Harmonic Extensions of Distributions

Josefina Alvarez

Department of MathematicsNew Mexico State University

Las Cruces, NM 88003, USAE-mail: jalvarez@nmsu.edu

Martha Guzman-Partida

Departamento de MatematicasUniversidad de Sonora

Hermosillo, Sonora 83000, Mexico

E-mail: martha@gauss.mat.uson.mx

Salvador Perez-Esteva∗

Instituto de Matematicas, Unidad Cuernavaca

Universidad Nacional Autonoma de MexicoCuernavaca, Morelos 62251, Mexico

E-mail: salvador@matcuer.unam.mx

Abstract

We obtain harmonic extensions to the upper half-space of distri-butions in the weighted spaces wn+1D′

L1 , which according to [1] arethe optimal spaces of tempered distributions S ′-convolvable with theclassical euclidean version of the Poisson kernel . We also characterizethe class of harmonic functions in the upper half-space with boundary

∗Partially supported by PAPIIT-IN105801.02000 Mathematics Subject Classification. Primary 46F20, 46F05, 46F12.0Key words and phrases. S ′-convolution, distributions in weighted spaces as boundary

values of harmonic functions, harmonic extensions of distributions.

1

values in wn+1D′L1 , generalizing in this way a classical result in the

theory of Hardy spaces. Some facts concerning harmonic extensions ofdistributions in D′

Lp , 1 < p ≤ ∞, are also approached in this paper, aswell as natural relations among these spaces and the weighted spaceswn+1D′

L1 . We can also obtain n-harmonic extensions of appropriateweighted integrable distributions associated to a natural product do-main version of the Poisson kernel.

1 Introduction

In this paper, we first solve the Dirichlet problem in the upper half-spaceassuming that the value at the boundary is a tempered distribution in anoptimal class. Then, we characterize those harmonic functions in the upperhalf-space that are Poisson integrals of distributions in this optimal class.

In order to define the Poisson integral of a distribution, we need to sayhow the convolution is defined. Regardless of the definition, any bonafideconvolution should be commutative, should commute with derivatives, andshould satisfy the Fourier exchange formula F (T ∗ S) = F(T )F(S). Sincethe Fourier transform of the Poisson kernel is not smooth at zero, we cannotexpect, in any case, that the Poisson integral of any tempered distributionwill be defined. The fact that the Poisson kernel is not convolvable with everytempered distribution, was observed already by E.M. Stein ([17], p. 91) inrelation to the definition of the real Hardy space using a non tangentialmaximal function.

We consider here the so-called S ′-convolution, developed by Y. Hirata andH. Ogata ([8]) and R. Shiraishi ([14]) with the purpose of extending to appro-priate pairs of tempered distributions the classical definition of convolutiongiven by L. Schwartz ([13]). The optimal space of tempered distributionsconvolvable in this sense with the Poisson kernel is identified in ([1]) as aweighted space of distributions. As we will explain later, the fact that weconsider tempered distributions is not a limiting factor in the identificationof the optimal space.

The work of P. Sjogren in [15], plays a crucial role in our characterizationof harmonic functions as Poisson integrals of distributions. His work providesa weighted alternative to results by E.M. Stein and G. Weiss ([18], p. 38).We point out that, to our knowledge, the problem of formulating a directextension of the results by E.M. Stein and G. Weiss is open. In Section 5 we

2

present a partial extension.The results we have mentioned so far are related to the Euclidean version

of the Poisson kernel. We can obtain as well results for the product domainversion of the Poisson kernel, which appears in relation to n-harmonic func-tions. In this context we also have an optimal space of distributions, whichis a weighted space with respect to a weight that has separable variables.

Our results on the Euclidean case extend results on the Dirichlet problemwith boundary values that are functions in Lp or finite Borel measures (seefor instance [3], [4], [16]), or distributions with compact support in R ([5], p.49). The results we mention in the product domain case extend results of H.Bremermann ([5], p. 152).

The organization of our paper is as follows: In Section 2 we recall severaldefinitions and results that we use later on. In Section 3 we solve the Dirich-let problem for the upper half-space with boundary values in an optimalweighted space of distributions.

In Section 4 we extend to this space of distributions the characterizationby P. Sjogren of harmonic functions in the upper half-space. The link betweenSjogren’s results and the weighted space of distributions that we consider,is the following simple observation: The distributions in the weighted spacecan be represented as finite sums of distributional derivatives of functionsbelonging to a weighted L1 space. Finally, in Section 5 we present partialextension of the results of E.M. Stein and G. Weiss.

The notation we use here is standard. The symbols C∞0 , S, C∞, Lp,

Lploc, D

′, S ′, E ′, etc., indicate the usual spaces of distributions or functionsdefined on Rn, with complex values. We refer to the functions in the spaceC∞ as smooth functions. With Lp (w−µ) we indicate those functions that arep-integrable with respect to the measure w−µdx for an appropriate weightw−µ. We denote |·| the Euclidean norm on Rn, while ‖·‖p indicates the normin the space Lp. With p′ we indicate the conjugate of the exponent p. Whenwe work on a specific space other than the generic space Rn, we will writeD′ (R), S (R2), ‖·‖Lp(K), etc. Partial derivatives will be denoted as ∂α, whereα is a multi-index (α1, ..., αn). If it is necessary to indicate on which variablewe are taking the derivative, we will do so by attaching a sub-index. With 4x

we indicate the Laplacian operator ∂2x1

+ ...+ ∂2xn

. We will use the standardabbreviations |α| = α1 + ... + αn, xα = xα1

1 ...xαnn . For a function g, we will

indicate with∨g the function x → g (−x) . Given a distribution T , we will

indicate with T the distribution ϕ →(T,

∨ϕ), where ϕ is an appropriate

3

test function. The Fourier transform will be denoted as F . The letter C willindicate a positive constant, that may be different at different occurrences. Ifit 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.

2 Preliminary definitions and results

Before explaining the notion of S ′-convolution that we will use, we will reviewbriefly the definition of related spaces of functions and distributions. Formore details, we refer to [13], p. 199.

With B we indicate the space of smooth functions ϕ : Rn → C such that∂αϕ is bounded in Rn for each multi-index α. We consider in B the topologyof the uniform convergence in Rn of each derivative. With B we indicate theclosed subspace of B that consists of those smooth functions ϕ : R

n → C

such that ∂αϕ → 0 as |x| → ∞, for each multi-index α. The space C∞0 is

dense in B but not in B.The space D′

L1 of so called integrable distributions is, by definition, the

topological dual of the space B, endowed with the strong dual topology.Since C∞

0 is dense in B, the space D′L1 is a subspace of D′. Every com-

pactly supported distribution T belongs to D′L1. Moreover, according to [13],

p. 201, each distribution T ∈ D′L1 can be represented as T =

∑finite ∂

αfα

where fα ∈ L1. This explains why the distributions in D′L1 are called inte-

grable distributions. Moreover, it shows that they belong to the space S ′.It is very easy to see that D′

L1 is closed under derivation as well as undermultiplication by functions in B.

