Sets of determination for parabolic functions on a half-space

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Sets of determination for parabolic functions on a half-space

Jarmila Ranoˇsov´a

Abstract. We characterize all subsetsM ofRⁿ×R⁺such that sup

X∈Rⁿ×R⁺

u(X) = sup

X∈M

u(X)

for every bounded parabolic function u on Rⁿ×R⁺. The closely related problem of representing functions as sums of Weierstrass kernels corresponding to points ofMis also considered. The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References. As a by-product the question of representability of probability continuous distributions as sums of multiples of normal distributions is investigated.

Keywords: heat equation, parabolic function, Weierstrass kernel, set of determination, decomposition ofL1(Rⁿ), normal distribution

Classification: 35K05, 35K15, 31B10, 60Exx

1. Preliminaries

In this paper the following notation is used: Small letters, such as x, y, will denote points in Rⁿ; capital letters, such as X, points in Rⁿ⁺¹, and t denotes

“time”. (We will writeX = (x, t) for x∈Rⁿ and t ∈R.) The setRⁿ× {0} is identified withRⁿ, and, when there is no danger of confusion, the point

(y,0)∈Rⁿ× {0} is denoted byy. The Lebesgue measure inRⁿ will be denoted byλ.

Further notation is either standard or introduced when it is used.

Definition. A real function u on an open set G ⊂ Rⁿ⁺¹ having continuous partial derivatives ^∂u_∂t and ^∂_∂x²^u2

i

fori= 1, ..., n, and satisfying the heat equation

∂u

∂t = Xn i=1

∂²u

∂x²_i

onGis called parabolic on G.

We will be interested in parabolic functions onRⁿ×R⁺.

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The Weierstrass kernel forRⁿ×R⁺with the pole in y inRⁿis given by p(X, y) = (4πt)⁻^n/2exp(−kx−yk²

4t ), whereX = (x, t)∈Rⁿ×R⁺.

A parabolic function which is the limit of an increasing sequence of bounded positive parabolic functions will be called quasi-bounded. The class of all functions uwhich can be expressed as a difference of two positive quasi-bounded parabolic functions will be denoted byP1.

Moreover, a parabolic functionuwill be called simple if there exists a λ-measurable subset A of Rⁿ× {0} such that u(X) = R

A

p(X, y)dy for any X ∈Rⁿ×R⁺.

The class of all simple parabolic functions will be denoted by Ps. Of course, Ps⊂ P1.

Theorem A. A function u on Rⁿ×R⁺ is a difference of two positive quasi- bounded parabolic functions if and only if there is a λ-measurable function f_u onRⁿ for which Z

Rⁿ

exp(−kyk²

4t )|fu(y)|dy <∞ for allt >0and

u(X) = Z

Rⁿ

p(X, y)fu(y)dy, X∈Rⁿ×R⁺;

uis positive, if and only iffuis positiveλ-almost everywhere.

It is clear that the functionf_uis uniquely determinedλ-almost everywhere. In what follows, for anyu∈ P1,fu will denote this function.

Proof: See [5, p. 291].

Definition. A pointY = (y,0) is called a parabolic limit (resp. a1-parabolic limit)of a sequence{X_k},X_k= (x_k, t_k), of points inRⁿ×R⁺, if{X_k}converges toY andlim inf_k_→∞t_kkx_k−yk⁻²>0 (that is, allX_kbelong to some paraboloid of revolution with vertexY and opening upward)

(resp.lim inf_k_→∞t_kkx_k−yk⁻²≥1).

Let M ⊂Rⁿ×R⁺. A point Y ∈ Rⁿ× {0} is called a parabolic limit point (resp. a1-parabolic limit point)of the setM if there exists a sequence{X_k}such that everyX_k∈M andY is a parabolic limit(resp. a1-parabolic limit)of{X_k}. A functionf onRⁿ×R⁺ is said to have a parabolic limit qat Y if {f(X_k)} converges toqwheneverY is a parabolic limit of{X_k}.

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Theorem B (The Fatou limit theorem). Letu be a parabolic function in P1. Then for λ-almost all y ∈ Rⁿ the function u has the parabolic limit fu(y) at Y = (y,0).

Proof: See [5, p. 292].

