B ◦ in Y is a zero neighborhood in this space and so there is s ∈ N such that B ◦ ⊃ Us , hence B is contained in the closure of sB(Ys ) in Y . Proposition 1. The following properties hold: 1. For each r in N, B(Yr ) is a closed subset of Y . 2. If B is a bounded subset of Y , there is r ∈ N such that B is contained in Yr and it is a bounded subset of this space. P ROOF.
1. We take v ∈ X ∗ and β ∈ Nk0 . Let u be the linear functional on Y such that u((xα )) = xβ , v,
(xα ) ∈ Y.
Given s ∈ N and (xα ) ∈ Ys , we have |u((xα ))| = |xβ , v| ≤ xβ · v =
xβ |β| s M|β| v ≤ (s|β| M|β| v)(xα )s s|β| M|β|
and so u belongs to Y . We take a net { xα,j : j ∈ J, ≥ } in B(Yr ) which converges to (xα ) in Y . We fix β ∈ Nk0 and take v ∈ X ∗ . Let w be the element of Y such that (yα ), w = yβ , v,
(yα ) ∈ Y.
Then 0 = lim(xα − xα,j ), w = limxβ − xβ,j , v j
j
and thus the net { xβ,j : j ∈ J, ≥ } converges weakly to xβ in X. Given that |xβ,j |r,β =
20
xα,j xβ,j ≤ sup |α| = (xα,j )r ≤ 1, |β| r M|β| α∈Nk0 r M|α|
On the structure of certain ultradistributions
it follows that |xβ |r,β ≤ 1, from where we deduce that sup |xα |r,α = sup
α∈Nk 0
α∈Nk 0
xα ≤ 1 r|α| M|α|
and hence (xα ) is an element of Yr that belongs to B(Yr ). 2. We know there is r ∈ N such that B is contained in the closure of rB(Yr ) in Y . Making use of part 1 the result follows. If u is an arbitrary element of Y , we put, for each r ∈ N, u(r) := sup{ |(xα ), u| : (xα ) ∈ B(Yr ) }. For every u ∈ Y and every β ∈ Nk0 , we identify in a natural manner the restriction of u to Y β with an element uβ of X ∗ . Proposition 2. For u ∈ Y and r ∈ N, we have sup r|α| M|α| uα ≤ u(r)
α∈Nk 0
and
(xα ), u =
xα , uα ,
(xα ) ∈ Y.
α∈Nk 0
P ROOF.
We fix β ∈ Nk0 . We then have that u(r) = sup{ |(xα ), u| : (xα ) ∈ B(Yr ) } ≥ sup{ |(xβα ), u| : (xα ) ∈ B(Yr ) } = sup{ |xβ , uβ | : (xα ) ∈ B(Yr ) } = |uβ |r,β = r|β| M|β|uβ
from where
sup r|α| M|α| uα ≤ u(r).
α∈Nk 0
Let us now take (xα ) in Yr and we see that the family (xβα ) : β ∈ Nk0 is summable to (xα ) in Ys for every integer s ≥ 2r. Given an arbitrary q ∈ N, we have that xα (xβα ) = sup |α| (xα ) − s |α|>q s M|α| |β|≤q
xα 1 xα = sup |α| |α| |α| M (2r) 2 r M|α| |α|>q |α| |α|>q
≤ sup ≤
xα 1 1 sup |α| = q (xα )r , q 2 α∈Nk0 r M|α| 2
and the conclusion is obtained. It then follows that, if (xα ) is any element of Y , we have that, in this space, (xβα ) (xα ) = β∈Nk 0
and thus (xα ), u =
α∈Nk 0
(xβα ), u =
xβ , uβ .
β∈Nk 0
21
M. Valdivia
Proposition 3. If M is a bounded subset of Y and r ∈ N, then sup r|α| M|α| uα < ∞.
α∈Nk 0 u∈M
P ROOF.
We have that
U := { v ∈ Y : v(r) ≤ 1 }
is a zero neighborhood in Y , hence there is b > 0 such that bM ⊂ U . If u ∈ M , then u(r) ≤ b−1 and, after the previous proposition, the result follows. Proposition 4. If ( zα : α ∈ Nk0 ) is a family of elements of X ∗ such that, for each h > 0, sup h|α| M|α| zα < ∞,
α∈Nk 0
then there is a unique element u of Y for which uα = zα , α ∈ Nk0 . P ROOF.
We fix β ∈ Nk0 , r ∈ N and (xα ) ∈ Yr . Then |xβ , zβ | ≤ xβ · zβ = ≤
xβ (2kr)|β| M|β| zβ (2kr)|β| M|β|
1 (xα )r sup (2kr)|α| M|α| zα (2k)|β| α∈Nk 0
and consequently β∈Nk 0
|xβ , zβ | ≤ (xα )r sup (2kr)|α| M|α| zα α∈Nk 0
β∈Nk 0
1 (2k)|β|
= 2 sup (2kr)|α| M|α| zα · (xα )r , α∈Nk 0
from where it follows that the complex function u defined in Y such that u (xα ) := xα , zα , (xα ) ∈ Y, α∈Nk 0
which is clearly linear, is also continuous. After Proposition 2, we have that (xα ), u = xα , uα , (xα ) ∈ Y. α∈Nk 0
We fix β ∈ Nk0 and take an arbitrary element xβ ∈ X. Then, if xβα := 0, α = β, and xββ := xβ , we have that xβ , uβ = (xβα ), u = xβα , zα = xβ , zβ α∈Nk 0
and so uβ = zβ , β ∈ Nk0 . The uniqueness of u follows from the density in Y of the linear span of ∪{ Y β : β ∈ Nk0 }. Proposition 5. Let E be a locally convex space such that E [β(E , E)] is a Fr´echet space. Let F be a linear subspace of E such that each closed bounded absolutely convex subset of F is locally weakly compact, then F is closed in E [σ(E , E )] and, provided with the Mackey topology μ(F, F ), it is an (LB)-space. 22
On the structure of certain ultradistributions
P ROOF. Let v be a linear functional on F , which is bounded on every bounded subset of F . Let A be a closed bounded absolutely convex subset of E [σ(E , E )]. We find a subset M of F , closed bounded and absolutely convex, such that A∩F is contained in FM and it is weakly compact in this space. Consequently, A ∩ F is weakly compact in E [σ(E , E )] and so v −1 (0) ∩ A is weakly compact in E [σ(E , E )]. We apply Krein-Smulyan’s theorem, [2, p. 246], and we have that v −1 (0) is closed in E [σ(E , E )]. Thus F is closed in E [σ(E , E )] and v is continuous in F . The result now is clear. Proposition 6. Let E be a locally convex space such that E [β(E , E)] is a Fr´echet space. If F is a subspace of E such that every closed bounded absolutely convex subset of F is locally compact, then F , with the topology induced by ρ(E, E ) is an (LB)-space. P ROOF. We apply the former proposition and obtain that F [μ(F, F )] is an (LB)-space and F is closed in E [σ(E , E )]. Let F ⊥ stand for the subspace of E orthogonal to F . Let ψ be the canonical mapping from E onto E /F ⊥ . Every closed bounded absolutely convex subset of F is locally compact in E [σ(E , E )], from where we have that E [β(E , E)]/F ⊥ is a Fr´echet-Montel space and thus μ(F, F ) = ρ(F, E /F ⊥ ). We now consider a closed absolutely convex zero neighborhood U in F . Let U ◦ be the polar of U in E /F ⊥ . It follows that U ◦ is compact in E [β(E , E)]/F ⊥ , hence we may find a compact absolutely convex subset P of E [β(E , E)] such that ψ(P ) = U ◦ , [2, p. 274]. If P ◦ is the polar set of P in E, we have that P ◦ ∩ F = U , and the result follows. Proposition 7. If X is reflexive, then Y is weakly locally compact. P ROOF. Given a positive integer r, we take a sequence (xα,m ), m = 1, 2, . . . in B(Yr ). For each α of Nk0 , the sequence xα,m , m = 1, 2, . . ., belongs to B(Xr,α ), hence we may find by means of a diagonal process a subsequence (yα,m ), m = 1, 2, . . ., of (xα,m ) such that, for each α ∈ Nk0 , the sequence yα,m , m = 1, 2, . . . , converges weakly in Xr,α to an element yα which will clearly lie in B(Xr,α ). We deduce then that (yα ) ∈ B(Yr ). We take now an integer s ≥ 2r and we show that (yα,m ) converges weakly to (yα ) in Ys . In the proof of Proposition 2, we saw that, given (xα ) in Yr and q ∈ N, we have that 1 (xβα ) ≤ q (xα )r . (xα ) − 2 s |β|≤q
Thus, given ε > 0, we find q0 ∈ N such that 1/2q0 < ε/4. Then 1 ε (yαβ ) ≤ q0 (yα )r < (yα ) − 2 4 s |β|≤q0
β β β ) is the element of Y such that yα,m = 0, if α = β, and yβ,m = yβ,m , it follows that and, if (yα,m
1 ε β (yα,m ) ≤ q0 (yα,m )r < . (yα,m ) − 2 4 s |β|≤q0
We now take u in B(Ys∗ ). We find m0 ∈ N such that
ε β
(yαβ ) − (yα,m ), u < , 2
m ≥ m0 .
|β|≤q0
Then, for those values of m, we have that
β β
(yα ) − (yα,m ), u
(yα ) − (yα,m ) +
(y β ) − (y β ), u
(yα,m ) − (yα ), u + α α,m
|β|≤q0
|β|≤q0
23
M. Valdivia
ε
β ≤ (yα ) − (yαβ ), u + (yα,m ) − (yα,m ), u + 2 |β|≤q0 |β|≤q0 ε β (yαβ ) + (yα,m ) − (yα,m ) + ≤ (yα ) − 2 s s |β|≤q0
|β|≤q0
< ε. We obtain from here that B(Yr ) is a weakly compact subset of Ys and the result follows. {Mn }
3 The space E0
(Ω)
(M ),h
Given h > 0, we denote by E0 n (Ω) the linear space over C of the complex functions f , defined and infinitely differentiable in Ω, which vanish at infinity and so do each of its derivatives of any order, that is, given β ∈ Nk0 and ε > 0, there is a compact subset K of Ω such that |Dβ f (x)| < ε,
x ∈ Ω \ K.