For future use, we will need to consider an alternative topology in thespace B. This topology is the finest locally convex topology that induces theusual topology of C∞, on the subsets of B that are bounded with respect tothe topology previously defined on B. We will denote with Bc the space Bendowed with this topology. A sequence ϕj converges to ϕ in Bc when, foreach multi-index α, supj ‖∂

αϕj‖∞ < ∞ and the sequence ∂αϕj convergesto ∂αϕ uniformly on compact sets of Rn. It can be proved that C∞

0 , and soB, is dense in Bc ([13], p. 203). Moreover, it can also be proved that thespace D′

L1 is the topological dual of Bc ([13], p. 203).Now we are ready to consider the notion of S ′-convolution. The first

definition was proposed by Y. Hirata and H. Ogata [8]. They wanted to

4

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 Cartesianproduct O′

M × S ′([13], p. 268). Subsequently, R. Shiraishi introduced in [14]an equivalent definition which is the one we will use in this paper. Namely,

Definition 1 [14] Given two tempered distributions T and S, it is said thatthey are S ′-convolvable if T

(S ∗ ϕ

)∈ D′

L1 for every ϕ ∈ S. If this is thecase, the map

S → C

ϕ→(T(S ∗ ϕ

), 1)

D′L1

,Bc

turns out to be linear and continuous. Thus, it defines a tempered distributionwhich will be denoted by T ∗ S.

In this definition, T(S ∗ ϕ

)indicates the multiplicative product of the

tempered distribution T with the function given by the regularization x →(St, ϕ (x− t)

)S′,S

. This regularization is a smooth function and each deriva-

tive has at most polynomial growth at infinity.R. Shiraishi proved in [14] that the S ′-convolution T ∗ S exists if and

only if S ∗ T exists, and that they coincide. Thus, the operation introducedin Definition 1 satisfies the commutativity property expected of a bonafideconvolution. This operation coincides with the classical convolution definedby L. Schwartz, in all the cases in which both make sense.

Let us point out that when the distributions T and S are integrablefunctions, we have

(T(S ∗ ϕ

), 1)

D′L1

,Bc=(T(S ∗ ϕ

), 1)

L1,L∞ =

∫

n

(T ∗ S) (x)ϕ (x) dx

where T ∗ S indicates the classical convolution of two integrable functions.On the other hand, given two tempered distributions T and S that are

S ′-convolvable, we can write

(T(S ∗ ϕ

), 1)

D′L1

,Bc=∑

finite

(−1)|α| (fα∂α1)L1,L∞ = (f0, 1)L1,L∞

where fα ∈ L1. Thus, we recapture the case of the classical pairing (L1, L∞).

5

In the paper [1], it is solved the problem of finding the optimal spaces oftempered distributions that are S ′-convolvable with the Euclidean version ofthe Poisson kernel,

Py(x) =c(n)

yn

1(|x|2 /y2 + 1

)n+1

2

, (1)

where c(n) =Γ(n+1

2 )π

n+12

, y > 0, as well as the product domain version of the

Poisson kernel,

P(y)(x) =n∏

i=1

Pyi(xi), (2)

where (y) > 0, meaning y1, ..., yn > 0. In this last case, the product domainconsidered is the Cartesian product R2

+ × ... × R2+ of n copies of the upper

half-plane. It is well known that the Poisson kernel given by (1) is closelyrelated to the study of harmonic functions, while the Poisson kernel given by(2) is closely related to the study of n-harmonic functions.

The authors of [1] identified appropriate weighted spaces of distributions,as those tempered distributions that are S ′-convolvable with each of theabove versions of the Poisson kernel. We summarize now these definitionsand results.

Definition 2 ([9],[10],[11]) Let w(x) =(1 + |x|2

) 1

2 , for x ∈ R. Then, givenµ ∈ R we consider

wµD′L1 =

T ∈ S ′ : w−µT ∈ D′

L1

with the topology induced by the map

wµD′L1 → D′

L1

T → w−µT .

It is immediate from Definition 2, that the space wµD′L1 is the topological

dual of the spaces w−µB and w−µBc.

Theorem 3 ([1]) Given T ∈ S ′, the following statements are equivalent:

1. T ∈ wn+1D′L1.

6

2. T is S ′-convolvable with Py, for each y > 0.

It is easy to see that there is a continuous and strict inclusion wµ1D′L1 ⊂

wµ2D′L1 when µ1 < µ2. Given T ∈ wn+1D′

L1 , it was proved in [1] that theS ′-convolution T ∗ Py is a function defined on Rn as

(T ∗ Py) (x) =(w−n−1(t)Tt, w

n+1(t)Py(x− t))

D′L1

,Bc(3)

for each y > 0. More precisely,

Lemma 4 The space wn+1D′L1 is closed under derivation. Moreover, given

T ∈ wn+1D′L1 , the function (T ∗ Py) (x) is smooth and ∂α (T ∗ Py) = ∂αT ∗Py

for each y > 0.

Proof. Using the representation of distributions in wn+1D′L1 in terms

of derivatives of integrable functions, it is very easy to see that the spacewn+1D′

L1 is closed under derivation. Moreover, as in the classical case ([13],p. 105), we can prove using (3), that the function (T ∗ Py) is smooth. Toshow that ∂α (T ∗ Py) = ∂αT ∗ Py for each y > 0, we could invoke generalproperties of the S ′-convolution. However, for the sake of completeness, wewill prove it directly.

We consider a cut-off function ψ ∈ C∞0 such that 0 ≤ ψ ≤ 1, ψ = 1 in

|x| ≤ 1 and ψ = 0 in |x| ≥ 2, and let ψn = ψ (x/n) for each n = 1, 2, ....Thus, for every ϕ ∈ S we can write

((Py ∗ ϕ) ∂αT, ψn)D′L1

,Bc= (∂αT, ψn (Py ∗ ϕ))S′,S

= (−1)|α| (T, ψn (Py ∗ ∂αϕ))S′,S

+ (−1)|α|Cγ,α

∑

0<γ≤α

(T, ∂γψn (Py ∗ ∂α−γϕ))S′,S

(4)Since T = wn+1S, for some distribution S ∈ D′

L1, we have that (4) canbe written as

(−1)|α| (T, ψn (Py ∗ ∂αϕ))S′,S

+ (−1)|α| Cγ,α

∑

0<γ≤α

(S,wn+1∂γψn (Py ∗ ∂α−γϕ))D′

L1,Bc.

We claim that each term in the sum above goes to zero as n → ∞. Infact, we know that ψn → 1 in Bc. Moreover, for each y > 0 there exists

7

C = Cy > 0 such that |Py ∗ ∂α−γϕ| ≤ Cw−n−1. Thus, for each 0 < γ ≤ α,

we have that wn+1∂γψn (Py ∗ ∂α−γϕ) → 0 in Bc. This implies that

((Py ∗ ϕ) ∂αT, 1)D′L1

,Bc= lim

n→∞(−1)|α| (T, ψn (Py ∗ ∂

αϕ))S′,S

or,((∂αT ) ∗ Py, ϕ)S′,S = lim

n→∞(−1)|α| (T, ψn (Py ∗ ∂

αϕ))S′,S (5)

Furthermore,

(∂α (T ∗ Py) , ϕ)S′,S = (−1)|α| (T ∗ Py, ∂αϕ)S′,S

= (−1)|α| ((Py ∗ ∂αϕ)T, 1)D′

L1,Bc

= limn→∞

(−1)|α| ((Py ∗ ∂αϕ)T, ψn)D′

L1,Bc

= limn→∞

(−1)|α| (T, ψn (Py ∗ ∂αϕ))S′,S .

= limn→∞

(−1)|α| (S,wn+1ψn (Py ∗ ∂αϕ))D′

L1,Bc

= (−1)|α| (T, (Py ∗ ∂αϕ))S′,S .

(6)

From (5) and (6) we obtain the desired conclusion. This completes theproof of Lemma 4.

Remark 5 As a consequence of (3), we can deduce that the convolutionT ∗ Py is the function given by

(T ∗ Py) (x) =∑

finite

(−1)|α|∫

n

fα (t) ∂αt

(wn+1 (t)Py (x− t)

)dt

where T =∑

finite

wn+1∂αfα with fα ∈ L1. It is easy to show that this formula

does not depend on the representation of T . Of course, this formula requiresto know that the distributions in wn+1D′

L1 are convolvable with Py.