Lemma 1. A parabolic functionuonRⁿ×R⁺is bounded if and only if there is a functionfu∈L_∞(Rⁿ)such that

u(X) = Z

Rⁿ

p(X, y)f_u(y)dy, X∈Rⁿ×R⁺.

If it is the case, then sup

X∈Rⁿ×R⁺

u(X) = ess sup

y∈Rⁿ fu(y) and sup

X∈Rⁿ×R⁺|u(X)| = kfukL∞(Rⁿ).

Proof: Since uis a bounded parabolic function, it belongs to P1. From Theo- rem A it follows that there is aλ-measurable functionfu such that

u(X) = Z

Rⁿ

p(X, y)f_u(y)dy, X∈Rⁿ×R⁺.

As for the constant function u = c we have fu = c (λ-almost everywhere), by Theorem A it follows thatc−u≥0 if and only ifc−fu≥0λ-almost everywhere.

Consequently, sup

X∈Rⁿ×R⁺

u(X) = ess sup

y∈Rⁿ

f_u(y). The rest is trivial.

The class of all bounded parabolic functions onRⁿ×R⁺will be denoted byPb. Of course,P^s⊂ Pb ⊂ P1.

Lemma 2. LetM ⊂Rⁿ×R⁺. Then the set of all parabolic limit points ofM isλ-measurable.

Proof: The pointY = (y,0) is a parabolic limit point ofM if and only if lim sup

t0→0+ sup

{x;(x,t0)∈M}

t₀

kx−yk² >0.

It follows from the definition of parabolic limits points. Remark that sup∅=−∞, as usual.

LetM0 be a countable dense subset of M. The point Y is a parabolic limit point ofM if and only ifY is a parabolic limit point ofM₀. That is why we can assume thatM is a countable set.

Fort∈R⁺,x∈Rⁿ andy∈Rⁿ, define g(t, x, y) = t

kx−yk², ifx6=y,

=∞, ifx=y.

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It is easy to see that the functiongis a continuous function fromR⁺×Rⁿ×Rⁿ into (0,∞]. Fixxand t. The function g(t, x, .) is measurable (even continuous) fromRⁿinto (0,∞].

For fixedt, let us put

gt(y) = sup

{x;(x,t)∈M}

t kx−yk².

Let us recall that M is countable. Then gt, being a supremum of a countable system of measurable functions, is measurable.

BecauseM is countable, only for countably manyt the functiong_tis strictly positive. Let us denote this set oftbyT.

So,Y is a parabolic limit point ofM if and only if lim sup

t→0,t∈T gt(y)>0.

But theng, being a limes superior of a countable system of measurable functions, is measurable.

Thus the set of all parabolic limit points, which is{g >0}, is measurable.

2. Sets of determination Theorem 1. LetM ⊂Rⁿ×R⁺.

If

sup

X∈Rⁿ×R⁺

u(X) = sup

X∈M

u(X)

for all bounded positive parabolic functions, thenλ-almost every point y∈Rⁿ× {0}is a parabolic limit point ofM.

Proof: Suppose it is not true.

The set of all parabolic limit points of M will be denoted by M_p. We know from the preceding lemma that the setMp is measurable. Then its complement M_p^′ is also measurable and by our assumptionλ(M_p^′)>0.

Fork∈Nandy∈Rⁿ× {0}, Γ^k_y will denote the set {(x, t)∈Rⁿ×R⁺; 1/k > t >kx−yk²}.

Then, for everyy∈M_p^′, there isky ∈Nsuch that Γ^k_y^y∩M is empty. Denote by D_k the set ofy∈M_p^′ for which Γ^k_y∩M is empty.

We will prove now that, for anyk, the setD_kis a measurable subset ofRⁿ×{0}. By definition

D_k=M_p^′∩ {y∈Rⁿ× {0}; M∩Γ^k_y =∅}. We know thatM_p^′ is measurable. Thus it remains to prove that {y∈Rⁿ× {0}; M∩Γ^k_y =∅}is measurable.

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ButM∩Γ^k_y=∅ if and only if for anyt∈(0,1/k):

sup

{x; (x,t)∈M}

t

kx−yk² ≤1.

LetM0 be a countable dense subset of M. The point Y is a parabolic limit point ofM if and only ifY is a parabolic limit point ofM0. That is why we can assume thatM is a countable set. Let us fixt. Then

y7−→ sup

{x; (x,t)∈M}

t kx−yk²

is a measurable function and

{y∈Rⁿ; sup

{x; (x,t)∈M}

t

kx−yk² ≤1} is a measurable set.

