Characterization of sets of determination for parabolic functions on a slab by coparabolic (minimal) thinness

Characterization of sets of determination for parabolic functions on a slab by coparabolic (minimal) thinness

Jarmila Ranoˇsov´a

Abstract. LetTbe a positive number or +∞. We characterize all subsetsMofRⁿ×]0, T[

such that

(i) inf

X∈Rⁿ×]0,T[u(X) = inf

X∈Mu(X)

for every positive parabolic functionuonRⁿ×]0, T[ in terms of coparabolic (minimal) thinness of the setMδ=∪(x,t)∈MB^p((x, t), δt), whereδ∈(0,1) andB^p((x, t), r) is the

“heat ball” with the “center” (x, t) and radiusr. Examples of different types of sets which can be used instead of “heat balls” are given.

It is proved that (i) is equivalent to the condition sup_X∈Rⁿ_×R⁺u(X) = sup_X∈Mu(X) for every bounded parabolic function onRⁿ×R⁺and hence to all equivalent conditions given in the article [7].

The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References.

Keywords: heat equation, parabolic function, Weierstrass kernel, set of determination, Harnack inequality, coparabolic thinness, coparabolic minimal thinness, heat ball Classification: 35K05, 35K15, 31B10

I. Preliminaries

In this paper the following notation is used: Small letters, such as x, y, will denote points inRⁿ; capital letters, such asX, points inRⁿ⁺¹, andtdenotes the

“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_n.

The Green functionGis defined onRⁿ⁺¹ by

G(X, Y) =G((x, t),(y, s)) = [4π(t−s)]^−n/2exp(−kx−yk²

4(t−s)) fort > s;

= 0 fort≦s.

The symbolB(x, r) denotes the closed ball centered atx∈Rⁿwith radiusr.

In this paper the following subsets ofRⁿ⁺¹ will be of special interest:

Let X ∈ Rⁿ×R⁺, X = (x, t) = (x₁, x₂, . . . , x_n, t), r ∈ R⁺, α, β, γ ∈ R⁺, γ≧β,a, b∈R⁺,δ∈R⁺.

Discs: D(X, r) =B(x, r)× {r}; D_X,α,β =B(x, α√

t)× {βt}; Cylinders: C(X, r,[a, b]) =B(x, r)×[a, b];

C_X,α,β,γ=B(x, α√

t)×[βt, γt];

Parabolic balls: B^p(X, r) ={Z ∈Rⁿ⁺¹:G(X, Z)≧(4πr)^−n/2} ∪ {X}; B_X,δ^p =B^p(X, δt);

Coparabolic balls: B^cp(X, r) ={Z∈Rⁿ⁺¹:G(Z, X)≧(4πr)^−n/2} ∪ {X}; B_X,δ^cp =B^cp(X, δt);

Intervals: I(X, r) = (x₁−r, x₁+r)×. . .×(x_n−r, x_n+r)×(t−r², t);

I_X,δ=I(X, δ√ t);

Paraboloids: P(X, a) ={(z, s)∈Rⁿ⁺¹:kz−yk²≦a(s−t)}; P(X, a, v) ={(z, s)∈Rⁿ⁺¹ :kz−yk²≦a(s−t)

ands≦t+v}; P_X,a,δ=P(X, a, δt).

Let T ∈]0,∞], M ⊂ Rⁿ×]0, T[ and let a set A_X be associated with every X ∈M. Then M_A will denote the set∪X∈MA_X ∩(Rⁿ×]0, T[). We will use the obvious notationM_Dα,β,M_Cα,β,γ,M_B^p

δ,M_B^cp

δ ,M_Iδ andM_Pa,δ.

ForM ⊂Rⁿ⁺¹the set{(y, s)∈Rⁿ⁺¹; (y,−s)∈M}will be called the reflection ofM and denoted byM^⊖.

Let T ∈]0,∞], M ⊂ Rⁿ×]0, T[ and Y ∈ Rⁿ× {0}. The set is coparabolic minimal thin atY, if and only ifM is coparabolic thin atY. (See section III.2.) We will writeM is coparabolic (minimal) thin.

Let 0< T ≦∞. A pointY = (y,0) is called a parabolic limit of a sequence {X_k},X_k= (x_k, t_k), of points inRⁿ×]0, T[, if{X_k} converges toY and

lim inf

k→∞ t_kkx_k−yk⁻²>0

(that is allX_kbelong to some paraboloid of revolution with vertexY and opening upward).