The distributions in the space wµD′L1 are exactly those tempered distri-

butions T that can be represented as

T =∑

finite

wµ∂αfα

where fα ∈ L1. In particular, wµD′L1 contains the weighted space L1 (w−µ).

As we will show in Section 4, the distributions in wn+1D′L1 are finite sums of

8

distributional derivatives of functions in L1 (w−n−1). A similar result appliesto wµD′

L1 for any µ. The same formula as in (3) holds with obvious modi-fications for distributions in the weighted space wµD′

L1 for 0 ≤ µ ≤ n + 1.This result makes evident the convenience of thinking of the space D′

L1 as

the dual of both B and Bc.

Definition 6 ([1]) Let wj =(1 + x2

j

) 1

2 , j = 1, ..., n. Then

w21...w

2nD

′L1 =

T ∈ S ′ : w−2

1 ...w−2n T ∈ D′

L1

with the topology induced by the map

w21...w

2nD

′L1 → D′

L1

T → w−21 ...w−2

n T .

Theorem 7 ([1]) Given T ∈ S ′, the following statements are equivalent:

1. T ∈ w21...w

2nD

′L1 .

2. T is S ′-convolvable with P(y), for each (y) > 0.

The proofs of Theorems 3 and 7 use simple representations of distribu-tions in the appropriate spaces as well as estimates involving well chosentest functions in the space C∞

0 . This choice of test functions turns out tobe important for the following reason: The definition of convolution given inDefinition 1 could be modified to state that two distributions T and S willbe convolvable if T

(S ∗ ϕ

)∈ D′

L1 for each ϕ ∈ C∞0 ([14]). It is known ([6],

[11]), that two tempered distributions can be convolvable in this sense with-out the convolution being a tempered distribution. The fact that the proofsof Theorems 3 and 7 only use test functions in the space C∞

0 implies thatthe spaces are optimal for this convolution as well as for the S ′-convolution.So in our context, there is no distinction between these two ways of definingthe convolution.

We point out ([1]), that there are distributions in the space wn+1D′L1 that

do not belong to the space w21...w

2nD

′L1 , as well as there are distributions in

the space w21...w

2nD

′L1 that do not belong to the space wn+1D′

L1. Both spacesare continuously and strictly included in the space w2nD′

L1 .We now introduce briefly a few other spaces ([13], p. 199), that will be

of interest later.

9

For 1 ≤ p <∞, let

DLp = ϕ ∈ C∞ : ∂αϕ ∈ Lp for every multi-index α ,

endowed with the topology defined by the family of norms

‖ϕ‖m,p =

∑

|α|≤m

‖∂αϕ‖pp

1/p

, m = 0, 1, 2, ....

The space DLp is a Frechet space and we have the dense and continuous strictinclusions C∞

0 ⊂ DLp ⊂ D′.The space DL∞ is the space B defined above. For 1 ≤ p ≤ q ≤ ∞ we

have the continuous strict inclusions DLp ⊂ DLq .When 1 < p ≤ ∞, we denote with D′

Lp the topological dual of DLp′ ,endowed with the strong dual topology. We have the continuous strict in-clusions D′

Lp ⊂ D′Lq for p ≤ q. Moreover, every distribution T ∈ D′

Lp canbe represented as T =

∑finite ∂

αfα, fα ∈ Lp ([13], p. 201). This impliesthat the space D′

Lp is continuously included in S ′. A necessary and sufficientcondition for a tempered distribution T to belong to D′

Lp, 1 ≤ p ≤ ∞, isthat the regularization T ∗ϕ belongs to DLp for every ϕ ∈ C∞

0 ([13], p. 201).The distributions in D′

L∞ are usually called bounded distributions. As wewill indicate soon, bounded distributions play an essential role in the studyof real Hardy spaces.

We also consider, for 1 ≤ p ≤ ∞, µ ∈ R, the weighted space wµD′Lp ,

endowed with the topology induced by the map

wµD′Lp → D′

Lp

T → w−µT

It is not surprising that the S ′-convolution with each of the Poisson kernelscarries a restriction. Indeed, since the Fourier transform in x of any of thesekernels is only continuous at zero, not every tempered distribution can beS ′-convolvable with them. This situation is observed also when studyingthe definition of the real Hardy spaces by means of maximal functions. Infact, E.M. Stein showed in [17], p. 91, that the space Hp is contained inD′

L∞, the space of bounded distributions. This space is strictly includedin wn+1D′

L1 . The convolution of a bounded distribution with the Poissonkernel, as defined in [17], coincides with the S ′-convolution when viewed inthe context of wn+1D′

L1 .

10

We conclude this section on preliminary definitions and results with thefollowing estimate, which will be used in the next section.

Lemma 8 The integral I (η) =

∫

n

(1 + |ξ|)r (1 + |η − ξ|)s dξ is finite, for

each η ∈ Rn, if r + s+ n < 0. Moreover, if this is the case, we have

I (η) ≤

Cn,r,s

(1 + |η|r+s+n) if r + n > 0 and s+ n > 0

Cn,r,s (1 + |η|)maxr,s if r + n < 0 or s + n < 0

Proof. Using the inequality,

(1 + |η − ξ|)s ≤ 2|s| (1 + |η|)|s| (1 + |ξ|)s

we can conclude readily that I (ξ) is finite when r + s + n < 0. So, we willassume in what follows that r + s+ n < 0.

If r + n > 0 and s+ n > 0, we consider the sets

Ω1 =

ξ ∈ R

n : |η − ξ| ≤|η|

2

,

Ω2 =

ξ ∈ R

n : |ξ| ≤|η|

2

,

Ω3 = Rn − (Ω1 ∪ Ω2)

and the integrals

Ii(η) =

∫

Ωi

(1 + |ξ|)r (1 + |η − ξ|)s dξ, i = 1, 2, 3.

If ξ ∈ Ω1 we have that |η| /2 ≤ |ξ| ≤ 3 |η| /2. Then

I1(η) ≤ Cr,s |η|r

∫

Ω1

(1 + |η − ξ|)s dξ

≤ Cn,r,s |η|r

∫ |η|/2

0

tn−1(1 + t)sdt

≤ Cn,r,s (1 + |η|)r+s+n ,

since s+ n > 0.

11

By symmetry, we also have that I2(η) ≤ Cn,r,s (1 + |η|)r+s+n. Finally,we notice that if ξ ∈ Ω3, then C1 |ξ| ≤ |η − ξ| ≤ C2 |ξ| . In fact, since|η − ξ| ≥ |η| /2 and |ξ| ≥ |η| /2, we have that

|ξ − η|

|ξ|≤

|ξ| + |η|

|ξ|≤ 3

and|ξ|

|ξ − η|≤

|ξ − η| + |η|

|ξ − η|≤ 3.

Then

I3(η) ≤ C

∫

n−(Ω1∪Ω2)

(1 + |ξ|)r+s dξ

= Cn

∫ ∞

|η|/2

rn−1(1 + r)r+sdr

≤ Cn,r,s |η|r+s+n .

Without loss of generality, we assume now that r > s and s+ n < 0. Asin the previous case, we obtain

I1(η) ≤ Cn,r,s (1 + |η|)r

Likewise, the integral I2(η) can be estimated by (1 + |η|)s or (1 + |η|)r+s+n,depending on whether r+n ≤ 0 or r+ n > 0. In both cases we have I2(η) ≤Cn,r,s (1 + |η|)r . Finally, I3(η) ≤ Cn,r,s (1 + |η|)r+s+n ≤ Cr,s,n (1 + |η|)r.

This concludes the proof of Lemma 8.

3 Harmonic extensions of distributions in

wn+1D′L1

We will prove in this section that every distribution T ∈ wn+1D′L1 has a

harmonic extension to the upper half-space Rn+1+ . Before we state and prove

the main result, we need to do some preparatory work.

Lemma 9 Given T ∈ wn+1D′L1 , the S ′-convolution T ∗ Py belongs to the

space wn+1DL1 , for each y > 0.

12

Proof. According to Lemma 4, it suffices to show that S∗Py ∈ L1 (w−n−1)for every S ∈ wn+1D′

L1 . In fact, if this is the case, then ∂α (T ∗ Py) =(∂αT ∗ Py) will belong to L1 (w−n−1) for every multi-index α. This will im-ply that T ∗ Py ∈ DL1(w−n−1) = Dwn+1L1 = wn+1DL1 .

So, without loss of generality we assume that w−n−1S = ∂αf , with f ∈ L1.Then,

(S ∗ Py) (x) =(w−n−1 (t)St, w

n+1 (t)Py (x− t))

D′L1

,Bc

=∑

β≤α

(−1)|α| Cα,β

∫

n

f (t)(∂βwn+1

)(t)(∂α−βPy

)(x− t) dt

Since ∣∣∂βwn+1(t)∣∣ ≤ Cn,β

(1 + |t|2

)n+1−|β|2

and

∣∣(∂α−βPy

)(x− t)

∣∣ ≤ Cn,α,β

yn+|α−β|

(1 +

|x− t|2

y2

)−n+1+|α−β|

2

we can write

|(S ∗ Py) (x)| ≤∑

β≤α

Cn,α,β

yn+|α−β|

∫

n

|f(t)|(1 + |t|2

)n+1−|β|2

(1 +

|x− t|2

y2

)−n+1+|α−β|

2

dt

Thus ∫

n

|(S ∗ Py) (x)|w−n−1(x)dx ≤∑

β≤α

Cn,α,β

yn+|α−β|Iαβ,

where

Iα,β =

∫

n

w−n−1(x)

∫

n

|f(t)|(1 + |t|2

)n+1−|β|2

(1 +

|x− t|2

y2

)−n+1+|α−β|2

dtdx.

Each integral Iα,β can be written as

∫

n

|f(t)|

∫

n

(1 + |x|2

)−n+1

2

(1 +

|x− t|2

y2

)−n+1+|α−β|

2

dx

(1 + |t|2

)n+1−|β|2 dt

≤ Cn,α,β

∫

n

|f(t)|

[∫

n

(1 + |x|)−n−1

(1 +

|x− t|

y

)−n−1−|α−β|

dx

](1 + |t|)n+1−|β| dt

13

where we have used that the quantities (1 + a) and (1 + a2)1/2

are equivalentfor any a > 0.

We observe that

(1 +

|x− t|

y

)−n−1−|α−β|

≤ Cn,α,β,y (1 + |x− t|)−n−1−|α−β|

(7)

where

Cn,α,β,y =

1 if 0 < y < 1

yn+1+|α−β|

if y ≥ 1

Then, according to Lemma 8, we have that

∫

n

(1 + |x|2

)−n+1

2

(1 +

|x− t|2

y2

)−n+1+|α−β|

2

dx ≤ Cn,α,β,y (1 + |t|)−n−1 (8)

Thus,

Iα,β ≤ Cn,α,β,y

∫

n

|f(t)| (1 + |t|)−n−1 (1 + |t|)n+1−|β| dt <∞.

This concludes the proof of Lemma 9.

Theorem 10 Given T ∈ wn+1D′L1 the S ′-convolution T ∗ Py converges to T

in wn+1D′L1 as y → 0+.

Proof. According to Lemma 9, T ∗ Py ∈ wn+1DL1 ⊂ wn+1D′L1 for each

y > 0. We must show that for each bounded subset A of B,

(w−n−1 (T ∗ Py) , ϕ

)D′

L1,B

→(w−n−1T, ϕ

)D′

L1,B

(9)

14

uniformly with respect to ϕ ∈ A. It is enough to assume that w−n−1T = ∂αf ,for some f ∈ L1. If we fix ϕ ∈ A, we can write(w−n−1 (T ∗ Py) , ϕ

)D′

L1,B

=

∫

n

(T ∗ Py) (x)w−n−1(x)ϕ(x)dx

=

∫

n

(w−n−1(t)Tt, w

n+1(t)Py(x− t))

D′L1

,Bcw−n−1(x)ϕ(x)dx

= (−1)|α|∫

n

∫

n

f(t)

[∑

β≤α

Cα,β

(∂βwn+1

)(t)∂α−β

t (Py(x− t)) dt

]w−n−1(x)ϕ(x)dx

=

∫

n

∫

n

f(t)

[∑

β≤α

Cα,β

(∂βwn+1

)(t) (−1)|β| ∂α−β

x (Py(x− t)) dt

]w−n−1(x)ϕ(x)dx

We claim that we can change the order of integration, to obtain

∑

β≤α

Cα,β

∫

n

f(t)

[(−1)|β|

∫

n

∂α−βx (Py(x− t))w−n−1(x)ϕ(x)dx

]∂βwn+1(t)dt

=∑

β≤α

(−1)|α| Cα,β

∫

n

f(t)

[∫

n

Py(x− t)∂α−β(w−n−1ϕ

)(x)dx

]∂βwn+1(t)dt.

Indeed, if we denote

s|α|(ϕ) = sup0≤|γ|≤|α|

‖∂γϕ‖∞ ,

then, according to (7) and (8), we can write∫

n

Py(x− t)∣∣∂α−β

(w−n−1ϕ

)(x)∣∣ dx ≤ Cn,α,β,ys|α|(ϕ)

∫

n

Py(x− t)w−n−1(x)dx

≤ Cn,α,β,ys|α|(ϕ)w−n−1(t),

which implies that∫

n

∫

n

|f(t)|Py(x− t)∣∣∂α−β

(w−n−1ϕ

)(x)∣∣ ∣∣∂βwn+1(t)

∣∣ dxdt

is finite. If we set

Φy(t) =∑

β≤α

Cα,β

[∫

n

Py(x− t)∂α−β(w−n−1ϕ

)(x)dx

]∂βwn+1(t),

15

in order to prove (9), it suffices to show that∫

n

|f (t)| |Φy(t) − ∂αϕ(t)| dt→ 0 (10)

as y → 0+, uniformly with respect to ϕ ∈ A.For M > 0 to be chosen later, we can write

∫

n

|f (t)| |Φy(t) − ∂αϕ(t)| dt

=

∫

|t|≤M

|f (t)| |Φy(t) − ∂αϕ(t)| dt

+

∫

|t|>M

|f (t)| |Φy(t) − ∂αϕ(t)| dt

= I1 + I2

Given ε > 0, we claim that we can choose M so that

I2 < CAε (11)

for every ϕ ∈ A and for every 0 < y < 1. To prove this claim we write

|Φy(t) − ∂αϕ(t)|

≤∑

β≤α

Cα,β

∫

n

Py(x− t)

∣∣∂α−β(w−n−1ϕ

)(x) − ∂α−β

(w−n−1ϕ

)(t)∣∣

×∣∣∂βwn+1(t)

∣∣ dx

≤ Cn,αs|α|(ϕ)

[∫

n

Py(x− t)w−n−1(x)dx+ w−n−1(t)

∫

n

Py(x− t)dx

]wn+1(t).

We observe that∫

n

Py(x− t)w−n−1(x)dx = Cn (Py ∗ P1) (t) = CnPy+1 (t) .

Thus, using (7), we have, for 0 < y < 1,

|Φy(t) − ∂αϕ(t)| ≤ Cn,αs|α|(ϕ).

Then

I2 ≤ Cns|α|(ϕ)∑

β≤α

∫

|t|>M

f (t) dt (12)

16

Since the set A is bounded in B, supϕ∈A s|α|(ϕ) = KA <∞. On the otherhand, we can choose M large enough so that

∫

|t|>M

|f(t)| dt < ε. (13)

This concludes the proof of the claim. We now need to estimate I1, forthe value of M we have just selected.

We can write,

∫

n

Py(x− t)∣∣∂α−β

(w−n−1ϕ

)(x) − ∂α−β

(w−n−1ϕ

)(t)∣∣ dx

=

∫

|x−t|<1

Py(x− t)∣∣∂α−β

(w−n−1ϕ

)(x) − ∂α−β

(w−n−1ϕ

)(t)∣∣ dx

+

∫

|x−t|>1

Py(x− t)∣∣∂α−β

(w−n−1ϕ

)(x) − ∂α−β

(w−n−1ϕ

)(t)∣∣ dx

= J1 + J2

We have

J2 ≤ Cn,α,βs|α|(ϕ)

∫

|u|>1/y

(1 + |u|)−n−1 du.

Concerning J1 we can write,

J1 =

∫

|x−t|<1

Py(x− t)

∣∣∣∣∫ 1

0

∇(∂α−β

(w−(n+1)ϕ

))(t + θ(x− t)) · (x− t)dθ

∣∣∣∣ dx

≤ Cn,α,βs|α|+1(ϕ)

∫

|x−t|<1

Py(x− t) |x− t| dx.

Moreover, for any 0 < σ < 1 we have

∫

|x−t|<1

Py(x− t) |x− t| dx

≤Cn

yn

∫

|x−t|<1

(1 +

|x− t|

y

)−n−1

|x− t|σ dx

≤ Cnyσ

∫

n

(1 + |u|)−n−1 |u|σ du = Cnyσ.

17

Since supϕ∈A s|α|+1(ϕ) = HA <∞ we obtain the estimate

I1 ≤ C

(HAy

σ +KA

∫

|x−t|>1/y

(1 + |u|)−n−1 dx

)∫

|t|≤M

|f (t)| dt. (14)

Finally, if we gather estimates (12), (13), and (14), we can write∫

n

|f (t)| |Φy(t) − ∂αϕ(t)| dt

≤ Cn,α

[KAε+ ‖f‖1

(HAy

σ +KA

∫

|x−t|>1/y

(1 + |u|)−n−1 dx

)]

for every 0 < y < 1. As a consequence,

lim supy→0+

∫

n

|f (t)| |Φy(t) − ∂αϕ(t)| dt ≤ Cn,αKAε,

for every ε > 0. Thus, we have proved (10).This concludes the proof of Theorem 10.

Remark 11 The above results imply that given a distribution T ∈ wn+1D′L1 ,

the function u = T ∗ Py is a solution of the Dirichlet problem(∂2

y + 4x

)u = 0 in R

n+1+

u|y=0 = T

where the boundary condition is interpreted in the sense of convergence inwn+1D′

L1 as y → 0+. The function u (x, y) = y is a solution of the homoge-neous problem, so there is no uniqueness.

Remark 12 The results presented in this section can be formulated also inthe product domain case. The techniques detailed here for the Euclidean caseapply with appropriate modifications to the product domain case. We willomit any details.

4 Harmonic functions in Rn+1+ as Poisson in-

tegrals of distributions in wn+1D′L1

In what follows, we indicate

W = W (x, y)def=(1 + |x|2 + y2

)1/2

18

where (x, y) ∈ Rn+1+ . Moreover, we recall that L1,∞

(R

n+1+

)denotes the linear

space of functions f that satisfy the weak-type inequality

meas((x, y) ∈ R

n+1+ : |f (x, y)| > λ

) ≤

C

λ(15)

for some C = C (f) > 0 and for all λ > 0, where the measure is the Lebesguemeasure. The quantity

inf C > 0 : C satisfies (15)

is a quasi-norm in L1,∞(R

n+1+

), which we will denote ‖•‖1,∞.

We obtain the following two characterizations of harmonic functions inthe upper half-space as Poisson integrals of distributions in wn+1D′

L1.

Theorem 13 Let u : Rn+1+ → C be a harmonic function. Then the following

two statements are equivalent.

1. The function u can be written as

u =∑

|α|≤k

∂αgα (16)

where gαis harmonic on Rn+1+ and W−n−1y−1gα ∈ L1,∞

(R

n+1+

)

2. There exist a distribution T ∈ wn+1D′L1 and a complex number a such

thatu = T ∗ Py + ay. (17)

Moreover, the functions gα belong to a bounded set in the quasi-norm‖W−n−1y−1•‖1,∞ and the number of terms is bounded, if and only if the dis-

tribution T belongs to a bounded subset of wn+1D′L1 and the constant a belongs

to a bounded subset of C. These bounded sets are described by constants thatare equivalent quantities.

Theorem 14 Let u : Rn+1+ → C be a harmonic function such that u (·, y)

defines by integration a tempered distribution for each y > 0. Then thefollowing two statements are equivalent.

1. For some l = 0, 1, 2, ... the Bessel potential v (x, y)of order 2l of u (·, y),satisfies the condition

W−n−1y−1v (x, y) ∈ L1,∞(R

n+1+

).

19

2. There exist a distribution T ∈ wn+1D′L1 and a complex number a such

thatu = T ∗ Py + ay.

Moreover, the function v belongs to a bounded set in the quasi-norm‖W−n−1y−1·‖1,∞ and the order 2l is bounded, if and only if the distribu-

tion T belongs to a bounded subset of wn+1D′L1 and the constant a belongs

to a bounded subset of C. These bounded subsets are described by constantsthat are equivalent quantities.

The proofs of these two results rely on the following theorem by PeterSjogren ([15]).

Theorem 15 [15] Let u : Rn+1+ → C be a harmonic function. Then the

following two statements are equivalent.

1. The function W−n−1y−1u belongs to L1,∞(R

n+1+

).

2. There exists a complex Borel measure µ satisfying∫

n

w−n−1 (x) d |µ| (x) <∞

and there exists a ∈ C such that

u = µ ∗ Py + ay

Moreover, the quantities∫

n w

−n−1 (x) d |µ| (x)+|a| and ‖W−n−1y−1u‖1,∞

are equivalent.

The constant a in the representation of U as u = µ ∗ Py + ay may bedifferent from zero. In fact if we consider u = y, thenW−n−1y−1u = W−n−1 ∈L1,∞

(R

n+1+

), as it is very easy to verify. So, according to Sjogren’s theorem,

there exists a complex Borel measure µ satisfying∫

n w

−n−1 (x) d |µ| (x) <∞and there exists a ∈ C, so that y = µ ∗ Py + ay. If a = 0, we would have

y = µ ∗ Py

We can interpret the left hand side of this equation as 1x ⊗ y. So, if we takethe Fourier transform of both sides in the variable x, we obtain

yδξ = µ (ξ) e−2πy|ξ|

20

where δξ indicates the Dirac measure acting on the variable ξ.Or,

µ (ξ) = ye2πy|ξ|δξ = yδξ,

which is not possible. As Sjogren points out, the constant a will be zero ifand only if W−n−1y−1u ∈ L1,∞

(R

n+1+

)and for each ε > 0,

∫BεW−n−1dxdy <

∞, where Bε =(x, y) ∈ R

n+1+ : |u (x, y)| > εy

. When u = y, the set Bε

coincides with the upper half-space, so the integral becomes infinite.The proof of Theorem 13 uses two lemmas that we will state and prove

now.The first lemma provides a new representation for distributions in the

space wn+1D′L1 .

Lemma 16

wn+1D′L1 =

T ∈ S ′ : T =

∑

finite

∂αgα, where gα ∈ L1(w−n−1

)

(18)

Proof. Let us temporarily indicate with V the right hand side of (18).Given T ∈ V, we can write T =

∑finite

∂α (wn+1fα), where fα ∈ L1. Or,

T =∑

finite

∑

0≤β≤α

Cαβ∂α−βwn+1∂βfα

= wn+1∑

finite

∑

0≤β≤α

Cαβw−n−1

(∂α−βwn+1

)∂βfα.

By definition, the distribution ∂βfα belongs to D′L1 . Moreover, the func-

tion Cαβw−n−1

(∂α−βwn+1

)belongs to the space B. Since D′

L1 is closedunder multiplication by functions in B, we conclude that T belongs town+1D′

L1 . Conversely, given T ∈ wn+1D′L1 we can write, by definition,

T = wn+1∑

finite

∂αfα, where fα ∈ L1. Or, T = wn+1∑

finite

∂α (w−n−1gα), where

gα ∈ L1 (w−n−1). Now, given ϕ ∈ S, the pairing (T, ϕ)S′S can be written as

∑

finite

(−1)|α|(gα, w

−n−1∂α(wn+1ϕ

))S′S

=∑

finite

∑

0≤β≤α

(−1)|α| Cαβ

(gα, w

−n−1(∂α−βwn+1

)∂βϕ

)S′S

.

21

We observe that for each multi-indexes α and β, the function

bαβ = (−1)|α| Cαβw−n−1

(∂α−βwn+1

)

belongs to B. Thus,

(T, ϕ)S′S =∑

α,β

(∂β((−1)|β| bαβgα

), ϕ)

S′S

or,

T =∑

α,β

∂β((−1)|β| bαβgα

)

To conclude that the distribution T belongs to V we only need to observethat L1 (w−n−1) is closed under multiplication by functions in B.

This completes the proof of Lemma 16.

Remark 17 With the same proof, we can obtain a similar representationfor distributions in the space wµD′

Lp, for µ ∈ R and 1 ≤ p ≤ ∞.

Remark 18 Lemma 16 implies that the space wn+1D′L1 can also be described

asT ∈ S ′ : T =

∑

finite

bα∂αfα,where fα ∈ L1

(w−n−1

)and bα ∈ B

.

Lemma 19 Every complex Borel measure µ satisfying∫

n

w−n−1 (x) d |µ| (x) <∞ (19)

defines a distribution T in wn+1D′L1 as

(T, ϕ)wn+1D′L1

,w−n−1B =

∫

n

ϕ (x) dµ (x)

for ϕ ∈ C∞0 .

Proof. If we fix ϕ ∈ w−n−1B, we can write∣∣∣∣∫

n

ϕ (x) dµ (x)

∣∣∣∣

≤∥∥wn+1ϕ

∥∥∞

∫

n

w−n−1 (x) d |µ| (x) .

22

So, the claim follows immediately. This completes the proof of Lemma 19.

We are now ready to prove Theorem 13.Proof. We assume that u =

∑|α|≤k

∂αgα, with W−n−1y−1gα ∈ L1,∞(R

n+1+

).

Then according to Theorem 15, for each multi-index α there exist a measureµα satisfying (19) and cα ∈ R, so that gα = µα ∗ Py + cαy. Then, we canwrite

u =∑

|α|≤k

∂α (µα ∗ Py + cαy)

=

(∑

|α|≤k

∂αµα

)∗ Py + c0y

According to Lemma 19 and Lemma 4, the distribution

(∑

|α|≤k

∂αµα

)belongs

to wn+1D′L1. Thus, u can be represented as (17).

Conversely, we assume that the function u can be written as

u = T ∗ Py + ay

with T ∈ wn+1D′L1 and a ∈ C. According to Lemma 16, we have that T =∑

|α|≤k

∂αfα, with fα ∈ L1 (w−n−1), for some k = 0, 1, 2, .... Thus,

u =∑

|α|≤k

∂α (fα ∗ Py) + ay (20)

Since the measure fαdx satisfies (19), using Theorem 15, we see thatthe harmonic function gα = fα ∗ Py satisfies the property W−n−1y−1gα ∈L1,∞

(R

n+1+

). It is clear that the term ay also belongs to L1,∞

(R

n+1+

). Thus

the function u can be represented as in (16).To prove the quantification condition, we first observe that the bounded

subsets of wn+1D′L1 are exactly those subsets B ⊂ wn+1D′

L1 for which thefollowing statement is true: There exist k = kB = 0, 1, 2, ... and CB > 0fixed, so that each T ∈ B is represented as

T =∑

|α|≤k

∂αfα

where fα ∈ L1 (w−n−1) and ‖fα‖L1(w−n−1) ≤ C.

23

The proof of this claim follows the proof of Theorem XXV in [13] p. 201,taking into account the method used in the proof of Theorem XXII in page195. For another instance of the same type of results, we refer to page 246in the same reference.

According to the equivalence of norms and quasi-norms stated in Theorem15, there exists a constant An > 0 so that if u = T∗Py+ay for T ∈ B, boundedsubset of wn+1D′

L1 , and |a| ≤ C ′, then sup|α|≤k

‖W−n−1y−1 (fα ∗ Py)‖L1,∞( n+1

+ ) ≤

An (C + C ′). Moreover, the number k of terms in (20) remains bounded.Conversely, let us fix k = 0, 1, 2, ... and a constant C > 0 and let us

consider the set C of functions u : Rn+1+ → C such that

u =∑

|α|≤k

∂αgα

for harmonic functions gα that satisfy sup|α|≤k

‖W−n−1y−1gα‖L1,∞( n+1

+ ) ≤ C.

The proof of 1) ⇒ 2) above and Theorem 15 show that there exists An > 0

so that if u ∈ C, then u =

(∑

|α|≤k

∂αµuα

)∗ Py for some µu

α satisfying

∫

n

w−n−1 (x) d |µuα| (x) ≤ AnC.

We claim that the set

∑

|α|≤k

∂αµuα

u∈C

is bounded in wn+1D′L1 in the sense

explained above.Given l = 1, 2, ..., we consider the solution E of the equation

(1 −

4

4π2

)l

El = δ (21)

where δ indicates the Dirac measure concentrated at zero. It is understoodthat E depends on l, although for simplicity we will not indicate that. Takingthe Fourier transform on both sides of (21), we can see that E is a boundedcontinuous function when l > n

2. Moreover, the identity

(∂αE)∧ = (2πiξ)α E = (2πiξ)α (1 + |ξ|2)−l

shows that the continuous function E has continuous derivatives of order≤ s, provided that l ≥ n+s

2.

24

On the other hand, given m = 0, 1, 2, ..., we have for each |α| ≤ s,

(1 + |x|2

)m∂αE

∧

=

(1 −

4

4π2

)m

∂αE

=

(1 −

4

4π2

)m

(2πiξ)α (1 + |ξ|2)−l

The estimate∣∣∣∣(

1 −4

4π2

)m [(2πiξ)α (1 + |ξ|2

)−l]∣∣∣∣ ≤ Cn,m,l,s

(1 + |ξ|2

)−l+ s2

shows that(1 + |x|2

)m∂αE is a bounded function for each m, which implies

that ∂αE ∈ L1 (wm) for each |α| ≤ s and for each m.Using the above observations with s = 0, we can see that for appropriate

values of m and l, the convolution E∗ µuα is well defined. Moreover, it belongs

to L1 (w−n−1). In fact,

∫

n

|(E ∗ µuα) (x)|w−n−1 (x) dx (22)

≤

∫

n

w−n−1 (x)

∫

n

|E (x− y)| d |µuα| (y) dx

≤ 2n+1

∫

n

∫

n

wn+1 (x− y)×

|E (x− y)|w−n−1 (y)d |µuα| (y)dx.

From this estimate, we can see that there exists a constant AC,n,m,l > 0such that ‖E ∗ µu

α‖L1(w−n−1) ≤ AC,n,m,lC.Then, we can write

∑|α|≤k

∂αµuα =

∑|α|≤k

∂α

(1 −

4

4π2

)l

fuα (23)

where fuα = E ∗ µu

α. The estimates above prove our claim.This completes the proof of Theorem 13.The work done to prove (23) allows us to obtain yet another character-

ization of wn+1D′L1 , this time using Bessel potentials. We will use it in the

proof of Theorem 14.

25

Lemma 20 A distribution T belongs to wn+1D′L1 , if and only if there exists

l = lT,n = 0, 1, 2, ... and a function f = fT,n ∈ L1 (w−n−1) so that

T =

(1 −

4

4π2

)l

f (24)

The proof of Lemma 20 uses the following result from [12].

Theorem 21 [12] Given µ, ν ∈ R, 1 ≤ p, q ≤ ∞, we assume that µ + ν ≥0, 1

p+ 1

q− 1 = 1

rfor some 1 ≤ r ≤ ∞. Then, if ρ = min µ, ν, the S ′-

convolution

wµD′Lp × wνD′

Lq → wρD′Lr

(S, T ) → S ∗ T

is well defined and continuous.We are now ready to prove Lemma 20.Proof. It is clear that any distribution represented as in (24) will be-

long to wn+1D′L1 . Conversely, given T =

∑|α|≤s

∂αfα ∈ wn+1D′L1 and given E

satisfying (21) for m > n + 1, we can conclude from Theorem 21 that theS ′-convolution

w−n−1D′L1 → w−n−1D′

L1

T → T ∗ E

is well defined and continuous.Then we can write

T = T ∗

(1 −

4

4π2

)l

E =

(1 −

4

4π2

)l

(T ∗ E)

=∑|α|≤s

∂αfα ∗

(1 −

4

4π2

)l

E =

(1 −

4

4π2

)l ∑|α|≤s

fα ∗ ∂αE.

As in the proof of Theorem 13, we can see that fα ∗ ∂αE ∈ L1 (w−n−1).So, T will have the representation (24), if we choose f =

∑|α|≤s

fα ∗ ∂αE.

This completes the proof of Lemma 20.

26

Remark 22 Let us observe that for l fixed, the function f is uniquely deter-mined by the distribution T . Moreover, according to the definition of f , wehave

f = T ∗ E

Actually, E = F−1[(

1 + |ξ|2)−l]. That is to say, f is the Bessel potential of

order 2l of T . This representation of wn+1D′L1 can be extended to wµD′

Lp forµ ∈ R and 1 ≤ p ≤ ∞. We only need to observe that the work done in theproof of Theorem 13, implies that for each |α| ≤ s the continuous function∂αE belongs to Lp (wm) for each m, so it belongs to w−mD′

Lp for each m.The characterization given by (24) extends a characterization proved by L.Schwartz ([13], p. 204) in the non-weighted case.

We are now ready to prove Theorem 14.Proof. We first assume that 1) holds. Then, according to Sjogren’s

work, there exists a measure µ satisfying (19) and a constant a ∈ C, bothdepending on u and l, such that

u (·, y) ∗ E = µ ∗ Py + ay

As in the proof of Lemma 20, we can write

u =

(1 −

4

4π2

)l

(u (·, y) ∗ E) =

(1 −

4

4π2

)l

(µ ∗ Py + ay)

=

(1 −

4

4π2

)l

µ ∗ Py + ay

So, we have proved that 2) holds.Conversely, if we assume 2), we can write

u = T ∗ Py + ay

for some T ∈ wn+1D′L1 , and a ∈ C.

According to Lemma 20,

T =

(1 −

4

4π2

)l

f (25)

for some f ∈ L1 (w−n−1) and l = 0, 1, 2, .... Or,

f = T ∗ E

27

Since T ∗ Py belongs to wn+1DL1 for each y > 0, we can compute theS ′-convolution of T ∗ Py with E using the associativity and commutativityproperties of the S ′-convolution. So, we have for some c ∈ C,

u (·, y) ∗ E = (T ∗ E) ∗ Py + cy = f ∗ Py + cy. (26)

According to Sjogren’s work, this implies that W−n−1 [u (·, y) ∗ E] y−1 ∈L1,∞

(R

n+1+

). So we have 1).

Concerning the quantification condition, if the functionW−n−1 [u (·, y) ∗ E] y−1

and the order 2l satisfy the conditions∥∥W−n−1 [u (·, y) ∗ E] y−1

∥∥L1,∞(

n+1

+ ) ≤ C

2l ≤ C

for some C > 0, then Theorem 15 shows that there exists An > 0 such thatthe measure µ and the constant a satisfy

∫

n

w−n−1 (x) d |µ| (x) ≤ AnC

|a| ≤ AnC

The work done to prove the quantification condition in Theorem 13 shows

that the set of distributions(

1 − 44π2

)lµ

is bounded in wn+1D′L1 , with the

appropriate control on the constants.Conversely, we fix a bounded set B in wn+1D′

L1 and we assume that thefunction u can be written as

u = T ∗ Py + ay

for some T ∈ B, and a in a bounded subset of C. Then, writing the distribu-tion T as in (25) we claim that the function f belongs to a bounded subset ofL1 (w−n−1), with appropriate bounds on the constants. In fact, the proof ofthis claim involves the same type of estimates as shown in (22). If we writethe Bessel potential of the function u as in (26), we can use Sjogren’s resultsto conclude that the function W−n−1 [u (·, y) ∗ E] y−1 belongs to a boundedsubset of L1,∞

(R

n+1+

), with the appropriate estimates.

This completes the proof of Theorem 14.

Remark 23 The quantification conditions proved in Theorem 13 and Theo-rem 14 imply that the characterizations of the space wn+1D′

L1 given in Lemma16 and Lemma 20 preserve bounded sets.

28

Remark 24 As in the previous section, the results presented here can beformulated also in the product domain case. The techniques used in the Eu-clidean case apply with appropriate modifications to the product domain case.

5 A partial extension of a theorem of Stein

and Weiss

We begin with the following lemmas.

Lemma 25 Given a distribution T ∈ wµD′Lp, 0 ≤ µ ≤ n + 1, 1 < p ≤ ∞,

the S ′-convolution of T with the Poisson kernel Py is well defined, for eachy > 0.

Proof. It is enough to show that Lp (w−µ) is contained in L1 (w−n−1) for1 < p ≤ ∞, 0 ≤ µ ≤ n+ 1.

Given f ∈ Lp (w−µ) we assume first that p <∞. We can write

∫

n

|f (x)|w−n−1dx =

∫

n

|f (x)|w− δpw

δpw−n−1dx.

Using Holder’s inequality, we can estimate the above integral as

[∫

n

(w

µpw−n−1

)p′

dx

]1/p′

‖f‖Lp(w−µ) .

The condition for the bracket to be finite is − (n+ 1) p′ + µpp′ < −n.The

left hand side of this inequality is an increasing function of µ ∈ [0, n+ 1].When µ = n+1, the left hand side is equal to − (n+ 1) p′+(n+ 1) (p′ − 1) =−n− 1. So the inequality holds for all the allowed values of µ.

When p = ∞, we have the estimate

∫

n

|f (x)|w−n−1dx ≤ ‖f‖∞∥∥w−n−1

∥∥1.

This completes the proof of Lemma 25.

Lemma 26 Given f ∈ Lp (w−µ) for 1 < p <∞, 0 ≤ µ ≤ n+1, the Poissonintegral of f is well defined.