On the other hand, f also satisfies that there is C > 0 such that |Dα f (x)| ≤ C h|α| M|α| , We put
α ∈ Nk0 .
x ∈ Ω,
|Dα f (x)| |α| M x∈Ω h |α| α∈Nk 0
|f |h := sup sup (Mn ),h
and assume that E0
(Ω) is provided with the norm | · |h . We write {Mn }
E0
(Ω) :=
∞
(Mn ),m
m=1
E0
{M }
(Ω)
(M ),m
and consider E0 n (Ω) as the inductive limit of the sequence (E0 n (Ω)) of Banach spaces. We assume {M } {M } that the topological dual E0 n (Ω) of E0 n (Ω) is endowed with the strong topology. We put C0 (Ω) for the linear space over C of the complex functions f which are defined and continuous in Ω and vanish at infinity. We write |f |∞ := sup|f (x)| x∈Ω
and assume that C0 (Ω) is provided with the norm | · |∞ . In this section, we substitute the space X of the previous section by C0 (Ω). Then, each element of Yr is a family ( fα : α ∈ Nk0 ) of elements of C0 (Ω) such that (fα )r := sup
α∈Nk 0
|fα |∞ |α| r M|α|
< ∞.
We denote by Wr the subspace of Yr formed by those families ( Dα f : α ∈ Nk0 ) such that (Mn ),r
f ∈ E0 Let
(Mn ),r
Φr : E0 be such that
24
Φr (f ) = (Dα f ),
(Ω).
(Ω) −→ Wr (Mn ),r
f ∈ E0
(Ω).
On the structure of certain ultradistributions
Then, Φr is an onto linear isometry. We put W for ∪{ Wr : r ∈ N } and we consider W as a subspace of Y . Let {M } Φ : E0 n (Ω) −→ W be such that
{Mn }
Φ(f ) = (Dα f ),
f ∈ E0
(Ω).
Clearly, Φ is a continuous one-to-one and onto linear map. Theorem 1. For each j of a certain set J, let ( μα,j : α ∈ Nk0 ) a family of complex Borel measures in Ω such that, for each h > 0, sup h|α| M|α| |μα,j |(Ω) < ∞. α∈Nk 0 j∈J
{M }
Then, there is a bounded subset { Sj : j ∈ J } in E0 n (Ω) such that {M } ϕ, Sj = Dα ϕ dμα,j , j ∈ J, ϕ ∈ E0 n (Ω), α∈Nk 0
Ω
where the series converges absolutely and uniformly when j varies in J and ϕ belongs to any given bounded {M } subset of E0 n (Ω). P ROOF.
We consider each μα,j as a linear functional on C0 (Ω) by means of the duality ϕ, μα,j = ϕ dμα,j , ϕ ∈ C0 (Ω). Ω
Then, the norm of this linear functional is |μα,j |(Ω). We apply Proposition 4 and obtain, for every j ∈ J, an element uj in Y such that its restriction to Yα coincides with μα,j , α ∈ Nk0 . Making use of Proposition 2 we obtain that (fα ), uj = fα dμα,j , (fα ) ∈ Y. (4) α∈Nk 0
Ω
We fix now a bounded subset B of Y . We find r ∈ N such that B is a bounded subset of Yr . We take (fα ) in B. It follows that
fα dμα,j ≤ |fα | d|μα,j |
Ω
Ω k α∈Nk α∈N 0 0 |fα |∞ |μα,j |(Ω) ≤ α∈Nk 0
1 |fα |∞ (2kr)|α| |M|α| |μα,j |(Ω) |α| |α| (2k) r M |α| α∈Nk 0 ⎞ ⎛ 1 ⎟ ⎜ ≤ fα r ⎝ sup (2kr)|β| M|β| |μβ,j |(Ω)⎠ . (2k)|α| β∈Nk k =
0
j∈J
α∈N0
We deduce from this that the series in (4) converges absolutely and uniformly when j varies in J and (fα ) varies in B. Besides
sup (fα ), uj < ∞, j∈J (fα )∈B
25
M. Valdivia
from where we get that { uj : j ∈ J } is a bounded subset of Y . If ψ is the map Φ considered from {M } E0 n (Ω) into Y , and t ψ is the transpose of ψ, we put Sj := t ψ(uj ),
j ∈ J.
{M }
{Mn }
Then { Sj : j ∈ J } is a bounded subset of E0 n (Ω). On the other hand, for each ϕ ∈ E0 have that (Dα ϕ), uj = ψ(ϕ), uj = ϕ,t ψ(uj ) = ϕ, Sj .
(Ω), we
{Mn }
(Ω) and j ∈ J, making use of (4), we obtain ϕ, Sj = (Dα ϕ), uj = Dα ϕ dμα,j .
Consequently, for each ϕ ∈ E0
α∈Nk 0
Ω
{M }
Finally, when ϕ varies in a bounded subset of E0 n (Ω), (Dα ϕ) varies in a bounded subset of Y , from where we deduce that the series above converges absolutely and uniformly when j varies in J and ϕ belongs {M } to any given bounded subset of E0 n (Ω). We shall need later the following result which is found in [3, p. 42]: Result c) Let K be a compact subset of Ω . If 0 < h < h < ∞, then the canonical injection from D(Mn ),h (K) into D(Mn ),h (K) is a compact map. For each compact subset K of Ω, we put D{Mn } (K) :=
∞
D(Mn ),r (K)
r=1
and assume that D{Mn } (K) is provided with the structure of (LB)-space as the inductive limit of the sequence (D(Mn ),r (K)) of Banach spaces. By D{Mn } (K) we denote the strong dual of D{Mn } (K). For the two next propositions, we fix a compact subset K of Ω. Given r ∈ N, let Vr be the subspace of Y whose elements have the form (Dα ϕ), with ϕ in D(Mn ),r (K). Let Λr : D(Mn ),r (K) −→ Vr be such that
Λr (ϕ) = (Dα ϕ),
ϕ ∈ D(Mn ),r (K).
It follows that Λr is an onto linear isometry. We put V := ∪{ Vr : r ∈ N } and we consider V as a subspace of Y . We write Λ : D{Mn } (K) −→ V such that
Λ(ϕ) = (Dα ϕ),
ϕ ∈ D{Mn } (K).
Clearly, Λ is linear continuous one-to-one and onto. Proposition 8. Λ is a topological isomorphism. P ROOF. We take a closed bounded absolutely convex subset A of V . Applying Proposition 1 we obtain −1 r ∈ N such that A is a bounded subset of Vr . Then Λ−1 (A) is a bounded absolutely convex r (A) = Λ (Mn ),r {Mn } subset of D (K), and is closed in D (K). We apply result 3 and obtain that Λ−1 r (A) is compact in D(Mn ),r+1 (K), from where we deduce that A is compact in Vr+1 . Applying now Proposition 6 we have that V is the inductive limit of the sequence (Vr ) of Banach spaces. Consequently, Λ is a topological isomorphism. 26
On the structure of certain ultradistributions
We consider now D{Mn } (K) as a subspace, clearly closed, of D{Mn } (Ω). If A is a closed bounded absolutely convex subset of D{Mn } (K), then A is a closed bounded absolutely convex subset of D{Mn } (Ω), hence there is m ∈ N such that A is a compact subset of D(Mn ),m (Km ), [3, p. 44], therefore A is locally compact in D{Mn } (K). Proposition 6 applies again to have that the topology induced by D{Mn } (Ω) in D{Mn } (K) coincides with the original topology of D{Mn } (K). In what follows we put Γ for the mapping Λ considered from D{Mn } (K) into Y . Then, if t Γ is the transpose of Γ, we have that t Γ : Y −→ D{Mn } (K) is onto.
Proposition 9. If { Sj : j ∈ J } is a bounded subset of D{Mn } (Ω), then there is, for each j ∈ J, a family ( μα,j : α ∈ Nk0 ) of complex Borel measures in Ω such that, for each h > 0, sup h|α| M|α| |μα,j |(Ω) < ∞
α∈Nk 0 j∈J
and ϕ, Sj =
α∈Nk 0
Ω
Dα ϕ dμα,j ,
j ∈ J,
ϕ ∈ D{Mn } (K).