This set can be different fromRⁿ only fortsuch that there isX = (x, t)∈M. Let us denote this set byT. So, the intersection

\

0<t<1/k

{y∈Rⁿ; sup

{x; (x,t)∈M}

t

kx−yk² ≤1} is equal to

\

0<t<1/k t∈T

{y∈Rⁿ; sup

{x; (x,t)∈M}

t

kx−yk² ≤1},

and, as an intersection of countable system of measurable sets, it is measurable.

It means thatD_kis measurable.

As S_∞

k=1D_k = M_p^′, the Lebesgue measure of at least one of the sets D_k is strictly positive. Let it be the set D_a where a ∈ N. Then there is a bounded subset ofD_awhich has strictly positive Lebesgue measure. Denote this set byD.

For a pointX = (x, t) ofRⁿ×R⁺ we define

A_X ={(y,0)∈Rⁿ× {0}; kx−yk²< t}. (A_X is the ball inRⁿ× {0}with the center (x,0) and radiust^1/2.) DenoteD^′= (Rⁿ× {0})\D.

LetX= (x, t)∈M∩(Rⁿ×(0,1/a)) and lety∈A_X. Then 1/a > t >kx−yk², so thatX ∈Γ^a_y. Consequently,y /∈Da, and soy /∈D. We conclude thatA_X ⊂D^′, wheneverX∈M ∩(Rⁿ×(0,1/a)).

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For any measurable setA⊂Rⁿ× {0}we define u_A(X) =

Z

A

p(X, y)dy, X∈Rⁿ×R⁺.

It means u_A ∈ Ps, thus u_A ∈ Pb and −u_A ∈ Pb. By Lemma 1 we get u_A is positive, and ifλ(A)>0, then sup

Rⁿ×R⁺

u_A(X) = 1.

Denote the Lebesgue measure of the unit ball in Rⁿ by α_n. Then for any X ∈Rⁿ×R⁺:

u_A_X(X) = Z

AX

p(X, y)dy= (4πt)⁻^n/2 Z

AX

exp(−kx−yk² 4t )dy

≥(4πt)⁻^n/2 Z

AX

exp(−t

4t)dy= (4πt)⁻^n/2λ(A_X) exp(−1/4)

= (4π)⁻^n/2αnexp(−1/4).

We see that

c= inf

X∈Rⁿ×R⁺u_A_X(X)>0.

The setDhas a positive measure, hence the functionu_D is positive, parabolic, bounded and

sup

X∈Rⁿ×R⁺

u_D(X) = 1.

Sinceu_D+u_D^′= 1, we get for everyN ⊂Rⁿ×R⁺ the equality sup

X∈N

u_D(X) = 1− inf

X∈Nu_D^′(X).

ButA_X is a subset ofD^′ for everyX ∈M ∩(Rⁿ×(0,1/a)).

It follows that

u_D^′(X)≥u_A_X(X)≥c for everyX ∈M∩(Rⁿ×(0,1/a)).

Then

sup

X∈M∩(Rⁿ×(0,1/a))

u_D(X)≤1−c.

Now sup

X∈M∩(Rⁿ×[1/a,∞))

u_D(X) will be estimated. AsD is bounded, there is

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d∈R⁺such thatD⊂[−d, d]ⁿ and so we have for any (x, t)∈Rⁿ×[1/a,∞) u_D(x, t) = (4πt)⁻^n/2

Z

D

≤(4πt)⁻^n/2 Z

[−d,d]ⁿ

exp(−kx−yk²

4t )dy≤(4πt)⁻^n/2 Z

[−d,d]ⁿ

≤(4πt)⁻^n/2 Z

[−d,d]ⁿ

exp(−kyk²

4t )dy=π⁻^n/2

Z

[−d/(2√ t),d/(2√

t)]ⁿ

exp(−kyk²)dy

≤π⁻^n/2

Z

[−d√a/2,d√a/2]ⁿ

exp(−kyk²)dy <1.

Thus

sup

X∈M∩(Rⁿ×[1/a,∞))

u_D(X)<1.

