MATEMATIQKI VESNIK

59 (2007), 197–204

UDK 517.982

originalni nauqni rad research paper

A NOTE ON BOUNDARY VALUES FOR THE POISSON TRANSFORM

Martha Guzmán-Partida

Abstract. We determine boundary values in distributional and pointwise sense for the Poisson transform of a certain class of weighted distributions.

1. Introduction

The authors of [1] have proved that the weighted space w^{n+1}D′_{L^{1}} is the optimal space of tempered distributions admitting S′-convolution with the euclidean version of the Poisson kernel P_{y}. Moreover, they have studied in [2] the boundary behavior of the S′-convolution T ∗ P_{y} when T ∈ w^{n+1}D′_{L^{1}}, obtaining in this way harmonic extensions of weighted integrable distributions to the upper half-space, where the convergence is understood in the sense of the strong topology of the space.

The goal of this note is to study the boundary values of T ∗ P_{y} when T is a function in the weighted space L^{p} (w^{−n−1}), 1 ≤ p < ∞, or a weighted distribution in an intermediate space between L^{p} (w^{−n−1}) and w^{n+1}D′_{L^{1}}. Since L^{p} (w^{−n−1}) ⊂ L^{1} (w^{−n−1}) ⊂ w^{n+1}D′_{L^{1}}, these considerations make sense.

First, we will introduce briefly some definitions and results that will be required along this work, trying to avoid technicalities that could be tedious the reading of the prerequisites.

With B we indicate the space of smooth functions φ : R^{n} → C such that ∂^{α}φ is bounded in R^{n} for each multi-index α. We consider in B the topology of the uniform convergence in R^{n} of each derivative. With Ḃ we indicate the closed subspace of B that consists of those smooth functions φ : R^{n} → C such that ∂^{α}φ → 0 as |x| → ∞, for each multi-index α. The space C_{0}^{∞} is dense in Ḃ but not in B.

The space D′_{L^{1}} of integrable distributions is, by definition, the topological dual of the space Ḃ, endowed with the strong dual topology.

AMS Subject Classification: 46F20, 46F05, 46F12.

Keywords and phrases: S′-convolution; weighted distributions; Poisson kernel.

Since C_{0}^{∞} is dense in Ḃ, the space D′_{L^{1}} is a subspace of D′. According to [5], each distribution T ∈ D′_{L^{1}} can be represented as

T = ∑_{finite} ∂^{α}f_{α}  (1)

where f_{α} ∈ L^{1}.

We can also consider an alternative topology in the space B. We will denote with B_{c} the space B endowed with the following notion of convergence: a sequence {φ_{j}} converges to φ in B_{c} if for each multi-index α, sup_{j} ∥∂^{α}φ_{j}∥_{∞} < ∞ and the sequence {∂^{α}φ_{j}} converges to ∂^{α}φ uniformly on compact sets of R^{n}. It can be proved that C_{0}^{∞}, and so Ḃ, is dense in B_{c} and also that the space D′_{L^{1}} is the topological dual of B_{c}.

Now we consider the notion of S′-convolution whose aim is to preserve the Fourier exchange formula.

Definition 1. [6] Given two tempered distributions T and S, it is said that

they are S′-convolvable if T (S^{∨} ∗ φ) ∈ D′_{L^{1}} for every φ ∈ S. If this is the case, the map

S → C
φ ↦ (T (S^{∨} ∗ φ), 1)_{D′_{L^{1}} ,B_{c}}  (2)

is linear and continuous. Thus, it defines a tempered distribution which will be denoted by T ∗ S.

This operation coincides with the classical convolution defined by L. Schwartz, in all the cases in which both make sense.

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

P_{y}(x) = \frac{c_{n}}{y^{n}} \frac{1}{(\frac{|x|^{2}}{y^{2}} + 1)^{\frac{n+1}{2}}} ,  (3)

where c_{n} = Γ (\frac{n+1}{2})/π^{\frac{n+1}{2}} , y > 0.

The authors of [1] identified distributions in appropriate weighted spaces, as those S′-convolvable with the euclidean version of the Poisson kernel. These distributions are defined as follows.

Definition 2. Let w(x) = (1 + |x|^{2})^{\frac{1}{2}} , for x ∈ R^{n}. Then, given μ ∈ R we consider

w^{μ}D′_{L^{1}} = {T ∈ S′ : w^{−μ}T ∈ D′_{L^{1}}}

with the topology induced by the map

w^{μ}D′_{L^{1}} → D′_{L^{1}}
T ↦ w^{−μ}T.

A note on boundary values for the Poisson transform 199

The space w^{μ}D′_{L^{1}} is the topological dual of the spaces w^{−μ}Ḃ and w^{−μ}B_{c}. Theorem 3. [1] Given T ∈ S′, the following statements are equivalent:

1. T ∈ w^{n+1}D′_{L^{1}}.

2. T is S′-convolvable with P_{y}, for each y > 0. Given T ∈ w^{n+1}D′_{L^{1}}, it was proved in [1] that the S′-convolution T ∗ P_{y} is a

function defined on R^{n} as

(T ∗ P_{y}) (x) = (w^{−n−1}(t)T_{t}, w^{n+1}(t)P_{y}(x − t))_{D′_{L^{1}} ,B_{c}} .  (4)

Also, in [2] it is shown that

w^{n+1}D′_{L^{1}} = {T ∈ S′ : T = ∑_{finite} ∂^{α}f_{α}, f_{α} ∈ L^{1} (w^{−n−1})}.  (5)

2. Main results According to [2], when T ∈ w^{n+1}D′_{L^{1}} the S′-convolution T ∗ P_{y} is a function

in L^{1} (w^{−n−1}). Thus, we can consider the family of operators Λ_{y} : L^{p} (w^{−n−1}) → L^{1} (w^{−n−1}), 1 ≤ p < ∞, such that Λ_{y} (f) = P_{y} ∗ f.

Lemma 4. For each y > 0, Λ_{y} is a bounded operator from L^{p} (w^{−n−1}) into itself, 1 ≤ p < ∞, and moreover ∥Λ_{y}∥ ≤ C_{n,p} (1 + y)^{1/p}.

Proof. Applying Jensen inequality and the semigroup property of the family (P_{η})_{η>0} we have

∥P_{y} ∗ f∥_{L^{p}(w^{−n−1})}^{p} ≤ _{∫_{R_{n}}}_{∫_{R_{n}}} |f(t)|^{p} P_{y} (x − t) dt \frac{dx}{(1 + |x|^{2})^{\frac{n+1}{2}}}

= C_{n}_{∫_{R_{n}}} |f(t)|^{p} (P_{y} ∗ P_{1}) (t) dt = C_{n}_{∫_{R_{n}}} |f(t)|^{p} (P_{1+y}) (t) dt

≤ C_{n} (1 + y)_{∫_{R_{n}}} |f(t)|^{p} \frac{dt}{(1 + |t|^{2})^{\frac{n+1}{2}}}

= C_{n} (1 + y) ∥f∥_{L^{p}(w^{−n−1})}^{p}.

This concludes the proof.

It has been proved in [2, Th. 3.6] that for f ∈ L^{1} (w^{−n−1}) the S′-convolution P_{y} ∗ f → f in L^{1} (w^{−n−1}) as y → 0^{+}. This is also true for 1 ≤ p < ∞ as we prove below.

Lemma 5. For each f ∈ L^{p} (w^{−n−1}), 1 ≤ p < ∞, P_{y} ∗ f → f in L^{p} (w^{−n−1}) as y → 0^{+}.

Proof. When f ∈ C_{c} it is clear that P_{y} ∗ f → f in L^{p}, hence in L^{p} (w^{−n−1}) as y → 0^{+}. Now, if f ∈ L^{p} (w^{−n−1}) since C_{c} is dense in L^{p} (w^{−n−1}), given ε > 0 there exists g ∈ C_{c} such that ∥f − g∥_{L^{p}(w^{−n−1})} < ε and also ∥P_{y} ∗ g − g∥_{L^{p}(w^{−n−1})} < ε. Thus

∥P_{y} ∗ f − f∥_{L^{p}(w^{−n−1})} ≤ ∥P_{y} ∗ (f − g)∥_{L^{p}(w^{−n−1})} + 2ε.  (6)

Jensen inequality and the semigroup property of the family (P_{η})_{η>0} imply

∥P_{y} ∗ (f − g)∥_{L^{p}(w^{−n−1})}^{p} ≤ C_{n}_{∫_{R_{n}}} |f − g|^{p} (x) P_{1+y} (x) dx.  (7)

For 0 < y ≤ 2 we have that

|f − g|^{p} (x) P_{1+y} (x) ≤ C_{n} |f − g|^{p} (x) P_{1} (x)
|f − g|^{p} (x) P_{1+y} (x) → |f − g|^{p} (x) P_{1} (x) as y → 0^{+}

and by Lebesgue Dominated Convergence Theorem we obtain that

lim_{y→0^{+}} _{∫_{R_{n}}} |f − g|^{p} (x) P_{1+y} (x) dx = C_{n} ∥f − g∥_{L^{p}(w^{−n−1})}^{p}.

Thus, choosing y > 0 small enough we can conclude from (6) and (7) that

∥P_{y} ∗ f − f∥_{L^{p}(w^{−n−1})} < 3ε + C_{n,p}ε.

This concludes the proof.

Remark 6. Lemmas 4 and 5 imply that the family of operators (Λ_{y})_{y>0} is a strongly continuous semigroup of convolution operators in the Banach space L^{p} (w^{−n−1}), 1 ≤ p < ∞, such that ∥Λ_{y}∥ ≤ C_{n,p} (1 + y)^{1/p}.

The previous results lead us to consider the following subspace of w^{n+1}D′_{L^{1}}:

A_{p} = {T ∈ S′ : T = ∑_{finite} ∂^{α}f_{α}, f_{α} ∈ L^{p} (w^{−n−1})}, 1 ≤ p < ∞.

Clearly, A_{p} is closed under differentiation and ∂^{α} (T ∗ P_{y}) = (∂^{α}T) ∗ P_{y} for each T ∈ A_{p} and each multi-index α.

Now, let T ∈ A_{p}. Using the fact that the S′-convolution T ∗P_{y} can be computed as

(T ∗ P_{y}, φ)_{S′,S} = ((P_{y} ∗ φ) T, 1)_{D′_{L^{1}} ,B_{c}}
= (T, P_{y} ∗ φ)_{w^{n+1}D′_{L^{1}} ,w^{−n−1}B_{c}}

(see [1]), we can readily see for T = ∑_{finite} ∂^{α}f_{α}, f_{α} ∈ L^{p} (w^{−n−1}) that

(T ∗ P_{y}) (x) = ∑_{finite} (−1)^{|α|} _{∫_{R_{n}}}f_{α} (t) ∂_{t}^{α}P_{y} (x − t) dt.  (8)

A note on boundary values for the Poisson transform 201

Thus, we can prove the following result.

Lemma 7. If T ∈ A_{p}, 1 ≤ p < ∞, then T ∗ P_{y} is a smooth function and ∂^{α} (T ∗ P_{y}) ∈ L^{p} (w^{−n−1}) for every multi-index α and each y > 0.

Proof. Since A_{p} is closed under differentiation, it suffices to prove that T ∗P_{y} ∈ L^{p} (w^{−n−1}) for every T ∈ A_{p}.

Without loss of generality let us assume that T = ∂^{α}f, f ∈ L^{p} (w^{−n−1}). Then, according to (8)

(T ∗ P_{y}) (x) = (−1)^{|α|} _{∫_{R_{n}}}f (t) ∂_{t}^{α}P_{y} (x − t) dt.

Since

|∂_{t}^{α}P_{y} (x − t)| ≤ \frac{C_{n,α}}{y^{n+|α|}} (1 + \frac{|x − t|^{2}}{y^{2}} )^{− \frac{(n+1+|α|)}{2}}

≤ \frac{C_{n,α}}{y^{n+|α|}} (1 + \frac{|x − t|^{2}}{y^{2}} )^{− \frac{(n+1)}{2}}

we have that

|(T ∗ P_{y}) (x)| ≤ \frac{C_{n,α}}{y^{n+|α|}}_{∫_{R_{n}}} |f(t)| (1 + \frac{|x − t|^{2}}{y^{2}} )^{− \frac{(n+1)}{2}} dt

= \frac{C_{n,α}}{y^{|α|}} (P_{y} ∗ |f|) (x) .

Therefore

∥T ∗ P_{y}∥_{L^{p}(w^{−n−1})} ≤ \frac{C_{n,α}}{y^{|α|}} ∥P_{y} ∗ |f|∥_{L^{p}(w^{−n−1})}
≤ \frac{C_{n,α,p}}{y^{|α|}} (1 + y)^{1/p} ∥f∥_{L^{p}(w^{−n−1})} .  (9)

This concludes the proof.

Remark 8. For T ∈ A_{p}, 1 ≤ p < ∞, T = ∑_{finite} ∂^{α}f_{α}, f_{α} ∈ L^{p} (w^{−n−1}) we

have

∥T ∗ P_{y}∥_{L^{p}(w^{−n−1})} ≤ ∑_{finite} \frac{C_{n,α,p}}{y^{|α|}} (1 + y)^{1/p} ∥f_{α}∥_{L^{p}(w^{−n−1})} .

We also have a version for the subspace A_{p} of [2, Th. 3.6].

Theorem 9. Given T ∈ A_{p}, 1 ≤ p < ∞, the S′-convolution T ∗ P_{y} → T as y → 0^{+} in A_{p}.

Proof. Since the topology in A_{p} is the inherited from w^{n+1}D′_{L^{1}}, L^{p} (w^{−n−1}) ↪ L^{1} (w^{−n−1}) ↪ w^{n+1}D′_{L^{1}}, and A_{p} is closed under differentiation, it suffices to show that P_{y} ∗ f → f in L^{p} (w^{−n−1}) as y → 0^{+}. But this last assertion was proved in Lemma 5.

Next, we will approach the problem of pointwise convergence of the S′- convolution P_{y} ∗ f when y → 0^{+} and f ∈ L^{1} (w^{−n−1}).

Theorem 10. Given f ∈ L^{1} (w^{−n−1}), (P_{y} ∗ f) (x) → f(x) as y → 0^{+} for every Lebesgue point x of f and therefore, almost everywhere on R^{n}.

Proof. For the first part of the proof, we will follow [3, Th. 8.15]. Let x be a Lebesgue point of f. Thus for every β > 0 there exists λ > 0 such that

∫|t|<r |f (x − t) − f (x)| dt ≤ Cβr^{n}, 0 < r ≤ λ.  (10)

Now, we consider the integrals

I_{1} = _{∫|t|<λ} |f (x − t) − f (x)| P_{y} (t) dt
I_{2} = _{∫|t|≥λ} |f (x − t) − f (x)| P_{y} (t) dt.

We will prove that I_{1} is bounded by Cβ, where C is a constant independent of y, and that I_{2} → 0 as y → 0^{+}. This will imply the desired conclusion.

To get an estimate for I_{1}, let y > 0 fixed. Select the unique integer N satisfying

\{ \begin{matrix}2^{N} ≤ \frac{λ}{y} < 2^{N+1} & if \frac{λ}{y} ≥ 1 \\ 0 & if \frac{λ}{y} < 1. \end{matrix}

We decompose the ball |t| < λ as the union of the annuli \frac{λ}{2^{j}} ≤ |t| < \frac{λ}{2^{j−1}} , 1 ≤ j ≤ N, and the ball |t| < \frac{λ}{2^{N}}. By means of (10) and using the estimates

|P_{y}(t)| ≤ Cy^{−n} [\frac{2^{−j}λ}{y}]^{−n−1} on the jth annulus

and

|P_{y}(t)| ≤ Cy^{−n} on the ball |t| < \frac{λ}{2^{N}},

we obtain exactly in the same way as in [3] that I_{1} ≤ 2^{n}Cβ.

To estimate I_{2} we proceed as follows. Since C_{c} is dense in L^{1} (w^{−n−1}), given ε > 0 we write f = h + g where g ∈ C_{c} and ∥h∥_{L^{1}(w^{−n−1})} < ε. Thus

I_{2} ≤ _{∫|t|≥λ} |h (x − t) − h (x)| P_{y} (t) dt +_{∫|t|≥λ} |g (x − t) − g (x)| P_{y} (t) dt

≤ _{∫|t|≥λ} |h (x − t)| P_{y} (t) dt + |h(x)|_{∫|t|≥λ}P_{y} (t) dt + 2 ∥g∥_{∞} _{∫|t|≥λ}P_{y} (t) dt

and the last two terms in the sum go to 0 as y → 0^{+}.

A note on boundary values for the Poisson transform 203

It remains to analyze the term

∫|t|≥λ |h (x − t)| P_{y} (t) dt.

Using the change of variable u = y^{−1}t we obtain

∫|t|≥λ |h (x − t)| P_{y} (t) dt = _{∫|u|≥λ/y} |h (x − yu)| P (u) du

= C_{n}_{∫|u|≥λ/y} \frac{|h (x − yu)|}{(1 + |x − yu|^{2})^{\frac{n+1}{2}}} \begin{matrix}(1 + |x − yu|^{2}) & \frac{n+1}{2} \\ (1 + |u|^{2}) & \frac{n+1}{2} \end{matrix} du

≤ C_{n} sup_{|u|≥λ/y} (\frac{1 + |x − yu|^{2}}{1 + |u|^{2}} )^{\frac{n+1}{2}} _{∫_{R_{n}}} \frac{|h (x − yu)|}{(1 + |x − yu|^{2})^{\frac{n+1}{2}}} du

= C_{n} sup_{|u|≥λ/y} (\frac{1 + |x − yu|^{2}}{1 + |u|^{2}} )^{\frac{n+1}{2}} y^{−n}_{∫_{R_{n}}} \frac{|h (s)|}{(1 + |s|^{2})^{\frac{n+1}{2}}} ds

≤ C_{n}εy^{−n} sup_{|u|≥λ/y} (\frac{1 + |x − yu|^{2}}{1 + |u|^{2}} )^{\frac{n+1}{2}} .

We need to check that

sup_{|u|≥λ/y} (\frac{1 + |x − yu|^{2}}{1 + |u|^{2}} )^{\frac{n+1}{2}} < ∞

and

C_{n}εy^{−n} sup_{|u|≥λ/y} (\frac{1 + |x − yu|^{2}}{1 + |u|^{2}} )^{\frac{n+1}{2}} → 0

as y → 0^{+}.

We observe that

C_{n}εy^{−n} sup_{|u|≥λ/y} (\frac{1 + |x − yu|^{2}}{1 + |u|^{2}} )^{\frac{n+1}{2}} = C_{n}εy sup_{|u|≥λ/y} (\frac{1 + |x − yu|^{2}}{y^{2} + |yu|^{2}} )^{\frac{n+1}{2}}

≤ C_{n}εy sup_{|u|≥λ/y} (\frac{1 + (|x| + |yu|)^{2}}{y^{2} + |yu|^{2}} )^{\frac{n+1}{2}}

≤ C_{n}εy (1 + |x|^{2})^{\frac{n+1}{2}} sup_{|u|≥λ/y} (\frac{(1 + |yu|^{2})}{y^{2} + |yu|^{2}} )^{\frac{n+1}{2}}

= C_{n}εy (1 + |x|^{2})^{\frac{n+1}{2}} sup_{|u|≥λ/y} (\frac{(\frac{1}{y^{2}} + |u|^{2})}{1 + |u|^{2}} )^{\frac{n+1}{2}}

≤ C_{n}εy (1 + |x|^{2})^{\frac{n+1}{2}} sup_{|u|≥λ/y} ( \frac{\frac{|u|^{2}}{λ^{2}} + |u|^{2}}{1 + |u|^{2}} )^{\frac{n+1}{2}}

≤ C_{n}εy (\frac{1}{λ^{2}} + 1)^{\frac{n+1}{2}} (1 + |x|^{2})^{\frac{n+1}{2}} sup_{|u|≥λ/y} (\frac{|u|^{2}}{1 + |u|^{2}} )^{\frac{n+1}{2}}

≤ C_{n}εy (\frac{1}{λ^{2}} + 1)^{\frac{n+1}{2}} (1 + |x|^{2})^{\frac{n+1}{2}} → 0 as y → 0^{+}.

This concludes the proof.

Remark 11. Using the fact that the function ϕ (x) = \frac{1}{(1+|x|^{2})^{s/2}} = w^{−s} with

0 < s < n is an A_{1} weight (see [2, Lemma 5.2], we can assure the boundedness of the Hardy-Littlewood maximal operator M from L^{p} (ϕ) into itself , 1 < p < ∞, as well as from L^{1} (ϕ) into weak L^{1} (ϕ). This fact, plus the standard estimate |(P_{y} ∗ f) (x)| ≤ CMf (x), and the arguments given in [4, Th. 4.12, p. 177], allow us to prove that (P_{y} ∗ f) (x) → f (x) as y → 0^{+} for almost every x ∈ R^{n} and f ∈ L^{p} (w^{−s}), 1 ≤ p < ∞. Since L^{p} (w^{−s}) ⊂ L^{p} (w^{−n−1}) ⊂ L^{1} (w^{−n−1}) and the inclusions are strict, Theorem 10 enlarge the class of functions for which the pointwise convergence occurs.

Remark 12. Given T ∈ w^{n+1}D′_{L^{1}} we know that T ∗ P_{y} ∈ L^{1} (w^{−n−1}) for each y > 0, thus, it seems natural to ask if T ∗ P_{y} has pointwise boundary values. However, this is not a relevant question. For example, if we consider the Dirac measure μ concentrated at 0, we have that P_{y} = μ ∗ P_{y} and P_{y} → 0 as y → 0^{+} a.e., thus the boundary value 0 does not determine P_{y}. In this case T = μ ∈ D′_{L^{1}}.

REFERENCES

[1] J. Alvarez, M. Guzmán-Partida, U. Skórnik, S′-convolvability with the Poisson kernel in the euclidean case and the product domain case, Studia Mathematica 156, (2003), 143–163.

[2] J. Alvarez, M. Guzmán-Partida, S. Pérez-Esteva, Harmonic extensions of distributions, Math. Nachr. 280, 13–14 (2007), 1443–1466.

[3] G. B. Folland, Real Analysis. Modern Techniques and Their Applications, 2nd. Ed. Wiley-Interscience, 1999.

[4] J. García-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North Holland, 1985.

[5] L. Schwartz, Théorie des distributions, Hermann, 1966.

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

(received 26.04.2007)

Departamento de Matemáticas, Universidad de Sonora, Rosales y Blvd. Encinas, Hermosillo, Sonora 83000, México

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