P ROOF. Let Sj∗ be the restriction of Sj to D{Mn } (K). We then have that { Sj∗ : j ∈ J } is a relatively compact subset of D{Mn } (K). Applying [2, p. 274], we obtain a relatively compact subset { Tj : j ∈ J } in Y such that t Γ(Tj ) = Sj∗ , j ∈ J. If (Tj )α is the element of C0 (Ω)∗ which identifies with the restriction of Tj to Y α , α ∈ Nk0 , making use of Riesz’s representation theorem, [6, p. 131], we have that there is a complex Borel measure μα,j in Ω such that ϕ dμα,j , ϕ ∈ C0 (Ω), ϕ, (Tj )α = Ω
and |μα,j |(Ω) is the norm of (Tj )α . After Proposition 3 we obtain sup
α∈Nk 0 , j∈J
h|α| M|α| |μα,j (Ω)| < ∞.
By using Proposition 2 we get, for (fα ) in Y and j ∈ J, (fα ), Tj =
Ω
α∈Nk 0
and, in particular, if ϕ belongs to D{Mn } (K), (Dα ϕ), Tj =
α∈Nk 0
Ω
fα dμα,j
Dα ϕ dμα,j .
On the other hand, if ϕ belongs to D{Mn } (K), we have (Dα ϕ), Tj = Γ(ϕ), Tj = ϕ,t Γ(Tj ) = ϕ, Sj∗ = ϕ, Sj and the result follows.
Before giving the proof of the next theorem, we need the following construction. We take a bounded subset { Sj : j ∈ J } of D{Mn } (Ω) so that there is a compact subset H of Ω which contains the support of Sj , j ∈ J, that is supp Sj ⊂ H, j ∈ J. 27
M. Valdivia ◦
Let K be a compact subset of Ω such that its interior K contains H. Applying Proposition 9, we obtain, for each j ∈ J, a family ( μα,j : α ∈ Nk0 ) of complex Borel measures in Ω such that, for each h > 0, sup h|α| M|α| |μα,j |(Ω) < ∞
α∈Nk 0 j∈J
and ϕ, Sj =
Dα ϕ dμα,j ,
Ω
α∈Nk 0
ϕ ∈ D{Mn } (K).
j ∈ J,
We take an element g of D{Mn } (Ω) which has value 1 in a neighborhood of H and with support contained ◦
in K. We find b > 0 and a positive integer s such that |Dα g(x)| ≤ b s|α| M|α| , {Mn }
We take ϕ ∈ E0
x ∈ Ω,
α ∈ Nk0 .
(Ω). Since gϕ belongs to D{Mn } (K), it follows that, for each j ∈ J, Dα (gϕ) dμα,j gϕ, Sj = =
α∈Nk 0
Ω
α∈Nk 0
Ω
β≤α
α! Dβ g · Dα−β ϕ dμα,j . β!(α − β)!
(Mn ),m
We take an integer m > s such that ϕ is in E0 β≤α
(Ω). It follows that, for each x ∈ Ω,
α! |Dβ g(x)| · |Dα−β ϕ(x)| β!(α − β)! ≤
β≤α
α! b s|β| M|β| m|α−β| |ϕ|m M|α−β| β!(α − β)!
≤ b m|α| |ϕ|m
β≤α
≤bm
and hence α∈Nk 0
(5)
β≤α
α! β!(α − β)!
Ω
|α|
α! M|β| M|α−β| β!(α − β)!
|ϕ|m 2|α| M|α|
|Dβ g · Dα−β ϕ| d|μα,j |
≤ b |ϕ|m
α∈Nk 0
≤ b |ϕ|m
α∈Nk 0
1 (4km)|α| M|α| |μα,j |(Ω) (2k)|α| 1 sup (4km)|δ| M|δ| | μδ,j |(Ω) (2k)|α| δ∈Nk 0
j∈J
from where we deduce that the series (5) is absolutely convergent, thus we may write, putting γ := α − β, (β + γ)! α! Dβ g · Dα−β ϕ dμα,j = Dβ g · Dγ ϕ dμβ+γ,j . (6) β!(α − β)! β!γ! Ω Ω k k k α∈N0 β≤α
28
γ∈N0 β∈N0
On the structure of certain ultradistributions
Theorem 2. Let { Sj : j ∈ J } be a bounded subset of D{Mn } (Ω) such that there is a compact subset H of Ω with supp Sj ⊂ H, j ∈ J. ◦
Let K be a compact subset of Ω such that K ⊃ H. Then, there is, for each j ∈ J, a family ( να,j : α ∈ Nk0 ) of complex Borel measures in Ω such that, for each h > 0, sup h|α| M|α| |να,j |(Ω) < ∞
α∈Nk 0 j∈J
◦
α ∈ Nk0 ,
supp να,j ⊂ K, and ϕ, Sj =
α∈Nk 0
Ω
Dα ϕ dνα,j ,
j ∈ J,
j ∈ J, ϕ ∈ D{Mn } (Ω),
where the series converges absolute and uniformly when j varies in J and ϕ varies in any given bounded subset of D{Mn } (Ω). P ROOF. For each j ∈ J, we find the family ( μα,j : α ∈ Nk0 ) of complex Radon measures in Ω with the properties above cited. We now fix γ ∈ Nk0 and take an arbitrary element η of C0 (Ω). Then
(β + γ)!
(β + γ)! β
D g · η dμβ+γ,j ≤ |Dβ g| · |η| d|μβ+γ,j |
β!γ! β!γ! Ω Ω
k k β∈N0
β∈N0
≤ b |η|∞ ≤ b|η|∞
(β + γ)! s|β| M|β| |μβ+γ,j |(Ω) β!γ! β∈Nk 0 2|β+γ| s|β| M|β| |μβ+γ,j |(Ω).
β∈Nk 0
On the other hand, 2|β+γ| s|β| M|β| |μβ+γ,j |(Ω) ≤ (2s)|β+γ| M|β+γ| |μβ+γ,j |(Ω) β∈Nk 0
β∈Nk 0
≤
β∈Nk 0
1 · (4ks)|β+γ| M|β+γ| |μβ+γ,j |(Ω) (2k)|β|
≤ 2 sup (4ks)|α| M|α| |μα,j |(Ω). α∈Nk 0 j∈J
Consequently, there is a constant C > 0 such that
(β + γ)! β
D g · η dμ β+γ,j ≤ C |η|∞ .
Ω
β∈Nk β!γ!
(7)
0
If we set vγ,j (η) :=
(β + γ)! Dβ g · η dμβ+γ,j , β!γ! Ω k
η ∈ C0 (Ω),
β∈N0
29
M. Valdivia
we have that vγ,j is a complex function which is clearly linear and, after (7), belongs to C0 (Ω)∗ . We apply Riesz’s representation theorem, [6, p. 131], and so obtain a complex Borel measure νγ,j in Ω such that vγ,j (η) = η · dνγ,j , η ∈ C0 (Ω). Ω
If M denotes the support of g, it is plain that ◦
supp νγ,j ⊂ M ⊂ K, {Mn }
For each ϕ ∈ E0
γ ∈ Nk0 .
j ∈ J,
(Ω), we have that (β + γ)! β γ D g · D ϕ · dμβ+γ,j = Dγ ϕ · dνγ,j β!γ! Ω Ω k
β∈N0
and, having in mind (6), gϕ, Sj =
γ∈Nk 0
Ω
{Mn }
Dγ ϕ · dνγ,j ,
ϕ ∈ E0
(Ω).
(8)
Let us now fix γ in Nk0 and j in J. We choose η in C0 (Ω) such that |η|∞ < 2 and vγ,j (η) = |νγ,j |(Ω). We take h > 1. Then h|γ| M|γ| |νγ,j |(Ω) = h|γ| M|γ| vγ,j (η) (β + γ)! b s|β| M|β| |η|∞ |μβ+γ,j |(Ω) ≤ h|γ| M|γ| β!γ! k β∈N0 (2hs)|β+γ| M|β+γ| |μβ+γ,j |(Ω) ≤ 2b β∈Nk 0
≤ 2b
β∈Nk 0
1 sup (4khs)|α| M|α| |μα,j |(Ω) (2k)|β+γ| α∈Nk 0
j∈J
≤ 4b sup (4ks)|α| M|α| |μα,j |(Ω). α∈Nk 0 j∈J
It follows from above that
sup h|γ| M|γ| |νγ,j |(Ω) < ∞.
