Weighted -Integral Representations of -Functions in
For -functions , given in the complex space , integral representations of the form are obtained. Here, is the orthogonal projector of the space onto its subspace of entire functions and the integral operator appears by means of explicitly constructed kernel Φ which is investigated in detail.
Let , , and , . Denote by the space of all measurable complex-valued functions , , satisfying the condition Let be the corresponding subspace of entire functions. It was established in [1, 2] (when ) and  (when ) that arbitrary function has an integral representation of the form where and is the Mittag-Leffler type function. Moreover, the integral operator generated by the right-hand side of the formula (1.2) is an orthogonal projection of the space onto its subspace . Certainly, the condition (1.1) and corresponding properties of ensure an absolute convergence of the integral in (1.2).
Note that for ,, , coincides with the well-known Fock space of entire functions and (1.2) takes the form
In  general weighted integral representations were obtained for differential forms. In particular, for functions (satisfying certain growth conditions), the following generalization of the formula (1.4) was established: where
In  a canonical operator is constructed for -solution in a space of differential forms square integrable with the weight .
The following natural question arises: as good as (1.4) is generalized for the case of smooth (not necessarily holomorphic) functions by the representation (1.5), is it possible to generalize the representation (1.2) in the similar way? Of course, such generalization should contain (1.5) as a particular case. Let us note (before we discuss this question) that in the case of bounded domains (and ) integral representations are well investigated: for unit ball of see [6–8]; for general strictly pseudoconvex domains see [9–13]; for Cartan matrix domain (“matrix disc”) see . The whole space essentially differs from strictly pseudoconvex and bounded symmetric domains (the last ones have rich group of automorphisms!), so the above-mentioned generalization requires other methods. For it was done in . For the case , two essentially different generalizations are possible. In  “polycylindric” weight function of the type , , was considered. In the present paper, the corresponding weighted integral representations are obtained for the case of radial weight function of the type , , (see the condition (1.1)). More precisely, for functions (satisfying certain growth conditions), the integral representation of the form is established. Moreover, the kernel is written in an explicit form. Also, we prove certain important differential and integral properties of this kernel. As it will be seen below, in the case of radial weight function, a new approach is requested and significant analytical difficulties are arised.
Remark 1.1. In , instead of , the case of weight function of the type was considered in the assumption that is a convex function of class . In this case, a formula of type (1.8) was obtained. But in that formula the operator of orthogonal projection of the space onto its subspace of entire functions does not appear, except of the special case when we again obtain (1.5).
2. Heuristic Argument: Revealing of the Kernel
In what follows, it is supposed that and , , .
We intend to reveal a formula of type (1.2) but this time for -functions. This means that the formula we search needs to have a second summand containing -“part” of functions. In other words, for functions satisfying certain (indefinite yet) growth conditions at infinity, we search a formula of the type (1.8). Besides, it will be desirable for the kernel , to have the following (or similar) properties:
Denote by (for all , ) the well-known Martinelli-Bochner kernel, which has the following useful properties (see, for instance, [17, Chapter 16]): for arbitrary function , where is an open set; for arbitrary function continuous in a neighborhood of ; where is the unit sphere in , and is the surface measure on .
Let us fix an arbitrary and consider the following differential form:
Then apply the Stokes formula to this form and to the domain : The integral (as ) due to the property (2.2) (our argument is heuristic !!!). When , then due to the properties (2.5) and (2.1). Thus after and , we have
Let us fix arbitrary , and consider the function
Then we have
The condition (2.2) implies ; hence due to Cauchy-Green-Pompeiju formula,
Since , we finally have This formula can be also written in the following (may be, more convenient) form: Now it is natural to investigate the properties of the kernel introduced. But first we need some auxiliary results.
3. Auxiliary Results
First of all let us put for brevity (see (1.3)) It is an entire function of order and of type 1. The same is true for its derivative , . Consequently, we have
Let us introduce a convenient notation If , we can suppose that in (3.3). Then obviously the function and Note that under additional assumptions , or , , the function and, moreover, In view of (3.1) and (3.3), the formula (2.16) can be written as follows:
Further, assume that , , , then evidently where
Note that and lies on the same complex “straight line” (i.e., complex plane) of passing through the origin.
Lemma 3.1. Assume that and , then uniformly in and with , , where are positive constants and depend, in general, on .
Proof. According to (3.2),
where is a positive number.
Further, if , then in view of (3.4) and (3.7) Due to the conditions on and , we have . Let us choose such that , then for Hence Combining (3.11) and (3.13), we obtain where is a positive number.
Combination of (3.10) and (3.14) easily implies (3.9) for the case . If , then similarly to (3.11)–(3.14) we have where is a positive number. Also, in view of (3.7), Combination of (3.10), (3.15), and (3.16) establishes (3.9) for the case . The proof is complete.