Consequently, sup

X∈M

u_D(X)≤max (1−c, sup

X∈M∩(Rⁿ×[1/a,∞))

u_D(X))<1,

contradicting our assumption.

In fact, we proved a bit more than the assertion of Theorem 1. Namely we proved

Theorem 1’. LetM ⊂Rⁿ×R⁺. If

sup

X∈Rⁿ×R⁺

u(X) = sup

X∈M

u(X)

for all simple parabolic functions, then λ-almost every point y ∈ Rⁿ× {0} is a1-parabolic limit point ofM.

Theorem 2. LetM be a subset ofRⁿ×R⁺and letλ-almost every point (y,0)∈Rⁿ× {0} be a parabolic limit point ofM. Then

sup

X∈Rⁿ×R⁺|u(X)|= sup

X∈M|u(X)| for every bounded parabolic functionuonRⁿ×R⁺.

Proof: Letube a bounded parabolic function onRⁿ×R⁺. By Lemma 1, there existsf ∈L_∞(Rⁿ) such that

u(X) = Z

Rⁿ

p(X, y)f(y)dy, X∈Rⁿ×R⁺

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and

sup

X∈Rⁿ×R⁺|u(X)|=kfkL∞. It is clear that

sup

X∈M|u(X)| ≤ sup

X∈Rⁿ×R⁺|u(X)|=kfkL∞.

By the hypothesis, λ-almost every point of Rⁿ× {0} is a parabolic limit point ofM. From the Fatou limit theorem, forλ-almost everyy∈Rⁿ,

limu(X_k) =f(y), wheneveryis a parabolic limit of {X_k}. Thus

sup

X∈M|u(X)| ≥ kfkL∞, so that

sup

X∈M|u(X)|.

3. A decomposition theorem forL₁(Rⁿ)

Let us introduce the following notation: The closure ofM ⊂Rⁿ×R⁺ in the setRⁿ×R⁺ (which is the setM∩(Rⁿ×R⁺)) will be denoted byMf.

The support of a measureνonRⁿ×R⁺with respect toRⁿ×R⁺(it means the complement inRⁿ×R⁺of the largest open setG⊂Rⁿ×R⁺such thatν(G) = 0) will be denoted bys(ν).

Theorem 3. Let M be a subset of Rⁿ×R⁺ and ν be a σ-finite measure on Rⁿ×R⁺ such thats(ν) =Mf. Let

sup

X∈M|u(X)| for every bounded parabolic functionuonRⁿ×R⁺.

Then, for anyf inL1(Rⁿ), there existsΦinL1(ν)such that

(1) f =

Z

Rⁿ×R⁺

Φ(X)p(X, .)dν(X)

λ-almost everywhere and

kfkL1(Rⁿ)= inf {kΦkL1(ν); (1)holds for someΦ∈L₁(ν)}.

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Further, there exists a sequence{X_k},X_k∈M and{λ_k} ∈l₁ such that

(2) f =

X∞ k=1

λ_kp(X_k, .)

λ-almost everywhere and kfkL1(Rⁿ)= inf {X

|λ_k|; (2)holds for some{X_k}inM}.

The second part is an easy consequence of the first one. Just take a dense countable subset ofM, denote it{X_k}, and takeν the counting measure on it.

We will need the following version of the closed range theorem (see [7, p. 97]).

LetX andY be Banach spaces,T a bounded linear mapping ofX intoY. If there exists a constant c >0 such that kT^∗y^∗k ≥cky^∗k for all y^∗∈ Y^∗ thenTX =Y. In our situation,X =L₁(ν),Y=L₁(Rⁿ) and for Φ∈L₁(ν) we define

TνΦ = Z

Rⁿ×R⁺

p(X, .)Φ(X)dν(X).

Lemma 3. The mappingT_ν is a bounded linear mapping ofL₁(ν)intoL₁(Rⁿ), kTνk= 1;T_ν^∗ is the parabolic extension mappingL_∞(Rⁿ)intoL_∞(ν).