γ∈Nk 0 j∈J
{M }
Theorem 1 now applies to obtain, for each j ∈ J, an element Tj ∈ E0 n (Ω) such that {M } ϕ, Tj = Dα ϕ · dνα,j , ϕ ∈ E0 n (Ω) α∈Nk 0
Ω
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } subset of E0 n (Ω). On the other hand, for each ϕ ∈ D{Mn } (K) and each j ∈ J, ϕ, Sj = gϕ, Sj = Dα (gϕ) dμα,j = Dα ϕ · dνα,j = ϕ, Tj . α∈Nk 0
30
Ω
α∈Nk 0
Ω
On the structure of certain ultradistributions ◦
Given an arbitrary element x in Ω, if x belongs to K, then this set is a neighborhood of x such that, if ◦
◦
ϕ ∈ D{Mn } (K), then ϕ ∈ D{Mn } (K), and so ϕ, Sj = ϕ, Tj . If x does not belong to K, we find an open neighborhood Ux of x such that Ux ∩ M = ∅. We take ϕ in D{Mn } (Ux ). Then, Ω Dα ϕ · dνα,j = 0, α ∈ Nk0 , thus ϕ, Tj = 0. Besides, Ux ∩ H = ∅, therefore ϕ, Sj = 0. We have thus proved that Sj and Tj coincide locally, from where it follows that Sj and Tj coincide in D{Mn } (Ω). The conclusion now follows.
4
Structure of the ultradistributions of Roumieu type
Theorem 3. For each j in a set J, let ( uα,j : α ∈ Nk0 ) be a family of Radon measures in Ω. If, given h > 0 and a compact subset K ⊂ Ω, we have that sup h|α| M|α| uα,j (K) < ∞,
α∈Nk 0 j∈J
then there is a bounded subset { Sj : j ∈ J } in D{Mn } (Ω) such that Dα ϕ, uα,j , ϕ ∈ D{Mn } (Ω) ϕ, Sj = α∈Nk 0
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded subset of D{Mn } (Ω). P ROOF.
◦
For each m ∈ N, we identify K(Km ) with C0 (Km ). We put um α,j for the restriction of uα,j to ◦
K(Km ). If μm α,j is the complex Borel measure in Km such that ◦ f, um = f dμm f ∈ C0 (Km ), α,j α,j , ◦ Km
we have that
◦
uα,j (Km ) = |μm α,j |(Km ). Therefore, given h > 0, it follows that sup
◦
α∈Nk 0 , j∈J
h|α| M|α| |μα,j |(Km ) < ∞, {Mn }
from where we obtain, applying Theorem 1, that there is a bounded subset { Sjm : j ∈ J } of E0 such that ◦ {M } ϕ, Sjm = Dα ϕ · dμm ϕ ∈ E0 n (Km ) α,j , ◦ α∈Nk 0
◦
(Km )
Km
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M }
◦
subset of E0 n (Km ). Given an arbitrary element ϕ of D{Mn } (Ω), we find m ∈ N such that ◦
supp ϕ ⊂ Km , we put
ϕ, Sj := ϕ, Sjm .
It is easy to see that { Sj : j ∈ J } is a bounded subset of D{Mn } (Ω) satisfying the requirements of the statement. 31
M. Valdivia
Theorem 4. If { Sj : j ∈ J } is a bounded subset of D{Mn } (Ω), then there is, for each j ∈ J, a family ( uα,j : α ∈ Nk0 ) of Radon measures in Ω such that, given h > 0 and a compact subset K of Ω, we have that sup h|α| M|α| uα,j (K) < ∞ α∈Nk 0 j∈J
and ϕ, Sj =
Dα ϕ, uα,j ,
ϕ ∈ D{Mn } (Ω),
j ∈ J,
α∈Nk 0
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded subset of D{Mn } (Ω). P ROOF. Let { Om : m ∈ N } be a locally finite open cover of Ω such that Om is relatively compact in Ω, m ∈ N. Let { gm : m ∈ N } be a partition of unity of class {Mn } subordinated to such covering. It follows that { gm Sj : j ∈ J } is a bounded subset of D{Mn } (Ω) whose elements have their supports contained in m a compact subset of Om . Applying Theorem 2, we obtain, for each j ∈ J, a family ( να,j : α ∈ Nk0 ) of complex Borel measures in Ω such that, for each h > 0, m |(Ω) < ∞ sup h|α| M|α| |να,j
α∈Nk 0 j∈J
m supp να,j ⊂ Om ,
and ϕ, gm Sj =
α∈Nk 0
Ω
j ∈ J,
m Dα ϕ · dνα,j ,
α ∈ Nk0 , ϕ ∈ D{Mn } (Ω)
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded subset of D{Mn } (Ω). Given an arbitrary element f of K(Ω), there is a finite number of subindex m such that Om ∩ supp f = ∅. Consequently, we may define, for each α ∈ Nk0 and j ∈ J, m f · dνα,j . uα,j (f ) := m∈N
Ω
We then have that uα,j is a linear functional in K(Ω). Given any compact subset K of Ω, there is a positive integer m0 such that K ∩ Om = ∅, m > m0 . Hence, if f has its support contained in K, it follows that, for each j ∈ J, m0 m0 m m |uα,j (f )| ≤ |f | d|να,j | ≤ |να,j |(Ω) · |f |∞ , m=1
Ω
m=1
from where we deduce that uα,j is a Radon measure in Ω. Besides uα,j (K) ≤
m0
m |να,j |(Ω),
j ∈ J,
m=1
and, if h is an arbitrary positive number, sup h|α| M|α| uα,j (K) ≤
α∈Nk 0 j∈J
32
m0 m=1 α∈Nk 0 j∈J
m h|α| M|α| |να,j |(Ω) < ∞.
On the structure of certain ultradistributions
We take now ϕ in D{Mn } (Ω) with support in K. Then m0 ∞ ϕ, Sj = ϕ g m , Sj = ϕ g m , Sj m=1
=
m0
ϕ, gm Sj =
m=1
=
m=1
m0
m=1 α∈Nk 0
m0 m=1 α∈Nk 0
Ω
m Dα ϕ · dνα,j =
Ω
m Dα ϕ · dνα,j
Dα ϕ, uα,j .
α∈Nk 0
It is now easy to show that the last series converges absolute and uniformly when j varies in J and when ϕ varies in any given bounded subset of D{Mn } (Ω). {M }
5 The space D(Lpn) (Ω) We put Lp (Ω) and Lp (Ω), 1 ≤ p ≤ ∞, to denote the classical Lebesgue spaces. If f ∈ f˜ ∈ Lp (Ω), 1 ≤ p < ∞, we write p1 ˜ f p = fp = |f | dx , Ω
and if f ∈ f˜ ∈ L∞ (Ω), then ˜ ∞ = supess{ |f (x)| : x ∈ Ω }. f ∞ = f By DLp (Rk ), 1 ≤ p < ∞, we represent the classical L. Schwartz space, [7, p. 199]. We put BLp (Ω) for the linear space over C of the complex functions f , defined and infinitely differentiable in Ω, such that Dα f belongs to Lp (Ω), α ∈ Nk0 . We assume that BLp (Ω) is endowed with the metrizable locally convex topology such that a sequence (fm ) in BLp (Ω) converges to the origin if and only if (Dα fm p ) converges to zero for every α ∈ Nk0 . We then have that BLp (Ω) is a Fr´echet space. We have that BLp (Rk ) coincides with DLp (Rk ). (M ),r Given r ∈ N and 1 ≤ p < ∞, we use BLp n (Ω) to denote the linear space over C of the functions f in BLp (Ω) which satisfy: Dα f p < ∞. f p,r := sup |α| r M|α| α∈Nk 0 (M ),r
(M ),r
We assume that BLp n (Ω) is provided with the norm · p,r . Given a Cauchy sequence in BLp n (Ω), it is immediate that (fm ) is also a Cauchy sequence in BLp (Ω), thus it converges in this space to a function f . For a given ε > 0, there is a positive integer m0 such that fm − fs p,r < ε,
m, s ≥ m0 .
Then, for those values of m and s, and for each α ∈ Nk0 , we have Dα fm − Dα fs p <ε r|α| M|α| and so, for m ≥ m0 ,
Dα fm − Dα f p ≤ ε, r|α| M|α| 33
M. Valdivia (M ),r
from where we deduce that f ∈ BLp n
(Ω) and
fm − f p,r ≤ ε, (M ),r
Consequently, BLp n
m > m0 .
(Ω) is a Banach space. We put ∞
{M }
BLp n (Ω) :=
(M ),r
r=1
BLp n
(Ω)
(M ),r {M } and assume that BLp n (Ω) is the inductive limit of the sequence BLp n (Ω) of Banach spaces. We {M }
{M }
assume that the topological dual BLp n (Ω) of BLp n (Ω) is endowed with the strong topology. In this section, we substitute the space X of Section 2 by the space Lp (Ω), 1 ≤ p < ∞. Then, each element of Yr is a family ( f˜α : α ∈ Nk0 ) of elements of Lp (Ω) with (f˜α )r = sup
α∈Nk 0
f˜α p < ∞. M|α|
r|α|
{M } ˜ α f for the element of Lp (Ω) to which Dα f belongs, α ∈ Nk . If f belongs to BLp n (Ω), we put D 0 ˜ α f : α ∈ Nk ) such that By Zr we denote the subspace of Yr formed by those families ( D 0 (M ),r
f ∈ BLp n Let be such that
(M ),r
(Ω).
Xr : BLp n
(Ω) −→ Zr
˜ α f ), Xr (f ) = (D
f ∈ BLp n
(M ),r
(Ω).
