N-harmonic extensions of weighted integrable distributions

We obtain n-harmonic extensions to the Cartesian product of n copies of the upper half-plane, of distributions in the weighted space w 1 . . . w 2 nD L1 , which is known to be the optimal space of tempered distributions that are S ′ convolvable with a natural product domain version of the Poisson kernel.


Introduction and notation
The aim of this article is twofold.First, we solve the following boundary value problem: For each tempered distribution T in an optimal class, we find a n-harmonic function u(x 1 , y 1 ; . . .; x n , y n ) on the product space R 2 + × . . .× R 2 + such that u converges to T in an appropriate sense when y 1 , . . ., y n → 0 + .Let us recall that a complex valued function u(x 1 , y 1 ; . . .; x n , y n ) is called n-harmonic if it is harmonic in each set of variables (x j , y j ) separately.Second, we characterize the class of nharmonic functions on R 2 + × . . .× R 2 + that can be written as finite sums of products of one-dimensional Poisson integrals of appropriate distributions.The work of P. Sjögren in [14] plays a crucial role in the proof of such characterization.
Let us observe that to define the Poisson integral of a distribution, one must specify the rule of convolution and the class of distributions that will be considered.In fact, the Poisson kernel cannot be convolved with every tempered distribution if one expects to preserve the Fourier exchange formula.We consider here the S -convolution developed by Y. Hirata and H. Ogata ( [7]) and R. Shiraishi ([13]), to extend to appropriate pairs of tempered distributions the classical definition of convolution ( [12]).
The results presented in this article concern a natural product domain version of the Poisson kernel, which is related to the notion of n-harmonic function.The optimal space of tempered distributions S -convolvable with this version of the Poisson kernel is identified in [1] as a weighted space of distributions where the weight is a particular function w with separable variables.The need to impose a weighted integrability condition in order to have a well defined Poisson integral, is already present in the case of functions.Indeed, as we show later, a measurable function f has a well defined and everywhere finite Poisson integral if and only if f is integrable with respect to the weight w .This observation extends to the product domain case the condition shown by T. M. Flett [6, p. 762] in the Euclidean case.
H. Bremermann studied in [4, p. 152] the boundary behavior of nharmonic functions and obtained representation formulas by means of Poisson integrals.Our results extend the work of H. Bremermann to a larger class of distributions that is optimal in the context of the S -convolution.
This article is organized as follows: In Section 2 we collect definitions and results to be used later on.In Section 3 we obtain n-harmonic extensions to R 2 + × . . .× R 2 + of appropriate weighted integrable distributions.In Section 4, we characterize a class of n-harmonic functions on R 2 + ×. ..×R 2 + as finite sums of products of one-dimensional Poisson integrals.We must point out that this characterization still leaves open the problem of characterizing any n-harmonic function in terms of the convolution of appropriate distributions with the product domain Poisson kernel.We close our article with a Section 5 where we establish S -convolvability results with Poisson like kernels, both in the Euclidean case and the product domain case.At the heart of this section is the observation that the S -convolvability results rely more on having the appropriate estimates than on having exact formulas for the kernels under consideration.
The notation used here is standard.The symbols C ∞ 0 , S , C ∞ , L p , D , S , E , etc., indicate the usual spaces of distributions or functions defined on R n , with complex values.With L p (ω) we indicate those functions that are p-integrable with respect to the measure ωdx for an appropriate weight ω .We denote |•| the Euclidean norm in R n , while • p indicates the norm in the space L p .When we work on a specific space other than R n , we will write D (R), S(R 2 ), • L p (K) , etc., as appropriate.Partial derivatives will be denoted as ∂ α , where α is a multi-index (α 1 , . . ., α n ).If it is necessary to indicate on which variable we are taking the derivative, we will do so by attaching a sub-index.We will use the standard abbreviations |α| = α 1 +. ..+α n , x α = x α1 1 . . .x αn n .For a function g , we will indicate with ∨ g the function x → g(−x).Given a distribution T , we will denote with Ť the distribution ϕ → (T, ∨ ϕ), where ϕ is an appropriate test function.The Fourier transform will be denoted as F .A smooth function is a continuous function with continuous derivatives of all orders.The letter C will indicate a positive constant, probably different at different occurrences.When it is important to indicate that the constant depends on certain parameters, we will do so by attaching sub-indexes to the constant.
Other notation will be introduced at the appropriate time.