LetM ⊂Rⁿ×]0, T[. A pointY ∈Rⁿ× {0} is called a parabolic limit point of the setM, if there exists a sequence{X_k} such that everyX_k∈M and Y is a parabolic limit of{X_k}.

II. The main results

Theorem. Let0< T ≦∞andM ⊂Rⁿ×]0, T[. Then the following statements are equivalent:

(i) inf

X∈Rⁿ×]0,T[u(X) = inf

X∈Mu(X) for all bounded parabolic functionsuonRⁿ×]0, T[;

(ii) inf

X∈Rⁿ×]0,T[u(X) = inf

X∈Mu(X) for all positive parabolic functionsuonRⁿ×]0, T[;

(iii)there existα, β, γ∈R⁺,γ≧β such that

the set of points ofRⁿ× {0}at whichM_Cα,β,γ is(minimal)coparabolic thin has Lebesgue measure zero;

(iv)for any α, β, γ∈R⁺,γ≧β

the set of points ofRⁿ× {0}at whichM_Cα,β,γ is(minimal)coparabolic thin has Lebesgue measure zero;

(v)the set of points ofRⁿ× {0}which are not parabolic limit points ofM has Lebesgue measure zero;

(vi)there exists δ∈(0,1) such that the set of points ofRⁿ× {0}at whichM_B^p

δ is(minimal)coparabolic thin has Lebesgue measure zero;

(vii)for anyδ∈(0,1)

the set of points ofRⁿ× {0}at whichM_B^p

δ is(minimal)coparabolic thin has Lebesgue measure zero;

Remark1. A set satisfying condition (i) will be called a set of determination.

Remark2. The equivalence of (i), (ii), (v) and (vi) was announced in the abstract.

Remark 3. The “cylinder” conditions (iii) and (iv) include “disc” conditions, becauseC_X,α,β,γ=D_X,α,β forβ=γ.

Corollary. Conditions(ix)–(xiii)are equivalent to(i)from previous Theorem:

(ix)there exists δ∈R⁺ such that the set of points ofRⁿ× {0}in whichM_B^cp

δ is(minimal)coparabolic thin has Lebesgue measure zero;

(x)for anyδ∈R⁺

the set of points ofRⁿ× {0}at whichM_B^cp

δ is(minimal)coparabolic thin has Lebesgue measure zero;

(xi)there exists δ∈(0,1) such that

the set of points ofRⁿ× {0}at whichM_Iδ is(minimal)coparabolic thin has Lebesgue measure zero;

(xii)for anyδ∈(0,1)

the set of points ofRⁿ× {0}at whichM_Iδ is(minimal)coparabolic thin has Lebesgue measure zero;

(xiii)there exista, δ∈R⁺ such that

the set of points ofRⁿ× {0}at whichM_Pa,δ is(minimal)coparabolic thin has Lebesgue measure zero;

(xiv)for anya, δ∈R

the set of points ofRⁿ× {0}at whichM_Pa,δ is(minimal)coparabolic thin has Lebesgue measure zero.

In Part III, results from parabolic and coparabolic potential theory needed in this paper are summarized.

The proof of Theorem is given in Part IV. First equivalence of (i), (ii), (iii), (iv) and (v) forα, β, γ∈R⁺,γ≧β andβ >1 will be proved.

The implications (ii)⇒(i) and (iv)⇒(iii) are trivial. The equivalence (i)⇔ (v) was established in [7] forT =∞. But any bounded parabolic functionuon Rⁿ×]0, T[ is a restriction of a bounded parabolic function on Rⁿ×R⁺ and for anyuparabolic onRⁿ×R⁺: inf

X∈Rⁿ×R⁺u(X) = inf

X∈Rⁿ×]0,T[u(X) (both assertions follow immediately from Theorem III.1), so (i)⇔(v) is true onRⁿ×]0, T[ as well.

We will prove (iii)⇒(ii) and (v)⇒(iv).

Then the assumptionβ >1 will be removed and in the end the rest of Theorem and Corollary will be proved.

III. Parabolic and coparabolic potential theory 1. The case of Rⁿ⁺¹.

Parabolic and coparabolic function (see [4, p. 263]).

A real functionuon an open setD⊂Rⁿ⁺¹ having continuous partial deriva- tives ^∂u_∂t and ^∂_∂x^2u2

i

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

∂u

∂t =

n

X

i=1

∂²u

∂x²_i (resp. ∂u

∂t =−

n

X

i=1

∂²u

∂x²_i ) onDis called parabolic (resp. coparabolic) on D.