Then, Xr is a linear onto isometry. By Z we mean ∪{ Zr : r ∈ N } considered as a subspace of Y . Let {M }
X : BLp n (Ω) −→ Z be such that
˜ α f ), X (f ) = (D
{M }
f ∈ BLp n (Ω).
Clearly, X is linear bijective and continuous. {M } We put W for the map X considered from BLp n (Ω) into Y . By t W we denote the map from Y into {M } BLp n (Ω) given by the transpose of W . Throughout what follows in this section, we fix 1 ≤ p < ∞ and write q for the conjugate value of p, i.e., p1 + 1q = 1. Theorem 5. For each j in a set J, let ( gα,j : α ∈ Nk0 ) be a family of elements of Lq (Ω) such that, for each h > 0, sup h|α| M|α| gα,j q < ∞. α∈Nk 0 j∈J
{M }
Then there is a bounded subset { Sj : j ∈ J } in BLp n (Ω) so that Dα ϕ · gα,j dx, j ∈ J, ϕ, Sj = α∈Nk 0
Ω
{M }
ϕ ∈ BLp n (Ω),
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } subset of BLp n (Ω). 34
On the structure of certain ultradistributions
P ROOF. We identify in the usual manner gα,j with a continuous linear functional on Lp (Ω), whose norm is gα,j q . We apply Proposition 4 to obtain, for each j ∈ J, an element uj in Y such that its restriction to Yα coincides with gα,j , α ∈ Nk0 . Applying now Proposition 2 we have that (f˜α ), uj = fα · gα,j dx, fα ∈ f˜α , (f˜α ) ∈ Y. (9) α∈Nk 0
Ω
Let us now fix a bounded subset B of Y . We find r ∈ N such that B is a bounded subset of Yr . We take (f˜α ) in B. It follows that
fα · gα,j dx ≤ |fα | · |gα,j | dx
α∈Nk 0
Ω
α∈Nk 0
≤
Ω
fα p · gα,j q
α∈Nk 0
1 fα p (2kr)|α| M|α| gα,j q |α| |α| (2k) r M |α| α∈Nk 0 ⎛ ⎞ ⎜ ⎟ ≤ (f˜α )r ⎝ sup (2kr)|β| M|β| gβ,j q ⎠ =
β∈Nk 0 j∈J
α∈Nk 0
1 . (2k)|α|
We deduce from here that the series (9) converges absolutely and uniformly when j varies in J and (f˜α ) varies in B. Besides,
sup (f˜α ), uj < ∞, j∈J (f˜α )∈B
from where it follows that { uj : j ∈ J } is a bounded subset of Y . We now write Sj := t W (uj ),
j ∈ J.
{M }
{M }
Then { Sj : j ∈ J } is a bounded subset of BLp n (Ω). On the other hand, for each ϕ ∈ BLp n (Ω), we have that α ˜ ϕ), uj = W (ϕ), uj = ϕ, t W (uj ) = ϕ, Sj . (D {M }
Consequently, for each ϕ ∈ BLp n (Ω) and each j ∈ J, we obtain, making use of (9), that α ˜ ϕ), uj = Dα ϕ · gα,j dx. ϕ, Sj = (D α∈Nk 0
Ω
{M } ˜ α ϕ) varies in a bounded subset of Y , from Finally, when ϕ varies in a bounded subset of BLp n (Ω), (D where we deduce that the above series converges absolutely and uniformly when j varies in J and ϕ varies {M } in any given bounded subset of BLp n (Ω). (M ),r
(M ),r
Given a compact subset K of Ω and r ∈ N, we put D(Lpn) (K) for the subspace of BLp n
(Ω)
(M ),r DLp n (K)
whose elements have their support in K. If (fm ) is a sequence in which converges to f (M ),r in BLp n (Ω), there is a subsequence (fmi ) of (fm ) which converges to f almost everywhere. Since (M ),r fmi (x) = 0, x ∈ Ω \ K, i ∈ N, it follows that f belongs to DLp n (K) and thus this space is complete. We write ∞ {M } (M ),r D(Lpn) (K) := D(Lpn) (K) r=1
35
M. Valdivia {M }
and we assume that DLp n (K) is endowed with the structure of (LB) space, as the inductive limit of (M ),r {M } the sequence (D(Lpn) (K)) of Banach spaces. We write D(Lpn) (K) for the strong topological dual of {M }
D(Lpn) (K). (M ),r
Proposition 10. The closed unit ball Br of D(Lpn)
(M ),r+1
(K) is a compact subset of D(Lpn)
(K).
(M ),r
P ROOF. Let μ be the Lebesgue measure in Rk . We assume the elements f of D(Lpn) (K) extended to Rk setting f (x) = 0, x ∈ Rk \ K. Given α ∈ Nk0 and f ∈ Br , we have that x1 x2 xk ∂ |α|+k f (t) α D f (x) = ... dt1 dt2 . . . dtk α1 +1 t ∂ α2 +1 t . . . ∂ αk +1 t 1 2 k −∞ −∞ −∞ ∂ and hence, if βj := αj + 1, j = 1, 2, . . . , k, we have that |Dβ f (t)| dt ≤ μ(K)1/q Dβ f p |Dα f (x)| ≤ K
≤ μ(K)1/q f p,r r|β| M|β| ≤ μ(K)1/q r|β| M|β| , and so the set of functions { Dα f : f ∈ Br } is uniformly bounded in Rk . Consequently, for each γ ∈ Nk0 , the set { Dγ f : f ∈ Br } is equicontinuous, therefore, applying Ascoli’s theorem and a diagonal process, given an arbitrary sequence (fm ) in Br , there is a complex function f , defined and infinitely differentiable in Rk , and a subsequence (fmi ) of (fm ) such that, for each α ∈ Nk0 , (Dα fmi ) converges uniformly a Dα f in Rk . Since, for each α ∈ Nk0 , Dα fmi p ≤ 1, i ∈ N, r|α| M|α| it follows that
Dα f p ≤ 1, r|α| M|α|
and thus f is in Br . We see next that (fmi ) converges to positive integer s0 such that s0 r < r+1 We determine a positive integer i0 such that
ε μ(K)1/p Dα fmi (x) − Dα f (x) < , 2
(M ),r+1
f in D(Lpn)
(K). Given ε > 0, we find a
ε . 4 x ∈ Rk ,
i ≥ i0 ,
|α| ≤ s0 .
Then, if i ≥ i0 , we have fmi − f p,r+1 = sup
α∈Nk 0
Dα fmi − Dα f p (r + 1)|α| M|α|
Dα fmi − Dα f p Dα fmi − Dα f p + sup |α| M |α| M |α|≤s0 (r + 1) |α| |α|>s0 (r + 1) |α| |α| Dα fmi − Dα f p r ≤ sup sup μ(K)1/p |Dα fmi (x) − Dα f (x)| + sup r+1 (r)|α| M|α| |α|≤s0 x∈Rk |α|>s0 s0 ε Dα fmi p + Dα f p r ≤ + sup 2 r+1 (r)|α| M|α| |α|>s0 ε ε ≤ + (fmi p,r + f p,r ) ≤ ε. 2 4 ≤ sup
36
On the structure of certain ultradistributions (M ),r+1
It then follows that (fmi ) converges to f in D(Lpn)
(K) and the result follows.
For the next two propositions, we are going to fix a compact subset K of Ω. Given r ∈ N, let Pr be the ˜ α ϕ), with ϕ ∈ D(Mpn ),r (K). Let subspace of Yr whose elements have the form (D (L ) (M ),r
ζr : D(Lpn) be such that
(K) −→ Pr (M ),r
˜ α ϕ), ζr (ϕ) = (D
ϕ ∈ D(Lpn)
(K).
We have that ζr is a linear onto isometry. We put P := ∪{ Pr : r ∈ N } and we consider it as a subspace of Y . We then write {M } ζ : D(Lpn) (K) −→ P such that
{M }
˜ α ϕ), ζ(ϕ) = (D
ϕ ∈ D(Lpn) (K).
Clearly, ζ is linear bijective and continuous. Proposition 11. ζ is a topological isomorphism. P ROOF. We take an absolutely convex closed and bounded subset A of P . Applying Proposition 1, we obtain r ∈ N such that A is a bounded subset of Pr . Then ζr−1 (A) = ζ −1 (A) is an absolutely convex (M ),r {M } bounded subset of D(Lpn) (K) which is closed in D(Lpn) (K). Making use of the former proposition, (M ),r+1
we obtain that ζr−1 (A) is compact in D(Lpn) (K), from where we have that A is a compact subset of Pr+1 . We apply Proposition 6 to have that P is the inductive limit of the sequence (Pr ) of Banach spaces. Consequently, ζ is a topological isomorphism. We now put {M }
D(Lpn) (Ω) := and assume that {M }
{M } D(Lpn) (Ω)
∞ r=1
(M ),r
D(Lpn) (Kr ) (M ),r
is the inductive limit of the sequence (D(Lpn) (Kr )) of Banach spaces. We {M }
write D(Lpn) (Ω) for the strong topological dual of D(Lpn) (Ω). It follows, from Proposition 10, that {M }
D(Lpn) (Ω) is Fr´echet-Schwartz space. {M }
{M }
We now consider D(Lpn) (K) as a subspace of D(Lpn) (Ω). If A is an absolutely convex closed bounded {M }
{M }
subset of D(Lpn) (K), then A is a bounded subset of D(Lpn) (Ω) and thus there is a positive integer m (M ),r
(M ),r
such that K ⊂ Kr and A is a relatively compact subset of D(Lpn) (Kr ). Clearly, D(Lpn) (K) is a closed (M ),r
subspace of D(Lpn)
(M ),r
(Kr ) and A is closed in the Banach space D(Lpn)
(M ),r A is compact in D(Lpn) (K). We {M } {M } D(Lpn) (Ω) in D(Lpn) (K) coincides
(K), from where we conclude that
apply Proposition 6 and so we obtain that the topology induced by with the original topology of this space. {M }
In the following, we put η for the mapping ζ considered from D(Lpn) (K) into Y . Then, if t η is the transpose of η, we have that {M } t η : Y −→ D(Lpn) (K) is an onto map. {M }
Proposition 12. If { Sj : j ∈ J } is a bounded subset of D(Lpn) (Ω), then there is, for each j ∈ J, a family ( gα,j : α ∈ Nk0 ) of elements of Lq (Ω) such that, for each h > 0, sup h|α| M|α| gα,j q < ∞
α∈Nk 0 j∈J
37
M. Valdivia
and ϕ, Sj =
α∈Nk 0
P ROOF.
Ω
Dα ϕ · gα,j dx,
j ∈ J,
{M }
ϕ ∈ D(Lpn) (K).
{M }
Let Sj∗ be the restriction of Sj to D(Lpn) (K). It follows that { Sj∗ : j ∈ J } is a relatively compact {M }
subset of D(Lpn) (K). Applying [2, p. 274], we obtain a relatively compact subset { Tj : j ∈ J } of Y such that t η(Tj ) = Sj∗ , j ∈ J. If (Tj )α is the element of Lq (Ω) which identifies with the restriction of Tj to Y α , α ∈ Nk0 , we obtain an element gα,j in Lq (Ω) such that ϕ · gα,j dx, j ∈ J, ϕ ∈ ϕ˜ ∈ Lp (Ω). ϕ, ˜ (Tj )α = Ω
Given h > 0, we take r ∈ N, r > h. Making use of Proposition 3 we obtain that sup
α∈Nk 0 , j∈J
h|α| M|α| gα,j q < ∞.
Having in mind Proposition 2, we have, for each (f˜α ) in Y and j ∈ J, fα · gα,j dx, fα ∈ f˜α , (f˜α ), Tj = α∈Nk 0
Ω
α ∈ Nk0 ,
{M }
and, in particular, if ϕ ∈ D(Lpn) (K), then
α ˜ ϕ), Tj = Dα ϕ · gα,j dx (D α∈Nk 0
and besides
Ω
α ˜ ϕ), Tj = η(ϕ), Tj = ϕ,t η(Tj ) = ϕ, S ∗ = ϕ, Sj , (D j
from where the result follows.
If g ∈ Lp1 (Rk ) and l ∈ Lp2 (Rk ), with 1 ≤ p1 ≤ ∞, 1 ≤ p2 ≤ ∞ and p11 + p12 ≥ 1, then there exists almost everywhere the convolution g ∗ l ∈ Ls (Rk ), being 1s := p11 + p12 − 1. Also having that g ∗ ls ≤ gp1 · lp2 .
(10)
This property will be used in the next proposition. {M }
Proposition 13. The linear space D{Mn } (Ω) is dense in D(Lpn) (Ω). ◦
◦
P ROOF. We may assume that K1 = ∅ and Ks ⊂ Ks+1 , s ∈ N. Given δ > 0, we put B(δ) for the closed ball in Rk with center in the origin and radius δ. We take {M } (M ),r f ∈ D(Lpn) (Ω). We find r ∈ N such that f ∈ D(Lpn) (Kr ). We choose a sequence (ψi ) in E {Mn } (Rk ) satisfying (i) ψi (x) ≥ 0, x ∈ Ω. (ii) Ω ψi (x) dx = 1. (iii) ψi ∈ D(Mn ),r (B(δi )), δ1 > δ2 > · · · > δi > . . . , ◦
lim δi = 0 and Kr + B(δi ) ⊂ Kr+1 . i
38
On the structure of certain ultradistributions
We consider f extended to Rk by setting f (x) = 0, x ∈ Rk \ Ω. We put fi := f ∗ ψi , i ∈ N. We shall (M ),r+1 see next that (fi ) is a sequence in D(Mn ),r+1 (Kr+1 ) which converges to f in D(Lpn) (Kr+1 ). For each α ∈ Nk0 , we have Dα fi (x) =
Rk
f (y)(Dα ψi )(x − y) dy,
x ∈ Rk ,
from where we get that fi belongs to D(Mn ),r+1 (Kr+1 ). We now take ε > 0, We find a positive integer s0 such that s0 r ε f p,r < . r+1 4 Given α ∈ Nk0 , we have that, for each x ∈ Rk ,
α
α
(D f )(x − y) − Dα f (x) ψi (y) dy
D fi (x) − Dα f (x) ≤ Rk
≤ sup{ |(Dα f )(x − y) − Dα f (x)| : y ∈ B(δi ) }. We find i0 ∈ N for which |Dα fi (x) − Dα f (x)| < Then
ε , 2μ(Kr+1 )
i ≥ i0 ,
x ∈ Rk ,
|α| ≤ s0 .
ε , i ≥ i0 . 2 We now apply (10) for p1 = p, p2 = 1, g = Dα f and l = ψi . Then Dα fi − Dα f p ≤
Dα fi p = (Dα f ) ∗ ψi p ≤ Dα f p · ψi 1 = Dα f p . Consequently, for i ≥ i0 , it follows that f − fi p,r+1 = sup
α∈Nk 0
Dα (f − fi )p (r + 1)|α| M|α|
Dα (f − fi )p Dα (f − fi )p + sup |α| |α| M |α| |α|≤s0 (r + 1) M|α| |α|≥s0 (r + 1) |α| α α D f p + D fi p r ε ≤ + sup 2 |α|≥s0 r + 1 (r)|α| M|α| s0 ε 2Dα f p r ≤ + sup |α| < ε. 2 r+1 r M|α| α∈Nk 0 ≤ sup
{M }
The last proposition tells us that the elements of D(Lpn) (Ω) may be considered as ultradistributions. In theorems 7 and 8, we shall characterize those ultradistributions. We proceed now in a similar way to the construction previous to Theorem 2. We take a bounded subset {M } { Sj : j ∈ J } in D(Lpn) (Ω) in such a way that there is a compact subset H of Ω with supp Sj ⊂ H,
j ∈ J.
◦
Let K be a compact subset of Ω with H ⊂ K. We apply Proposition 12 to obtain, for each j ∈ J, a family ( fα,j : α ∈ Nk0 ) in Lq (Ω) such that, for each h > 0, sup
α∈Nk 0 , j∈J
h|α| M|α| · fα,j q < ∞
39
M. Valdivia
Dα ϕ · fα,j dx, ϕ, Sj =
and
α∈Nk 0
Ω
j ∈ J,
{M }
ϕ ∈ D(Lpn) (K).
We take an element g of D{Mn } (Ω) which takes value one in a neighborhood of H and whose support is ◦
contained in K. We find b > 0 and a positive integer s such that
α
D g(x) ≤ b s|α| M|α| , x ∈ Ω, {M }
α ∈ Nk0 . {M }
We take ϕ in BLp n (Ω). It means no difficulty to see that gϕ belongs to D(Lpn) (K) and thus we have, for each j ∈ J, gϕ, Sj = Dα (gϕ) · fα,j dx =
α∈Nk 0
Ω
α∈Nk 0
Ω
β≤α
α! Dβ g · Dα−β ϕ · fα,j dx. β!(α − β)!
(11)
(M ),m
We take now a positive integer m > s such that ϕ is in BLp n (Ω). We then have α! |Dβ g| · |Dα−β ϕ| · |fα,j | dx β!(α − β)! Ω β≤α
≤
β≤α
≤
β≤α
≤
β≤α
α! b s|β| M|β| Dα−β ϕp · fα,j q β!(α − β)! α! b s|β| M|β| ϕp,m m|α−β| M|α−β|fα,j q β!(α − β)! α! b m|α| M|α| ϕp,m fα,j q β!(α − β)!
= 2|α| b m|α| M|α| ϕp,m fα,j q 1 ≤ b ϕp,m sup (4km)|δ| M|δ| fδ,j q (2k)|α| δ∈Nk0 , j∈J from where we deduce that the series (11) is absolutely convergent, hence, putting γ := α − β, we may write (β + γ)! α! β α−β D g·D ϕ · fα,j dx = Dβ g · Dγ ϕ · fβ+γ,j dx. (12) β!(α − β)! β! γ! Ω Ω k k k α∈N0 β≤α
γ∈N0 β∈N0
{M }
Theorem 6. Let { Sj : j ∈ J } be a bounded subset of D(Lpn) (Ω) such that there is a compact subset H of Ω with j ∈ J. supp Sj ⊂ H, ◦
Let K be a compact subset of Ω such that H ⊂ K. Then there is, for each j ∈ J, a family ( gα,j : α ∈ Nk0 ) of elements of Lq (Ω) such that, for each h > 0, we have sup h|α| M|α| gα,j q < ∞,
α∈Nk 0 j∈J
40
On the structure of certain ultradistributions ◦
supp gα,j ⊂ K, and ϕ, Sj =
α∈Nk 0
Ω
j ∈ J,
Dα ϕ · gα,j dx,
α ∈ Nk0 , {M }
j ∈ J,
ϕ ∈ D(Lpn) (Ω),
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } subset of D(Lpn) (Ω). P ROOF. For each j ∈ J, we obtain the family ( fα,j : α ∈ Nk0 ) of elements of Lq (Ω) with the properties cited above. We fix γ ∈ Nk0 and take ρ ∈ ρ˜ ∈ Lq (Ω). Then
(β + γ)!
(β + γ)! β
≤ D g · ρ · f dx |Dβ g| · |ρ| · |fβ+γ,j | dx β,j
β!γ! β!γ! Ω Ω
k k β∈N0
β∈N0
(β + γ)! b s|β| M|β| ρp fβ+γ,j q β!γ! k β∈N0 ≤ bρp 2|β+γ|s|β| M|β|fβ+γ,j q ≤
β∈Nk 0
≤ b ρp
β∈Nk 0
≤ b ρp
β∈Nk 0
1 (4ks)|β+γ| M|β+γ|fβ+γ,j q (2k)|β| 1 (2k)|β|
sup
α∈Nk 0 , j∈J
(4ks)|α| M|α| fα,j q
= 2 b ρp sup (4ks)|α| M|α| fα,j q , α∈Nk 0 j∈J
from where we deduce that there is C > 0 such that
(β + γ)!
β
D g · ρ · fβ+γ,j dx
≤ Cρp .
β!γ! Ω
β∈Nk
(13)
0
If we put, for each ρ ∈ ρ˜ ∈ Lp (Ω), ρ) := vγ,j (˜
(β + γ)! Dβ g · ρ · fβ+γ,j dx, β!γ! Ω k
β∈N0
we have that vγ,j is a complex function, clearly linear, such that, after (13), belongs to Lq (Ω). Then, there is gγ,j ∈ Lq (Ω) for which vγ,j (˜ ρ) = ρ · gγ,j dx, ρ ∈ ρ˜ ∈ Lp (Ω). Ω
If M is the support of g, it is plain that ◦
supp gγ,j ⊂ M ⊂ K,
j ∈ J,
γ ∈ Nk0 .
{M }
It follows that, for each ϕ ∈ BLp n (Ω), (β + γ)! Dβ g · Dγ ϕ · fβ+γ,j dx = Dγ ϕ · gγ,j dx β! γ! Ω Ω k β∈N0
41
M. Valdivia
and, having in mind (11), gϕ, Sj =
Ω
γ∈Nk 0
Dγ ϕ · gγ,j dx,
{M }
ϕ ∈ BLp n (Ω).
(14)
We fix γ in Nk0 and j in J. We choose ρ˜ in Lp (Ω) such that ρ˜p < 2 and vγ,j (˜ ρ) = gγ,j q . We take h > 1. Then, if ρ ∈ ρ˜, h|γ| M|γ|gγ,j q = h|γ| M|γ| vγ,j (˜ ρ)
(β + γ)!
Dβ g · ρ · fβ+γ,j dx
≤ h|γ| M|γ|
Ω
β∈Nk β! γ! 0
≤ h|γ| M|γ| ρ˜p b ≤2b
(β + γ)! s|β| M|β| fβ+γ,j q β! γ! k
β∈N0
2
|β+γ|
(sh)|β+γ| M|β+γ|fβ+γ,j q
β∈Nk 0
≤2b
β∈Nk 0
from where we deduce that
1 sup (4ksh)|α| M|α| fα,j q (2k)|β| α∈Nk 0
j∈J
sup h|γ| M|γ|gγ,j q < ∞.
γ∈Nk 0 j∈J
{M }
We apply Theorem 5 to obtain, for each j ∈ J, an element Tj in BLp n (Ω) such that {M } ϕ, Tj = Dα ϕ · gα,j dx, ϕ ∈ BLp n (Ω), α∈Nk 0
Ω
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } {M } subset of BLp n (Ω). On the other hand, for each ϕ ∈ D(Lpn) (K) and each j ∈ J, ϕ, Sj = g ϕ, Sj = =
α∈Nk 0
Ω
α∈Nk 0
Ω
Dα (gϕ) · fα,j dx
Dα ϕ · gα,j dx = ϕ, Tj .
Finally, it can be shown in the same way that it was done in the proof of Theorem 2 that Sj and Tj coincide {M } in D(Lpn) (Ω) for every j ∈ J, so the result follows. We put Lploc (Ω) for the linear space over C of the complex functions f defined in Ω such that, for each compact subset K of Ω, f|K belongs to Lq (K). Theorem 7. For each j in a set J, let ( fα,j : α ∈ Nk0 ) be a family of elements of Lqloc (Ω) such that, for each h > 0 and each compact subset K of Ω, we have that sup h|α| M|α| fα,j|K q < ∞.
α∈Nk 0 j∈J
42
On the structure of certain ultradistributions {M }
Then, there is a bounded subset { Sj : j ∈ J } of D(Lpn) (Ω) such that ϕ, Sj = Dα ϕ · fα,j dx,
{M }
j ∈ J,
Ω
α∈Nk 0
ϕ ∈ D(Lpn) (Ω),
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } subset of D(Lpn) (Ω). P ROOF.
For each m ∈ N, we put m fα,j := f
It follows that, for each h > 0,
◦
α,j|Km
,
α ∈ Nk0 ,
j ∈ J.
m q < ∞. sup h|α| M|α| fα,j
α∈Nk 0 j∈J
{M }
◦
We apply now Theorem 5 to obtain a bounded subset { Sjm : j ∈ J } of BLp n (Km ) such that ϕ, Sjm = α∈Nk 0
◦ Km
m Dα ϕ · fα,j dx,
j ∈ J,
{M }
◦
ϕ ∈ BLp n (Km ),
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M }
◦
subset of BLp n (Km ). {M } Given an arbitrary element ϕ in D(Lpn) (Ω), we find m ∈ N such that ◦
supp ϕ ⊂ Km and we put
ϕ, Sj := ϕ, Sjm .
It is easy to verify that Sj is well defined, j ∈ J, and that { Sj : j ∈ J } is a bounded subset of D{Mn } (Ω), which satisfies the statement of our theorem. {M }
Theorem 8. If { Sj : j ∈ J } is a bounded subset of D(Lpn) (Ω), then there is, for each j ∈ J, a family ( fα,j : α ∈ Nk0 ) in Lqloc (Ω) such that, for each h > 0 and each compact subset K of Ω, we have that sup h|α| M|α| fα,j|K q < ∞
α∈Nk 0 j∈J
and ϕ, Sj =
α∈Nk 0
Ω
Dα ϕ · fα,j dx,
j ∈ J,
{M }
ϕ ∈ D(Lpn) (Ω),
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } subset of D(Lpn) (Ω). P ROOF. Let { Om : m ∈ N } be a locally finite open cover of Ω such that Om is relatively compact in Ω, m ∈ N. Let { gm : m ∈ N } be a partition of unity of class {Mn } subordinated to that open cover. We then {M } have that { gm Sj : j ∈ J } is a bounded subset of D(Lpn) (Ω) whose elements have their supports contained 43
M. Valdivia
m in a compact subset of Om . We apply Theorem 6 to obtain, for each j ∈ J, a family ( fα,j : α ∈ Nk0 ) of q elements of L (Ω) such that, for each h > 0, m q < ∞, sup h|α| M|α| fα,j
α∈Nk 0 j∈J
m ⊂ Om , supp fα,j
m ϕ, gm Sj = Dα ϕ · fα,j dx,
and
j ∈ J,
Ω
α∈N0k
α ∈ Nk0 ,
j ∈ J,
{M }
ϕ ∈ D(Lpn) (Ω),
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } subset of D(Lpn) (Ω). We put, for each x ∈ Ω, α ∈ Nk0 and j ∈ J, ∞
fα,j (x) :=
m fα,j (x).
m=1
If K is a compact subset of Ω, there is a positive integer m0 such that K ∩ Om = ∅, and hence fα,j is well defined and belongs to fα,j|K q ≤
Lqloc (Ω). m0
m ≥ m0 , Besides, we have
m fα,j|K q ≤
m=1
m0
m fα,j q
m=1
and thus, given h > 0, it follows that sup
α∈Nk 0 , j∈J
h|α| M|α| fα,j|K q < ∞. {M }
Applying now Theorem 7, we obtain a bounded subset { Tj : j ∈ J } of D(Lpn) (Ω) such that {M } Dα ϕ · fα,j dx, j ∈ J, ϕ ∈ D(Lpn) (Ω), ϕ, Tj = Ω
α∈Nk 0
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } subset of D(Lpn) (Ω). {M }
We now choose ϕ ∈ D(Lpn) (Ω). We find m0 ∈ N such that Om ∩ supp ϕ = ∅, Then
m > m0 .
m0 m ϕ, Tj = Dα ϕ · fα,j dx = Dα ϕ · fα,j dx α∈Nk 0
=
Ω
m0
m=1 α∈Nk 0
Ω
m=1 α∈Nk 0 m Dα ϕ · fα,j dx =
m0
Ω
ϕ, gm Sj =
m=1
m0
ϕ · g m , Sj
m=1
= ϕ, Sj . Consequently, Sj = Tj , j ∈ J, and the result now follows.