Preliminaries
In order to introduce the notion of S -convolution that we will extensively use, we shall first give a brief review of the spaces of functions and distributions related to this notion.All the notions and results can be found in [12] and [3].
The space of integrable distributions D L 1 is, by definition, the strong dual of the space Ḃ of smooth functions ϕ : R n → C for which ∂ α ϕ → 0 as |x| → ∞, for each multi-index α .The space Ḃ is a closed subspace of the space B consisting of those smooth functions ϕ with the property that ∂ α ϕ is bounded for every multi-index α , endowed with the topology of the uniform convergence on R n of each derivative.The space C ∞ 0 of compactly supported smooth functions defined on R n , is dense in Ḃ but not in B .Thus, the space D L 1 is a subspace of D , the space of distributions.Moreover, each integrable distribution is uniquely determined by its restriction to C ∞ 0 .The space E of compactly supported distributions is a subspace of D L 1 .Moreover, according to [12], p. 201, each T ∈ D L 1 can be represented as where f α ∈ L 1 .As a consequence, we have the (strict) inclusions E ⊂ D L 1 ⊂ S .The space D L 1 is closed under multiplication by functions in B , a fact that we will use repeatedly.
It is possible to consider D L 1 as the strong dual of the space B , provided that we endow B with a topology that gives rise to the following notion of sequence convergence: A sequence {ϕ j } converges to ϕ if, for each multiindex α , sup j ∂ α ϕ j ∞ < ∞ and the sequence {∂ α ϕ j } converges to ∂ α ϕ uniformly on compact sets.If we denote as B c the resulting topological space, it can be proved that C ∞ 0 , and so Ḃ , is dense in B c ( [12], p. 203).Since every distribution T ∈ D L 1 is well defined on C ∞ 0 and it is continuous on C ∞ 0 with respect to the topology of B c , it turns out that D L 1 is also the dual of B c ( [12], p. 203).
We now use these spaces to define the notion of S -convolution.Y. Hirata and H. Ogata [7] introduced the S -convolution to extend the validity of the Fourier exchange formula F (T * S) = F (T ).F (S), originally proved by L. Schwartz for pairs of distributions in the Cartesian product O c × S ( [12]).Later, R. Shiraishi proved in [13] an equivalent definition of S -convolution, which is the one we shall use.

Definition 1 ([13]
).Given two tempered distributions T and S , it is said that the S -convolution of T and S exists if T ( Š * ϕ) ∈ D L 1 for every ϕ ∈ S .When the S -convolution exists, the map is linear and continuous.Thus, it defines a tempered distribution which will be denoted by T * S .
Here, T ( Š * ϕ) is the multiplicative product of the distribution T with the convolution Š * ϕ.This convolution is a smooth function and furthermore, the function and each derivative has at most polynomial growth at infinity.R. Shiraishi proved in [13] that T * S exists if and only if S * T exists, and they coincide.It is important to remark the fact that Definition 1 coincides with the classical definition in all the cases in which the latter makes sense.
The authors of [1] identified optimal spaces of tempered distributions admitting S -convolution with two possible n-dimensional versions of the Poisson kernel: the Euclidean version (2) , where c , y > 0 , and the following product domain version where (y) > (0), meaning that y 1 > 0, . . ., y n > 0 .The product domain considered here is the Cartesian product R 2 + × . . .× R 2 + of n copies of the upper half-plane.
We summarize now these definitions and results.
and let us fix μ ∈ R. Then with the topology induced by the map The next theorem characterizes those tempered distributions which are S -convolvable with the classical Poisson kernel P y defined in (2).

Theorem 3 ([1]
).Given T ∈ S , the following statements are equivalent: It is not difficult to see that there is a continuous and strict inclusion for each y > 0 (see [1]).Also, it can be proved ( [2]) that the space w n+1 D L 1 is closed under differentiation, and that if T ∈ w n+1 D L 1 , then the function In fact, we have the following formula to compute the function T * P y (x): [2]).The intrinsic definition given in Definition 1 shows that the formula (4) is independent of the representation of the distribution T as in (1).The distributions T in the space w μ D L 1 are exactly those tempered distributions that can be represented as In particular, the weighted space L 1 (w −μ ) is contained in w μ D L 1 .Moreover, the distributions in w μ D L 1 can be written as finite sums of distributional derivatives of functions in L 1 (w −μ ) (see [2]).
At this point, it is pertinent to mention that T.M. Flett obtained in [6, Th. 6] a necessary and sufficient condition for a measurable function f on R n to have a well defined Poisson integral, finite everywhere on R n+1 + .This condition is exactly that f ∈ L 1 (w −(n+1) ).The following lemma extends this observation to the product domain case.Lemma 4. Given a measurable function f : R n → C, the following statements are equivalent.
(1) The Poisson integral of f is well defined and finite everywhere in Proof.The proof of this lemma closely resembles the proof of the Euclidean version found in page 762 of [6] and it will be omitted.
Motivated by these arguments, we recall the following definition, which introduces the space of weighted integrable distributions relevant to the product domain case.