Lemma 3.2. Assume that and . Then there exist positive constants (depending, in general, on ) such that uniformly in and with , .(a)If , then(b) If , or , , then
Lemma 3.3. Let , , be an arbitrary function of class such that it together with its first-order partial derivatives decreases (modulo) at infinity. For instance, the decreasing of type , , (for arbitrary small ) is quite sufficient for us. Then the function is of class and , .
This assertion is of standard type so we omit the proof. Similar results one can find in [18, page 10, Theorem ] for bounded open sets or in [19, page 300, Lemma] for arbitrary simply connected domains, but for -functions.
Lemma 3.4. Assume that , , is continuously differentiable (i.e., of class ) and , , is continuous. Then the following two relations are equivalent:
Proof. Let us fix an arbitrary ; then This immediately gives the implication (3.20)⇒(3.21). On the contrary, if (3.21) is valid, then Substitution of into the last relation gives (3.20). Thus, the assertion is proved.
4. The Main Properties of the Kernel
Proposition 4.1. If , then for fixed , and for arbitrary where Moreover, assume that and . Then there exist positive constants (depending, in general, on ) such that uniformly in and with , .(a) If , then is a continuous function in and(b) If , or , , then is a -function in and, in addition to (4.3),
Proposition 4.2. If , then the kernel is continuous in , .
Proof. Let us write as follows:
For arbitrary fixed positive numbers , it suffices to construct a function , such that uniformly in and with , . According to Lemma 3.1 (the case ), the function is suitable for a function we seek.
Proposition 4.3. If and is arbitrary, then for arbitrary function continuous in a neighborhood of .
Proof. For sufficiently small , put
Taking into account the explicit formula (2.16) for and the well-known explicit form of the Martinelli-Bochner kernel , we obtain
In view of (2.6), we obtain
After the change of variable in the inner integral in (4.13), we have
Obviously, without loss of generality, we can suppose that .
According to Lemma 3.1 (the case ) or, equivalently, to Lemma 3.2(a), there exist positive constants (depending on ) such that uniformly in , . Hence, due to the Lebesgue dominated convergence theorem, we can conclude that(i)the functions are continuous in ;(ii), (uniformly in );(iii)For for all , where Note that the function is also continuous in and , .
Now remember (see (4.14)) that Therefore, the application of the Lebesgue dominated convergence theorem once again gives in view of (1.2). Thus (4.10) is established.
Remark 4.4. Assume for a moment that can be defined at such that and, moreover, that after this becomes continuous at (unfortunately, this is not, in general, true, as it will be mentioned below). Then the relation (4.10) is a simple consequence of (2.5). Hence (4.10) can be considered as a substitute for the natural (and “very desired”) property .
Proposition 4.5. If , or , , the kernel is a function of class in , .
Proof. In view of (4.6)-(4.7), we have to show that is of class in , . In other words, for arbitrary fixed positive numbers , it suffices to construct a function , such that
uniformly in and with , , and .
Explicitly computing the corresponding partial derivatives and taking note of Lemma 3.2, we find that we are reduced to the question of the finiteness of integrals of the type which is evident.
Proposition 4.6. If , or , , then for arbitrary fixed we have or, equivalently,
Proof. We intend to use Lemma 3.4. To this end, let us put for It suffices to establish (3.21) for the introduced functions and . Fix , then in view of Proposition 4.1 we have where is defined by (4.2) and satisfies (4.3)-(4.3’). Consequently, satisfies all the conditions of Lemma 3.3. Hence By this (3.21) has been established for the pair of introduced functions and . The proof is complete.
The next assertion describes the behaviour of the kernel when or . Since these properties will not be used in what follows, we omit the proof, which, by the way, is not easy.
Proposition 4.7. If , then(a)for arbitrary and for arbitrary , we have (b) if and is the complex plane generated by the vector , then and for arbitrary and for (arbitrary small) , there exist a positive constant such that uniformly in with ;(c) if , then where the coefficients can be written in an explicit form;(d) for arbitrary , there exist positive constants (depending, in general, on ) such that uniformly in and with . In particular, the kernel remains bounded (uniformly in with ) as .
Remark 4.8. As it follows from (4.28) and (4.30), in general, (when tends to zero arbitrarily) cannot be properly defined (i.e., this limit does not exist). In view of this fact, it seems surprising the existence of the limit in (4.27). In fact, it only means that, nevertheless, the restrictions of the kernel on complex planes (generated by arbitrary ) have limit values at the origin.
5. The Main Integral Representation
Now we are ready to formulate and prove the main result: an integral representation of the type (1.8). To this end, we have to repeat the heuristic argument of Section 2, but this time it should be well reasoned.
In what follows, we need a function , , satisfying the following conditions:(i); (ii), ;(iii), ;(iv), ;(v); (vi), and (obviously) otherwise.
The existence of such functions is evident. Then put Note that and
Theorem 5.1. Assume that , , and , where either , or , . If the kernel ,for all , for all , is defined by the formula (2.15), then the integral representation of the form is valid for each function satisfying the following conditions:(a);(b)for any fixed, (c) for any fixed
Proof. Let us fix an arbitrary , and for consider the following differential form: