Surjectivity of the -operator between spaces of weighted smooth vector-valued functions
We derive sufficient conditions for the surjectivity of the Cauchy-Riemann operator between spaces of weighted smooth Fréchet-valued functions. This is done by establishing an analog of Hörmander’s theorem on the solvability of the inhomogeneous Cauchy-Riemann equation in a space of smooth -valued functions whose topology is given by a whole family of weights. Our proof relies on a weakened variant of weak reducibility of the corresponding subspace of holomorphic functions in combination with the Mittag-Leffler procedure. Using tensor products, we deduce the corresponding result on the solvability of the inhomogeneous Cauchy-Riemann equation for Fréchet-valued functions.
Key words and phrases:Cauchy-Riemann, weight, smooth, surjective, solvability, Fréchet
2010 Mathematics Subject Classification:Primary 35A01, 32W05, Secondary 46A32, 46E40
defDefinition LABEL:#1 \newrefformatremRemark LABEL:#1 \newrefformatsectSection LABEL:#1 \newrefformatsubSection LABEL:#1 \newrefformatpropProposition LABEL:#1 \newrefformatthmTheorem LABEL:#1 \newrefformatchapChapter LABEL:#1 \newrefformatcorCorollary LABEL:#1 \newrefformatparParagraph LABEL:#1 \newrefformatexExample LABEL:#1 \newrefformatpartPart LABEL:#1 \newrefformatfigFigure LABEL:#1 \newrefformatappAppendix LABEL:#1 \newrefformatqueQuestion LABEL:#1 \newrefformatcondCondition LABEL:#1 \newrefformatconvConvention LABEL:#1
We study the Cauchy-Riemann operator between spaces of weighted smooth functions with values in a Fréchet space. Let be a locally convex Hausdorff space over , open and the space of infinitely continuously partially differentiable functions from to . It is well-known that the Cauchy-Riemann operator
is surjective (see e.g. [H3, Theorem 1.4.4, p. 12]). Since , equipped with the usual topology of uniform convergence of partial derivatives of any order on compact subsets, is a nuclear Fréchet space by [meisevogt1997, Example 28.9 (1), p. 349], we have the topological isomorphy by [Treves, Theorem 44.1, p. 449] where is the completion of the projective tensor product. Due to classical theory of tensor products, the surjectivity of implies the surjectivity of
for Fréchet spaces over (see e.g. [Kaballo, Satz 10.24, p. 255]) where is the space of infinitely continuously partially differentiable functions from to and is the Cauchy-Riemann operator for -valued functions. In other words, given there is a solution of the -problem, i.e.
Now, we consider the following situation. Denote by a system of seminorms inducing the locally convex Hausdorff topology of . Let fulfil some additional growth conditions given by an increasing family of positive continuous functions on an increasing sequence of open subsets of with , namely,
for every , and . Let us call the space of functions having this growth . Then there is always a solution of (1). Our aim is to derive sufficient conditions such that there is a solution of (1) having the same growth as the right-hand side . So we are interested under which conditions the Cauchy-Riemann operator
The difficult part is to solve the -problem in the scalar-valued case, i.e. in . In the case that and for all there is a classical result by Hörmander [H3, Theorem 4.4.2, p. 94] on the solvability of the -problem (in the distributional sense) in weighted spaces of -valued square-integrable functions of the form
The solvability of the -problem in weighted -spaces and its subspaces of holomorphic functions has some nice applications (see [Hoermander2003]) and the properties of the canonical solution operator to are subject of intense studies [Bonami1990], [Charpentier2014], [Haslinger2001], [Haslinger2002], [Haslinger2007].
If there is a whole system of weights , i.e. the -problem is considered in the projective limit spaces or where
then solving the -problem becomes more complicated since a whole family of - resp. -estimates has to be satisfied. Such a -problem is usually solved by a combination of Hörmanders classical result with the Mittag-Leffler procedure. However, this requires the projective limit resp. , where
to be weakly reduced, i.e. for every there is such that is dense in with respect to the topology of for resp. , see [Epifanov1992, Theorem 3, p. 56], [Langenbruch1994, 1.3 Lemma, p. 418] and [Polyakova2017, Theorem 1, p. 145]. Unfortunately, the weak reducibility of the projective limit is not easy to check. Furthermore, in our setting we have control the growth of the partial derivatives as well and the sequence usually consists of more than one set.