Definition 5 ([1]). Let w
with the topology induced by the map Proposition 13 in [1] shows that neither w 2 1 . . .w 2 n D L 1 is included in w n+1 D L 1 nor the other way around.It also shows that both spaces are strictly and continuously included in the space w 2n D L 1 .
The next result characterizes those tempered distributions that are Sconvolvable with the kernel P (y) .

Theorem 6 ([1]
).Given T ∈ S , the following statements are equivalent: The proofs of Theorems 3 and 6 use simple representations of distributions in the appropriate spaces as well as estimates involving well chosen test functions in C ∞ 0 .The choices of functions in C ∞ 0 imply that the spaces of distributions obtained are optimal for the S -convolution as well as for the notion of convolution that results from requesting that Definition 1 holds only for functions ϕ in C ∞ 0 .Lastly, we introduce the space D L 1 of test functions ( [12], p. 199).
endowed with the topology defined by the family of norms The space D L 1 is a Fréchet space and we have the dense and continuous strict inclusions C ∞ 0 ⊂ D L 1 ⊂ D .We conclude this section stating the following known technical lemma, whose detailed proof can be seen in [2].

N -harmonic extensions of distributions in
In this section we obtain n-harmonic extensions to the Cartesian product of n copies of the upper half-plane, of distributions in the weighted space It is shown in [1] that if T ∈ D L 1 , the S -convolution with P (y) is given by the integrable function whose value at x is Also, it is proved that this S -convolution defines a function G = G(x 1 , y 1 ; . . .; x n , y n ) that is smooth and n-harmonic in the product domain In other words, G is a harmonic function of each pair (x j , y j ) separately ( [4]).Moreover, ( [4], [1]) T * P (y) converges to T in D L 1 as (y) → (0) + , which means y j → 0 + for each j = 1, . . ., n.These results imply that we can construct n-harmonic extensions of distributions in D L 1 by considering the S -convolution with the kernel P (y) .
In this section we will obtain these results for distributions in the weighted space w 2 1 . . .w 2 n D L 1 .First, we need to find a convenient formula to compute the S -convolution of a distribution T ∈ w 2 1 . . .w 2 n D L 1 with the kernel P (y) .We claim that this convolution coincides with the function whose value at x is given by ( 7) for each (y) > (0).Indeed, if we write We next notice that the function w 2 1 . . .w 2 n ( ∨ P (y) * ϕ) belongs to B .In fact, there exists a constant C > 0 not depending on (y) such that where the sum is taken over all the possible k -tuples (i 1 , . . ., i k ) with Thus, the right hand side of ( 8) can be expanded using Fubini's theorem to obtain the claimed result.
One can prove, as in the Euclidean case (see [2]), the following characterization for the space w 2 1 . . .w 2 n D L 1 .
We can also obtain the following result.
Lemma 10.Given T ∈ w 2 1 . . .w 2 n D L 1 the S -convolution T * P (y) belongs to the space w 2 1 . . .w 2 n D L 1 for each (y) > (0).Proof.According to Lemma 9, it is enough to prove that S * P Then, according to (7) we have For each j = 1, . . ., n we have and .
Then we have that where This concludes the proof of Lemma 10.

Ḃ
uniformly with respect to ϕ ∈ A.

Without loss of generality let us assume that w
We claim that we can change the order of integration to obtain Indeed, if we denote then, according to Lemma 7 we can estimate and this implies that is finite.Now, to prove (10) it suffices to show that (11) as (y) → (0) + , uniformly with respect to ϕ ∈ A, where So, let us prove (11).For M > 0 to be chosen later, we can write Given ε > 0 , we claim that we can choose M so that ( 12) for every ϕ ∈ A and each (0) < (y) < (1), meaning 0 < y j < 1 for every j .To prove this claim we can write where C does not depend on (y).Thus Then ( 13) Since the set A is bounded in Ḃ , sup ϕ∈A s |α| (ϕ) = K A < ∞.On the other hand, we can choose M large enough so that (14) |t|≥M |f (t)|dt < ε and this concludes the proof of (12).
Next, we need to estimate I 1 for the value of M previously selected.First, we write Concerning J 1 we can write Moreover, for any 0 < σ < 1 we have where C does not depend on (y).Since sup ϕ∈A s |α|+1 (ϕ) = H A < ∞ we obtain the estimate Finally, gathering estimates ( 13), ( 14) and ( 15) we can write for each (0) < (y) < (1).As a consequence, lim sup for every ε > 0 , proving in this way (11).This concludes the proof of Theorem 11.
Remark 12.By Theorem 11, every distribution given by the S -convolution T * P (y) (x).This extension is not unique, since the n-harmonic function y 1 . . .y n extends the zero distribution.

A characterization for a class of n-harmonic functions.
Before stating this characterization, we need to introduce some relevant notation.We set where (x, y) ∈ R n+1 + .We also indicate With L 1,∞ (R n+1 + ) we denote the linear space of functions f that satisfy the weak-type inequality ( 16) for some C > 0 and for every λ > 0, where meas means the Lebesgue measure in R n+1 + .The space L 1,∞ (R n+1 + ) is endowed with the quasi-norm f 1,∞ = inf{C > 0 : C satisfies (16)}.
Theorem 13.Let u : R 2 + × . . .× R 2 + → C be an n-harmonic function.Then the following two statements are equivalent: (1) The function u = u(z 1 , . . ., z n ) where z j = (x j , y j ), can be written as a finite sum of functions of the form where the function This characterization relies on the following theorem by P. Sjögren [14].

Proof.
We first prove that 1) implies 2).According to the onedimensional version of Theorem 14, for each multi-index α there exist a measure μ αj satisfying (18) and c αj ∈ C such that for each j = 1, . . ., n the function G αj can be written as where the sum collects all the possible cross-products containing at least one factor of the form c αi s y is , without repetition.We set Using Lemma 9 and the fact that for each j = 1, . . ., n, we can readily see that Thus, the function u can be represented as in (17) and the proof of 1) =⇒ 2) is complete.
Conversely, let us prove that 2) implies 1).We assume that u can be written as in (17) and we fix a term in (17), say According to the one-dimensional version of Lemma 8, the distribution T j 1 ,...,j n−k can be written as a finite sum of terms of the form Then the representation stated in 1) follows for y i1 . . .y for i ∈ I and G j (z j ) = y j , for j ∈ {1, 2, . . ., n}\I, where I is defined as before.
This concludes the proof of Theorem 13.
Remark 15.P. Sjögren observes in [14] that the quantities R n w −n−1 (x)d × |μ|(x)+|a| and W −n−1 y −1 u 1,∞ in Theorem 14 are equivalent, that is to say they are either both zero or their ratio remains bounded from below and from above by positive constants.Using this observation, it can be proved as in the Euclidean case ( [2]) that there is a common bound on the number of terms, the order of the derivatives and the quasi-norms W −2 i y −1 i G j 1,∞ in the representation given in 1), if and only if the distribution T ji belongs to a bounded set in w 2 j l D L 1 (R) for each j l , and c I belongs to a bounded subset of C.These common bounds and bounded sets can be described by constants that are equivalent quantities.

S -convolvability with Poisson like kernels.
As the results that follow show, in order to establish S -convolvability, it is not as important to assume an exact formula for a kernel as it is to assume that the kernel satisfies the appropriate estimates.We will illustrate this heuristic principle in both the Euclidean case and the product domain case.
We begin with the Euclidean case, where we consider a kernel of the form φ y (x) = 1 y n φ( x y ), y > 0 , for a function φ ∈ C ∞ (R n ) fixed.We assume that φ is a non negative and integrable function such that R n φ > 0.Moreover, we assume that φ satisfies the following estimates: (a) There exists μ > n such that for every multi-index α for some constant We shall denote by G μ the class of functions φ satisfying the properties described above.
It is clear that the classical Poisson kernel P y belongs to G n+1 .Moreover, we have the following result, which turns out to be crucial in the formulation of necessary and sufficient conditions for the existence of the S -convolution with the one-parameter family {φ y } y>0 , for φ ∈ G μ fixed.
Proposition 16.Let φ ∈ G μ fixed.Then the following statements hold.Conversely, if we assume that 2) holds, in order to conclude that T ∈ w μ D L 1 it suffices to show that T can be written as where T 1 ∈ E , T 2 ∈ D L 1 and T 2 is zero in a neighborhood of zero ( [2]).That this is the case follows quite readily from (22).
This concludes the proof of Theorem 17.