Proof: Using the Fubini theorem we arrive at kTνΦkL1(Rⁿ)=

Z

Rⁿ

|TνΦ|dλ= Z

Rⁿ

| Z

Rⁿ×R⁺

p(X, y)Φ(X)dν(X)|dλ(y)

≤ Z

Rⁿ

( Z

Rⁿ×R⁺

p(X, y)|Φ(X)|dν(X))dλ(y)

= Z

Rⁿ×R⁺

( Z

Rⁿ

p(X, y)|Φ(X)|dλ(y))dν(X)

= Z

Rⁿ×R⁺

|Φ(X)|( Z

Rⁿ

p(X, y)dλ(y))dν(X)

= Z

Rⁿ×R⁺

|Φ(X)|dν(X) =kΦkL1(ν). So, the first part of Lemma is proved.

Letg∈L_∞(Rⁿ) and Φ∈L1(ν). Using again the Fubini theorem we have [Φ, T_ν^∗g] = [T_νΦ, g] =

Z

Rⁿ

gT_νΦdλ= Z

Rⁿ

g(y)(

Z

Rⁿ×R⁺

Φ(X)p(X, y)dν(X))dλ(y) = Z

Rⁿ×R⁺

Φ(X)(

Z

Rⁿ

g(y)p(X, y)dλ(y))dν(X) = [Φ, Z

Rⁿ

p(X, y)g(y)dλ(y)].

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Proof of Theorem 3. We shall prove the existence of a constantc >0 such that kT_ν^∗gkL∞(ν)≥ckgkL∞(Rⁿ) for allg∈L_∞(Rⁿ) and the first part of the theorem will be proved.

The functionT_ν^∗gis bounded and parabolic onRⁿ×R⁺. Then, by hypothesis, Lemma 3 and Lemma 1,

sup

X∈M|(T_ν^∗g)(X)|= sup

X∈Rⁿ×R⁺|(T_ν^∗g)(X)|=kgkL∞(Rⁿ). SinceT_ν^∗gis a continuous function onRⁿ×R⁺ ands(ν) =Mf,

kT_ν^∗gkL∞(ν)= sup

X∈M|(T_ν^∗g)(X)|. Consequently,

kT_ν^∗gkL∞(ν)=kgkL∞(Rⁿ).

So, we can takec= 1. The first part of Theorem 3 is proved.

To prove the other part, define the space Z =L₁(ν)/kerT_ν. Forz∈ Zand Φ∈zput Sz=TνΦ.

ThenS is an invertible bounded linear mapping ofZ into L₁(Rⁿ) and so its adjointS^∗ is an invertible bounded linear mapping ofL_∞(Rⁿ) intoZ^∗

(see [7, p. 94]).

Letz∈ Z, Φ∈zandg∈L_∞(Rⁿ). Then we have (S^∗g)(z) = [Sz, g] = [TνΦ, g] = [Φ, T_ν^∗g].

Ifε >0, there exists Φ₀ ∈L₁(ν) withkΦ₀kL1(ν)= 1 and

|[Φ0, T_ν^∗g]|>kT_ν^∗gkL∞(ν)−ε.

Letz₀ denote the coset of Φ₀ inZ. Then

|(S^∗g)(z₀)|>kT_ν^∗gkL∞(ν)−ε, and

kz₀k_Z ≤ kΦ₀kL1(ν)= 1.

Therefore, the norm of the functionalS^∗gsatisfies

kS^∗gkZ^∗>kT_ν^∗gkL∞(ν)−ε=kgkL∞(Rⁿ)−ε.

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Sinceεwas arbitrary, we proved that

kS^∗gk_Z^∗≥ kgkL∞(Rⁿ)

for any g ∈ L_∞(Rⁿ), and so, using the fact that the norm of any operator is the same as the norm of its adjoint (see [7, p. 93]) and the obvious fact that (S^∗)⁻¹= (S⁻¹)^∗, we have

kS⁻¹k=k(S^∗)⁻¹k ≤1.

Fixf ∈L1(Rⁿ) and putz=S⁻¹f. Then kzkZ≤ kfkL1(Rⁿ), that is

inf {kΦkL1(ν);T_νΦ =f} ≤ kfkL1(Rⁿ). By Lemma 3 we have

kfkL1(Rⁿ)=kT_νΦkL1(Rⁿ)≤ kT_νk.kΦkL1(ν)=kΦkL1(ν).

So, the opposite inequality holds as well.