Let us outline our strategy to solve the -problem in for Fréchet spaces over . In the first part of this paper we phrase sufficient conditions (see \prettyrefcond:weights) such that there is an equivalent system of -seminorms on (see \prettyreflem:switch_top). If they are fulfilled for , then is a nuclear Fréchet space by [kruse2018_4, 3.1 Theorem, p. 12]. If they are fulfilled for as well, we can use Hörmander’s -machinery and the hypoellipticity of to solve the scalar-valued equation (1) on each with given and a solution satisfying for every . The solution is then constructed from the by using the Mittag-Leffler procedure in \prettyrefthm:scalar_CR_surjective which requires a density result given in \prettyrefthm:dense_proj_lim. This density result can be regarded as a weakened variant of weak reducibility of the subspace of consisting of holomorphic functions. In \prettyrefcond:dense we state sufficient conditions on and for our density result to hold that are more likely to be checked. Due to [kruse2017, 5.10 Example c), p. 24] we have if \prettyrefcond:weights holds for and are able to lift the surjectivity from the scalar-valued to the Fréchet-valued case in \prettyrefcor:frechet_CR_surjective. The stated results are obtained by generalising the methods in [ich, Chap. 5] where the special case and, amongst others, is treated (see [ich, 5.16 Theorem, p. 80] and [ich, 5.17 Theorem, p. 82]).
2. Notation and Preliminaries
We define the distance of two subsets , , w.r.t. a norm on via
Moreover, we denote by the sup-norm, by the Euclidean norm, by the usual scalar product on and by the Euclidean ball around with radius . We denote the complement of a subset by , the set of inner points of by , the closure of by and the boundary of by . Further, we also use for a notation of mixed-type
hence identify and as (normed) vector spaces. For a function and we denote by the restriction of to and by
the sup-norm on .
By we always denote a non-trivial locally convex Hausdorff space over the field or equipped with a directed fundamental system of seminorms . If , then we set Further, we denote by the space of continuous linear maps from a locally convex Hausdorff space to . If , we write for the dual space of .
We recall the following well-known definitions concerning continuous partial differentiability of vector-valued functions (c.f. [kruse2018_2, p. 4]). A function on an open set to is called continuously partially differentiable ( is ) if for the -th unit vector the limit
exists in for every and is continuous on ( is ) for every . For a function is said to be -times continuously partially differentiable ( is ) if is and all its first partial derivatives are . A function is called infinitely continuously partially differentiable ( is ) if is for every . The linear space of all functions which are is denoted by . Let . For we set if , and
if as well as
Due to the vector-valued version of Schwarz’ theorem is independent of the order of the partial derivatives on the right-hand side, we call the order of differentiation and write . Now, the precise definition of the spaces of weighted smooth vector-valued functions from the introduction reads as follows.
2.1 Definition ([kruse2018_2, 3.1 Definition, p. 5]).
Let be open and a family of non-empty open sets such that and . Let be a countable family of positive continuous functions such that for all . We call a (directed) family of continuous weights on and set
The subscript in the notation of the seminorms is omitted in the scalar-valued case. The notation for the spaces in the scalar-valued case is and .
The space is a projective limit, namely, we have
where the spectral maps are given by the restrictions
The space of scalar-valued infinitely differentiable functions with compact support in an open set is defined by the inductive limit
Every element of can be regarded as an element of just by setting on and we write for the support of . Moreover, we set for and
By we denote the space of (equivalence classes of) -valued Lebesgue integrable functions on , by , , the space of functions such that and by the corresponding space of locally integrable functions. For a locally integrable function we denote by the regular distribution defined by
For the partial derivatives of a distribution are defined by
The convolution of a distribution and a test function is defined by
In particular, we have for the Dirac distribution and
for and . Furthermore, is valid for and . For more details on the theory of distributions see [H1].
By we denote the space of -valued holomorphic functions on an open set and for we often use the relation
between real partial derivatives and complex derivatives of a function (see e.g. [ich, 3.4 Lemma, p. 17]).
3. From - to -seminorms
For applying Hörmander’s solution of the weighted -problem (see [H3, Chap. 4]) it is appropriate to consider weighted -(semi)norms and use them to control the seminorms of solutions of in weighted -spaces on for given .
Throughout this section let be a polynomial in real variables with constant coefficients in , i.e. there are and for , such that
where , and be the linear partial differential operator associated to .
Let be open, a compact exhaustion of , a hypoelliptic partial differential operator and . Then
is a topological isomorphism where is the equivalence class of , the first space is equipped with the system of seminorms defined by
and the latter with the system
defined for by
First, let us remark the following. The derivatives in the definition of are considered in the distributional sense and means that there exists such that . The definition of the seminorm does not depend on the chosen representative and we make no strict difference between an element of and its representative.
, equipped with the system of seminorms (4), is known to be a Fréchet space. The space , equipped with the system of seminorms (5), is a metrisable locally convex space. Let be a Cauchy sequence in . By definition of we get for all that there exists a sequence in such that . Therefore we conclude from (5) that and , , are Cauchy sequences in , which is a Fréchet space by [F/W/Buch, 5.17 Lemma, p. 36], so they have a limit resp. in this space. Since resp. converges to resp. in , it follows that resp. converges to resp. in . Here is the space equipped with the weak-topology. Hence we get
in implying and the convergence of to in with respect to the seminorms (5) as well. Thus this space is complete and so a Fréchet space.
is obviously linear and injective. It is continuous as for all and we have
and there exists , only depending on the coefficients and the number of summands of , such that
for all where denotes the Lebesgue measure.
The next step is to prove that is surjective. Let . Then we have where
and so by the Sobolev embedding theorem [H1, Theorem 4.5.13, p. 123] in combination with [H1, Theorem 3.1.7, p. 59]. To be precise, this means that the regular distribution has a representative in . Due to the hypoellipticity of we obtain , more precisely, that has a representative in , so is surjective.
Finally, our statement follows from (i)-(iii) and the open mapping theorem. ∎
Let be a hypoelliptic partial differential operator, and . Then we have
where , .
Let . Then the sets , , form a compact exhaustion of and there exists such that . Since is continuous by \prettyreflem:iso_sobolev, there are , and such that
for all , , and . ∎
Due to this corollary we can switch to types of -seminorms which induce the same topology on as the -seminorms and we get an useful inequality to control the growth of the solutions of the weighted -problem by the right-hand side under the following conditions.
Let be a family of continuous weights on an open set and a family of non-empty open sets such that and . For every let there be such that for all and let there be such that for any there is , , and and such that for any :
If , then these conditions are a special case of [kruse2018_4, 2.1 Condition, p. 3] by [kruse2018_4, 2.3 Remark b), p. 3] and modifications of the conditions - in [L4, p. 204]. They guarantee that is a nuclear Fréchet space by [kruse2018_4, 3.1 Theorem, p. 12] and [kruse2018_4, 2.7 Remark, p. 5] if which we use to derive the surjectivity of from the one of for Fréchet spaces over .
3.4 Convention ([L4, 1.1 Convention, p. 205]).
We often delete the number counting the seminorms (e.g. or ) and indicate compositions with the functions only in the index (e.g. ).
Let be open and a family of non-empty open sets such that and . Let be a (directed) family of continuous weights on . For we define the locally convex Hausdorff spaces
Let \prettyrefcond:weights be fulfilled for some .
Let be a hypoelliptic partial differential operator, and such that and . Then and
Then as locally convex spaces.
Due to [kruse2018_4, 2.11 Lemma (p.1), p. 8-9], [kruse2018_4, 2.7 Remark, p. 5] and [kruse2018_4, 2.3 Remark b), p. 3] there are and a sequence , for , in such that the balls
form an open covering of with
Let , , , and . By \prettyrefcor:iso_sobolev there exist and , and independent of and , such that
and so we get
Let and be the Laplacian. Then satisfies the conditions of a) for all because
for every . So for every and there exist and such that