From this and up to the end of this section we shall assume that the sequence Mn , n ∈ N, satisfies condition (3), that is, it is stable for differential operators. 44
On the structure of certain ultradistributions {M }
Proposition 14. The canonical injection from D{Mn } (Ω) into D(L1n) (Ω) is a topological isomorphism. P ROOF.
Clearly, {M }
ζ : D{Mn } (Ω) −→ D(L1n) (Ω) such that f ∈ D{Mn } (Ω)
ζ(f ) = f,
is well defined, linear and continuous. It is immediate that there are b > 0 and l > 0 for which Mn+k ≤ b ln Mn ,
n ∈ N0 .
{M }
We now take an arbitrary element ϕ ∈ D(L1n) (Ω). We extend ϕ to Rk by putting ϕ(x) = 0, x ∈ Rk \ Ω. {M },r
We find r ∈ N such that ϕ is in D(L1n) Dα ϕ(x) =
x1
−∞
x2
(Rk ). Given α ∈ Nk0 and x ∈ Rk , we have
...
−∞
xk
−∞
∂ α1 +1 t
1
∂ |α|+k ϕ(t) dt1 dt2 . . . dtk ∂ α2 +1 t2 . . . ∂ αk +1 tk
and thus
α
D ϕ(x) ≤
Kr
∂ |α|+k ϕ(t)
∂tα1 +1 ∂tα2 +1 . . . ∂tαk +1 dt 1
2
k
≤ ϕ1,r · r|α|+k M|α|+k ≤ ϕ1,r r|α|+k b l|α| M|α| , from where we deduce that, if s is an integer greater than rl, sup sup
x∈Kr α∈Nk 0
|Dα ϕ(x)| ≤ ϕ1,r rk b s|α| M|α|
and so ϕ ∈ D(Mn ),s (Kr ) ⊂ D{Mn } (Ω). It follows that ζ is onto. Applying now Grothendieck’s theorem, [1, p. 17], the result follows.
Theorem 9. If { Sj : j ∈ J } is a bounded subset of D{Mn } (Ω), then there is, for each j ∈ J, a family ( fα,j : α ∈ Nk0 ) of elements of L∞ loc (Ω) such that, for each h > 0 and each compact subset K of Ω, we have that sup h|α| M|α| |fα,j |K,∞ < ∞ α∈Nk 0 , j∈J
and ϕ, Sj =
α∈Nk 0
Ω
Dα ϕ · fα,j dx,
j ∈ J,
ϕ ∈ D{Mn } (Ω),
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded subset of D{Mn } (Ω). P ROOF.
It is an immediate consequence of the former proposition and Theorem 8.
45
M. Valdivia
{M }
6 The space DLp n (Ω) We shall use in this section the same notation as in the previous one. In particular, 1 ≤ p < ∞ and q is the conjugate element of p. (M ),r (Mn ),r For each r ∈ N, we put DLp n (Ω) for the Banach space given by the closure of ∪∞ (Km ) m=1 DLp (Mn ),r in BLp (Ω). We put ∞ {M } (M ),r DLp n (Ω) DLp n (Ω) := r=1
{M } DLp n (Ω)
and we assume that is provided with the structure of (LB)-space as the inductive limit of the se (Mn ),r {M } {M } (Ω) of Banach spaces. We put DLp n (Ω) for the strong topological dual of DLp n (Ω). quence DLp {M }
It is immediate that the canonical injection from D{Mn } (Ω) into DLp n (Ω) is continuous, therefore we {M } may consider the elements of DLp n (Ω) as ultradistributions. We shall characterize later these ultradistributions for the case p > 1. ˜ α f : α ∈ Nk0 ) such that f ∈ We write Qr for the subspace of Zr formed by those families ( D {Mn } DLp (Ω). Let (M ),r τr : DLp n (Ω) −→ Qr be such that
˜ α f ), τr (f ) = (D
f ∈ D(Mn ),r (Ω).
Then τr is an onto linear isometry. We put Q to denote ∪{ Qr : r ∈ N } considering it as a subspace of Y . Let {M } τ : DLp n (Ω) −→ Q be such that
˜ α f ), τ (f ) = (D
{M }
f ∈ DLp n (Ω). {M }
Clearly, τ is linear bijective and continuous. We put λ for the map τ considered from DLp n (Ω) into Y . By t λ we mean as usual the transpose map of λ. Theorem 10. For each j of a set J, let ( gα,j : α ∈ Nk0 ) be a family of elements of Lq (Ω) such that, for each h > 0, sup h|α| M|α| gα,j q < ∞. α∈Nk 0 j∈J
{M }
Then there is a bounded subset { Sj : j ∈ J } of DLp n (Ω) such that {M } ϕ, Sj = Dα ϕ · gα,j dx, j ∈ J, ϕ ∈ DLp n (Ω), α∈Nk 0
Ω
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } subset of DLp n (Ω). P ROOF.
{M }
{M }
It is analogous to the proof of Theorem 5, just replacing BLp n (Ω) by DLp n (Ω).
If 1 < p, we apply Proposition 7 to obtain that Qr is reflexive, r ∈ N. After Proposition 5, Q with the Mackey topology is an (LB)-space. It follows from this that t
is an onto map. 46
{M }
λ : Y −→ DLp n (Ω)
On the structure of certain ultradistributions {M }
Theorem 11. If p > 1 and {Sj : j ∈ J } is a bounded subset of DLp n (Ω), then there is, for each j ∈ J, a family gα,j : α ∈ Nk0 of elements of Lq (Ω) such that, for each h > 0, sup h|α| M|α| gα,j q < ∞
α∈Nk 0 j∈J
ϕ, Sj = Dα ϕ · gα,j dx,
and
α∈Nk 0
j ∈ J,
Ω
{M }
ϕ ∈ DLp n (Ω).
where the series converges absolutely and uniformly when j varies in J and ϕ varies in any given bounded {M } subset of DLp n (Ω). P ROOF. We apply [2, p. 274] and so obtain a relatively compact infinite subset { Tj : j ∈ J } of Y such that t λ(Tj ) = Sj . If (Tj )α is the element of Lq (Ω) which identifies with the restriction of Tj to Y α , α ∈ Nk0 , we obtain an element gα,j in Lq (Ω) such that ϕ · gα,j dx, j ∈ J, ϕ ∈ ϕ˜ ∈ Lp (Ω). ϕ, ˜ (Tj )α = Ω
Given h > 0, we take r in N such that r > h. From Proposition 3, we obtain that sup h|α| M|α| gα,j q < ∞.
α∈Nk 0 j∈J
Having in mind Proposition 2, it follows that, for each (f˜α ) in Y and j ∈ J, ˜ fα · gα,j dx, fα ∈ f˜α , α ∈ Nk0 , (fα ), Tj = α∈Nk 0
Ω
{M }
and, in particular, if ϕ is in D(Lpn) (Ω), then
α ˜ ϕ), Tj = Dα ϕ · gα,j dx (D α∈Nk 0
and besides
Ω
α ˜ ϕ), Tj = λ(ϕ), Tj = ϕ,t λ(Tj ) = ϕ, Sj . (D
The result now follows without difficulty.
Acknowledgement. The author has been partially supported by MEC and FEDER Project MTM200803211.
References [1] G ROTHENDIECK , A., (1955). Produits Tensoriels Topologiques et Espaces Nucl´eaires, Memoirs of the American Mathematical Society, 16. ´ , J., (1966). Topological Vector spaces and Distributions, Volume I, Addison-Wesley, Reading, Mas[2] H ORV ATH sachussets. [3] KOMATSU , H., (1973). Ultradistributions I. Structure theorems and characterizations, J. Fac. Sci. Uni. Tokyo, 20, 25–105.
47
M. Valdivia
[4] ROUMIEU , C., (1960). Sur quelques extensions de la notion de distribution, Ann. Sci. Ecole Norm. Sup. Paris 3, Ser. 77, 41–121. [5] ROUMIEU , C., (1962-63). Ultradistributions d´efinies sur Rn et sur certain classes de vari´et´es differentiables, J. Analyse Math., 10, 153–192. [6] RUDIN , W., (1970). Real and Complex Analysis, McGraw-Hill, London New York. [7] S CHWARTZ , L., (1966). Th´eorie des distributions, Hermann, Paris.
Manuel Valdivia Departamento de An´alisis Matem´atico, Facultad de Matem´aticas Universidad de Valencia Calle Doctor Moliner, 50 46100 Burjassot (Valencia, Spain).
48