Theorem 4. Letν be aσ-finite measure onRⁿ×R⁺ and s(ν) =Mf. Assume that for every functionf ∈L₁(Rⁿ)there exists ΦinL₁(Rⁿ)such that

(1) f =

Z

Rⁿ×R⁺

Φ(X)p(X, .)dν(X)

kfkL1(Rⁿ)= inf{kΦkL1(ν); (1)holds for someΦinL₁(ν)}. Then

sup

X∈Rⁿ×R⁺

u(X) = sup

X∈M

u(X)

for any quasi-bounded positive parabolic functionuonRⁿ×R⁺. Proof: Put c = sup

X∈M

u(X). If c =∞, sup

X∈Rⁿ×R⁺

u(X) = ∞. So, suppose that c <∞.

Letε >0. If we fix X₀ ∈Rⁿ×R⁺, then p(X₀, .)∈L₁(Rⁿ) and

kp(X₀, .)kL1(Rⁿ)= 1. By assumptions there is a function Φ∈L₁(ν) such that p(X0, .) =

Z

Rⁿ×R⁺

Φ(X)p(X, .)dν(X)≤ Z

Rⁿ×R⁺

|Φ(X)|p(X, .)dν(X)

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and

kΦkL1(ν)<1 +ε.

Asuis a quasi-bounded positive parabolic function,u∈ Pand we can integrate the first inequality with respect tof_udλ. Using the Fubini theorem and the fact thatu≤cons(ν), we have

u(X₀) = Z

Rⁿ

p(X₀, y)fu(y)dy≤ Z

Rⁿ

( Z

Rⁿ×R⁺

|Φ(X)|p(X, y)dν(X))fu(y)dy= Z

Rⁿ×R⁺

|Φ(X)|( Z

Rⁿ

p(X, y)f_u(y)dy)dν(X) = Z

Rⁿ×R⁺

|Φ(X)|u(X)dν(X)≤ Z

Rⁿ×R⁺

c.|Φ(X)|dν(X) =ckΦkL1(ν)≤c(1 +ε).

SinceX0 andεwere arbitrary, we have sup

X∈Rⁿ×R⁺

u(X) =c.

Of course, the following special form of Theorem 4 holds:

Theorem 4’. Let M be a subset ofRⁿ×R⁺. Assume that for every function f ∈L₁(Rⁿ)there exist {λ_k}^∞_k=1 in l₁ and a sequence {X_k}^∞_k=1 of points in M such that

(2) f =

X∞ k=1

λ_kp(X_k, .) λ−almost everywhere and

kfkL1(Rⁿ)= inf { X∞ k=1

|λ_k|; (2)holds for some{X_k}in M}. Then

sup

X∈Rⁿ×R⁺

u(X) = sup

X∈M

u(X) for any quasi-bounded positive parabolic functionu.

4. The main results

Theorem 5. LetM ⊂Rⁿ×R⁺. Then the following statements are equivalent:

(i)

sup

X∈Rⁿ×R⁺

u(X) = sup

X∈Mu(X) for all simple parabolic functionsu;

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(ii)

sup

X∈Rⁿ×R⁺

u(X) = sup

X∈M

u(X) for all bounded positive parabolic functionsu;

(iii)

sup

X∈Rⁿ×R⁺

u(X) = sup

X∈M

u(X) for all bounded parabolic functionsu;

(iv)

sup

X∈Rⁿ×R⁺

u(X) = sup

X∈M

u(X) for all quasi-bounded positive parabolic functionsu;

(v)

sup

X∈M|u(X)| for all bounded parabolic functionsu;

(vi)forλ-almost every pointY ∈Rⁿ× {0} there is a sequence of points ofM for whichY is a parabolic limit;

(vii)forλ-almost every pointY ∈Rⁿ× {0}there is a sequence of points ofM for whichY is a1-parabolic limit;

(viii)ifνis aσ-finite Borel measure withs(ν) =Mf, then for everyf ∈L₁(Rⁿ) there existsΦ∈L₁(ν)such that

(1) f =

Z

Rⁿ×R⁺

Φ(X)p(X, .)dν(X)

kfkL1(Rⁿ)= inf {kΦkL1(ν); (1)holds for someΦ∈L₁(ν)};

(ix)for every f ∈L₁(Rⁿ), there is a sequence {X_k},X_k ∈M and {λ_k} ∈l₁ such that

(2) f =

X∞ k=1

λ_kp(X_k, .)

λ-almost everywhere and kfkL1(Rⁿ)= inf {X

|λ_k|; (2)holds for some{X_k}inM}.

Proof: (i)⇒(vii) by Theorem 1’; (vii)⇒(vi) is clear; (vi)⇒(v) by Theorem 2;

(v)⇒(viii) by Theorem 3; (viii) ⇒(ix) is easy; (ix) ⇒(iv) by Theorem 4’;

(iv)⇒(ii) is clear; (ii)⇒(iii) is easy; (iii)⇒(i) is clear.

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Theorem 6. Let ν be a σ-finite Borel measure on Rⁿ×R⁺. Then for every f ∈L₁(Rⁿ)there existsΦ∈L₁(ν)such that

(1) f =

Z

Rⁿ×R⁺

Φ(X)p(X, .)dν(X)

and

kfkL1(Rⁿ)= inf {kΦkL1(ν); (1)holds for someΦ∈L1(ν)},

if and only ifλ-almost every pointY ∈Rⁿ×{0}is a parabolic limit point ofs(ν).

Proof: The result is obtained by combining Theorem 4, Theorem 1, Theorem 2

and Theorem 3.

5. Application

Definition. Normal distribution N(µ, σ²) on R with parameters µ and σ² is a continuous probability distribution onRwith a density

φ(y;µ, σ²) = 1 σ√

2πexp(−(y−µ)² 2σ² );

normal distribution Nn(µ,Σ) on Rⁿ with parameters µ and Σ, where µ ∈ Rⁿ and Σ = (σ_k,j) is a positive definite matrix n×n, is a continuous probability distribution with density

φ(y;µ,Σ) =φ(y1, y2, ..yn;µ,Σ) =

(2π)⁻^n/2|Σ|⁻^n/2exp(−1 2

Xn j=1

Xn k=1

(y_j−µ_j)(y_k−µ_k)σ^j,k), where|Σ|is the determinant ofΣand(σ^j,k) = Σ⁻¹.

If Σ is a σ²-multiple of the unit matrix U we have |Σ| =σ²ⁿ, Σ⁻¹ =σ⁻²U and

φ(y;µ,Σ) =φ(y;µ, σ²) = (2π)⁻^n/2σ⁻ⁿexp(−ky−µk² 2σ² ).

Such normal distribution will be denoted byN_n(µ, σ²) and the class of all such normal distributions will be denoted bySU. Let us remark that, if n= 1, then SU is a class of all normal distributions.

(We recall that a continuous probability distribution on Rⁿ is a probability measure onRⁿwhich is absolutely continuous with respect toλ.)

Definition. A setN is called non-tangentially dense inRⁿ×R⁺ if forλ-almost every point(y,0) ofRⁿ× {0}there is a sequence {(µ_k, σ_k)} of points of N such that(y,0)is a limit of this sequence andlim inf_k_→∞σ_k⁻¹kµ_k−yk>0. (It means that there is a coneC_y ⊂Rⁿ×R⁺ with vertex in (y,0) and open upward such thatM is a limit point of M∩Cy.)

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Theorem 7. LetS ⊂ SU. Then the setN ={(µ, σ)∈Rⁿ×R⁺; N_n(µ, σ²)∈ S}

is non-tangentially dense if and only if for any continuous probability distribution P there is a sequence{Nn(µ_k, σ²_k)} of normal distributions onRⁿ ofS and {λ_k} ∈l₁ such thatP can be expressed in the form

(3) P =

X∞ k=1

λ_kNn(µ_k, σ²_k), and

(4) inf {

X∞ k=1

|λ_k|; (3)holds for someN_n(µ_k, σ_k²)inS}= 1.

Proof: A functionf is a density of some continuous distribution if and only if f ≥0,f ∈L₁(Rⁿ) andkfkL1(Rⁿ)= 1.

Let us introduce substitutiont=σ²/2,µ=x. Then φ(y, µ, σ²) =p((x, t), y).

DenoteM ={(x, t)∈Rⁿ×R⁺; (µ, σ)∈N}.