29

Proof. It is enough to observe that∫

n

Py (x− t) |f (t)| dt ≤ Cx,y

∥∥w−µ/pf∥∥

p

×∥∥wµ/p −n−1

∥∥p′<∞

for 1 < p <∞, 0 ≤ µ ≤ n + 1.

Lemma 27 For any µ > 0, the weight w−µ belongs to the Muckenhoupt classA1.

Proof. If M denotes the Hardy-Littlewood maximal function, we needto show that there exists C > 0 such that

M(w−µ

)(x) ≤ Cw−µ (x)

for every x ∈ Rn (see for instance [7], p. 151).Since the weight w−µ is a bounded function and for some a > 0 is

w−µ (x) ≥ a

for |x| ≤ 1, we are left to consider the case |x| > 1. If we write the maximalfunction M as

M (f) (x) = supr>0

1

|B (x, r)|

∫

B(x,r)

|f (t)| dt,

we can consider the cases r < |x| /2 and r > |x| /2 > 1/2.In the first case we have

1

|B (x, r)|

∫

B(x,r)

w−µ (t) dt ≤ w−µ (t/2)

≤ Cw−µ (t)

for some C > 0.In the second case, we can write

1

|B (x, r)|

∫

B(x,r)

w−µ (t) dt

≤ Cr−n

∫

B(0,2r)

w−µ (t) dt

= Cr−n

∫

B(0,1)

w−µ (t) dt+ Cr−n

∫

B(0,2r)\B(0,1)

w−µ (t) dt

≤ Cr−n(1 + rn−µ

)≤ C

(1 + r−µ

)≤ Cw−µ (x) .

This completes the proof of Lemma 27.

30

Proposition 28 Let u : Rn+1+ → C be a harmonic function. We assume

thatu = T ∗ Py

for some distribution T ∈ wµD′Lp , where 1 < p < ∞ and 0 ≤ µ ≤ n + 1.

Then the following two conditions hold.

1. For each y > 0, the function u (·, y) belongs to Lp (w−µ).

2. There exists l = 0, 1, 2, ... such that

supy>0

‖v (·, y)‖Lp(w−µ) <∞,

where v (·, y) is the Bessel potential of order 2l of the function u (·, y).

Proof. We first assume that the distribution T is given by a function fin the space Lp (w−µ). Then

|u (x, y)| ≤ CM (f) (x)

for some C > 0. According to Lemma 27, since the weight w−µ belongs tothe class A1 that is contained in Ap for all p > 1, we have the inequality (see[7], p. 387)

‖u (·, y)‖Lp(w−µ) ≤ C ‖M (f)‖Lp(w−µ)

≤ C ‖f‖Lp(w−µ) .

So, we are done in this particular case.In the general case, we invoke Remark 22 to write the distribution T as

T =

(1 −

4

4π2

)l

f

with f ∈ Lp (w−µ). Then, we have that v (·, y) is the Poisson integral of thefunction f . Thus, according to what we did in the particular case, we canconclude that condition 2) holds. We can actually say that

supy>0

‖v (·, y)‖Lp(w−µ) ≤ C ‖f‖Lp(w−µ) .

31

Finally,

u (·, y) = f ∗

(1 −

4

4π2

)l

Py.

Since ∣∣∣∣∣

(1 −

4

4π2

)l

Py

∣∣∣∣∣ ≤ Cl,yPy

we conclude that condition 1) holds.This completes the proof of Proposition 28.

Remark 29 It is clear from the proof of Proposition 28 that we could add aquantization condition controlling the order 2l and the supy>0 ‖v (·, y)‖Lp(w−µ),for T in a bounded subset of wµD′

Lp.

Proposition 30 Given a harmonic function u : Rn+1+ → C , we assume that

the following conditions hold.

1. For each y > 0, the function u (·, y) belongs to Lp (w−µ).

2. There exists l = 0, 1, 2, ... such that

supy>0

‖v (·, y)‖Lp(w−µ) <∞,

where v (·, y) is the Bessel potential of order 2l of the function u (·, y).

In addition, we assume that the function u “evolves correctly”. That isto say, for each y1, y2 > 0, we have

u (x, y1 + y2) = u (·, y1) ∗ Py2.

Then, there exists a distribution T in wµD′Lp such that

u (·, y) = T ∗ Py.

Proof. We observe that condition 2) implies that the set v (·, y)y>0 isbounded and relatively compact in Lp (w−µ). Hence, there exists a net yλconverging to zero in (0,∞) and a function f in Lp (w−µ) such that

v (·, yλ) → f

32

as yλ → 0, in the weak-star topology of Lp (w−µ), the dual of Lp′ (w−µ).Thus,

v (x, y) = limyλ→0

v (x, yλ + y)

= limyλ→0

∫

n

Py (x− t) v (t, yλ) dt

=

∫

n

Py (x− t) f (t) dt.

Thus,

u (·, y) =

(1 −

4

4π2

)l

v (·, y) =

(1 −

4

4π2

)l

f ∗ Py.

This completes the proof of Proposition 30.

Remark 31 In the proof of Proposition 30 we really need that the functionv (·, y) evolves correctly. We do not know if this is implied by conditions 1)and 2).

References

[1] J. Alvarez, M. Guzman-Partida, U. Skornik, S ′-convolvability with thePoisson kernel in the Euclidean case and the product domain case, StudiaMathematica 156 (2003), 143-163.

[2] J. Barros-Neto, An introduction to the theory of distributions, MarcelDekker, 1973.

[3] S. Bochner, Harmonic analysis and the theory of probability, Universityof California Press, 1955.

[4] S. Bochner, K. Chandrasekaran, Fourier transforms, Princeton UniversityPress, 1949.

[5] H. Bremermann, Distributions, complex variables, and Fourier transform,Addison-Wesley, 1965.

[6] P. Dierolf, J. Voigt, Convolution and S ′-convolution of distributions, Col-lectanea Math. 29 (1978), 185-196.

33

[7] J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted norm inequalities andrelated topics, North-Holland, 1985.

[8] Y. Hirata, H. Ogata, On the exchange formula for distributions, J. Sci.Hiroshima Univ., Ser. A 22 (1958), 147-152.

[9] J. Horvath, Composition of hypersingular integral operators, ApplicableAnalysis 7 (1978), 171-190.

[10] J. Horvath, Convolution de noyaux hypersinguliers, Seminaire Initiationa l’Analyse, (G. Choquet, M. Rogalski, J. Saint-Raymond) (1979/1980),8, January 1980, 1-17.

[11] N. Ortner, Sur quelques proprietes des espaces D′Lpde Laurent Schwartz,

Boll. Un. Mat. Ital. B (6) 2 (1983), no. 1, 353-375.

[12] N. Ortner, P. Wagner, Applications of weighted D′Lp-spaces to the convo-

lution of distributions, Bull. Polish Acad. Sci. Math. 37 (1990), 579-595.

[13] L. Schwartz, Theorie des distributions, Hermann, 1966.

[14] R. Shiraishi, On the definition of convolutions for distributions, J. Sci.Hiroshima Univ., Ser. A 23 (1959),19-32.

[15] P. Sjogren, Weak L1 characterizations of Poisson integrals, Green po-tentials, and Hp spaces, Trans. Amer. Math. Soc. 233 (1977), 179-196.

[16] E. M. Stein, Singular integrals and differentiability properties of func-tions, Princeton University Press, 1970.

[17] E. M. Stein, Harmonic Analysis. Real-variable methods, orthogonalityand oscillatory integrals, Princeton University Press, 1993.

[18] E. M. Stein, G. Weiss, On the theory of harmonic functions on severalvariables, I. The theory of Hp spaces, Acta Mathematica 103 (1960), 25-62.

34