The setN is non-tangentially dense if and only if for almost every (y,0) there is a sequence (µ_k, σ_k)∈N, the sequence converges to (y,0) and

lim inf_k_→∞σ_kkµ_k−yk⁻¹ >0, it means lim inf_k_→∞σ²_k/2kµ_k−yk⁻² >0.Using the above substitution we have that it is equivalent to the existence of the sequence (x_k, t_k) of elements ofM which converges to (y,0) and

lim inf_k_→∞t_kkx_k−yk⁻²>0.It meansN is non-tangentially dense if and only ifλ-almost every point ofRⁿ× {0}is a parabolic limit point ofM.

Then, by Theorem 5, for any densityf there is a sequence (x_k, t_k) of elements ofM (it means (µ_k, σ_k)∈N) and{λ_k} ∈l₁

(5) f =

X∞ k=1

λ_kp(X_k, .) λ-almost everywhere and

(6) 1 =kfkL1(Rⁿ)= inf { X∞ k=1

|λ_k|; (5) holds for some {x_k, t_k}in M} and so

f = X∞ k=1

φ(., µ_k, σ²_k) (7)

1 = inf { X∞ k=1

|λ_k|; (7) holds for someN_n(µ_k, σ_k²) inS}. (8)

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Integrating (7) with respect to λ over any λ-measurable set A and using the Lebesgue Convergence Theorem (can be used thanks to (8)) we have (3) and (4).

Let us suppose that any continuous distribution can be expressed in (3) and that (4) is true. LetP be a continuous distribution andf its density. Then (3) can be written in the form for anyλ-measurable setA

Z

A

f dλ= X∞ k=1

λ_k Z

A

φ(., µ_k, σ²_k)dλ.

Then, using the Lebesgue Convergence Theorem again (thanks to (4)), we have Z

A

f dλ= Z

A

X∞ k=1

λ_kφ(., µ_k, σ_k²)dλ.

ChoosingA be a ball with the centery and radiusr, dividing the equality by the λ-measure of the ball and letting r → 0, we have (7) and (8) and thus (5) and (6).

Let us remark that for anyf ∈L₁(Rⁿ) there are real numbersa₁,a₂and func- tionsf₁,f₂ which are densities of some distributions such thatf =a₁f₁−a₂ f₂ andkfkL1(Rⁿ) =a₁ kf₁kL1(Rⁿ)+a₂ kf₂kL1(Rⁿ). (Really, puta₁ =kf⁺kL1(Rⁿ)

and a₂ =kf⁻kL1(Rⁿ). If a₁ 6= 0, put f₁ = _a¹₁f⁺, else f₁ =π⁻^n/2exp(−k.k²), and ifa₂6= 0, then putf₂= _a¹

2f⁻, else f₂=π⁻^n/2exp(−k.k²).)

Then (5) and (6) are true for anyf ∈L₁(Rⁿ) (of course, without 1) and, by Theorem 5,λ-almost every point ofRⁿ× {0}is a parabolic limit point ofM, and

then, as we know,N is non-tangentially dense.

6. Remark

Similar problems as in Sections 1–4 have been recently investigated for classical harmonic functions on a ball in [2], [3], [4], [6] and for more general domains in [1].

The proofs in Section 2 were inspired by [6]; those in Section 3 by [4]. Sufficiency of the non-tangential density of the set N in Theorem 7 can be obtained as a special case of Theorem 2.9 in [8]. (From this, using substitution σ = √

2t, x=µ, Theorem 5 (vi) ⇒(viii) follows.)

References

[1] Aikawa H.,Sets of determination for harmonic function in an NTA domains, preprint, 1992.

[2] Bonsall F.F.,Decomposition of functions as sums of elementary functions, Quart J. Math.

Oxford (2)37(1986), 129–136.

[3] ,Domination of the supremum of a bounded harmonic function by its supremum over a countable subset, Proc. Edinburgh Math. Soc.30(1987), 441–477.

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[4] ,Some dual aspects of the Poisson kernel, Proc. Edinburgh Math. Soc.33(1990), 207–232.

[5] Doob J.L.,Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, New York, 1984.

[6] Gardiner S.J.,Sets of determination for harmonic function, Trans. Amer. Math. Soc.338 (1993), 233–243.

[7] Rudin W.,Functional Analysis, McGraw-Hill Book Company, 1973.

[8] Dudley Ward N.F.,Atomic Decompositions of Integrable or Continuous Functions, D. Phil Thesis, University of York, 1991.

Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovsk´a 83, 186 00 Praha 8, Czech Republic

(Received September 15, 1993)