A function (x, t) → u(x, t) is coparabolic on D, if and only if the function (x, t)→u(x,−t) is parabolic onD^⊖.

The Green function ofRⁿ⁺¹ (see [4, p. 266]). Letbbe a function onRⁿ⁺¹defined as

b(x, t) = (4πt)^−n/2exp(−kxk²

4t ) fort >0,

= 0 fort≦0.

The Green functionGis defined onRⁿ⁺¹×Rⁿ⁺¹ by G(X, Y) =G((x, t),(y, s)) =b(x−y, t−s).

The functionG(., Y) is the Green function with pole Y for the heat equation.

This function is positive, parabolic onRⁿ⁺¹\{Y} and vanishes belowY and the limit at the point∞is zero.

The functionG(X, .) is the Green function with poleXfor the adjoint equation.

This function is positive, coparabolic on Rⁿ⁺¹\{X} and vanishes aboveX and the limit at the point∞is zero.

Ifµis a measure onRⁿ⁺¹ the functionsGµandµGdefined by Gµ(X) =

Z

Rⁿ⁺¹

G(X, Y)dµ(Y) and µG(Y) = Z

Rⁿ⁺¹

G(X, Y)dµ(X)

will be called potential and copotential onRⁿ⁺¹, respectively.

The Green function of an interval (see [4, p. 272]).

Let

I= (a₁, b₁)×(a₂, b₂)× · · · ×(an, b_n)×(t₁, t₂) andc_j =b_j−a_j.

The Green functionG_I is defined by G_I(X, Y) =

n

Y

j=1

X∞ i=−∞

[b(2ic_j−x_j+y_j, t−s)−b(2ic_j+ 2a_j−x_j−y_j, t−s)]

forX = (x₁, x₂, . . . , x_n, t) andY = (y₁, y₂, . . . , y_n, s) and−∞< t₁ < t < t₂<∞. Parabolic measure for an interval (see [4, p. 273]).

LetX∈I,µ_I(X, .) be supported by the part of∂I strictly belowX:

- on the lower boundary µ_I(X, .) is absolutely continuous relative to λ_n with continuous density

Y →G_I(X, Y);

- on the part of the lateral boundary withjth coordinateb_j

Y → − ∂

∂y_jG_I(X, Y);

- on the part of the lateral boundary withjth coordinatea_j

Y → ∂

∂y_jG_I(X, Y).

Parabolic averages (see [4, p. 275]).

LetX∈Rⁿ⁺¹ andδ >0. Recall that

I(X, δ) = (x₁−δ, x₁+δ)×(x₂−δ, x₂+δ)× · · · ×(x_n−δ, x_n+δ)×(t−δ², t).

Ifuis a Borel measurable function on∂I(X, δ), we define L(u, X, δ) =µ_I(X,δ)(X, u).

Ifuis a parabolic function onD andI(X, δ)⊂D, then u(X) =L(u, X, δ).

Superparabolic, subparabolic, cosuperparabolic and cosubparabolic functions (see [4, p. 277]).

A functionufrom an open setDinto ]− ∞,∞] is called superparabolic if (a)uis lower semicontinuous;

(b)uis finite on a dense subset ofD;

(c)u(X)≧L(u, X, δ) ifI(X, δ)⊂D.

A subparabolic function is defined as the negative of a superparabolic function.

A cosuperparabolic (resp. cosubparabolic) function is defined as a function on an open set D for which the function (x, t) →u(x,−t) is superparabolic (resp.

subparabolic) onD^⊖.

Let u be a function defined on D. The greatest subparabolic minorant, if there is one, is denoted byGM_Du. Ifuis a superparabolic function which has a subparabolic minorant, thenGM_Duexists and is parabolic. (See [4, p. 295].)

The Green function of an open set D (see [4, p. 298]).

LetD be a nonempty open subset ofRⁿ⁺¹ andY a point ofD. The parabolic Green function with poleY is defined onD by

G_D(., Y) =G(., Y)−GM_DG(., Y).

The functionG_D(., Y) is positive and superparabolic onD, parabolic onD\{Y} and differs fromG(., Y) by a continuous function andGM_DG_D(., Y) = 0.

The Riesz decomposition (see [4, p. 305]).

LetDbe a nonempty open subset ofRⁿ⁺¹. Ifvis a superparabolic function on Dwhich has a subparabolic minorant onD, then there exist a parabolic function uonD and a measureµonD such thatv=G_Dµ+uonD.

Parabolic reduction operation (see [4, p. 310]).

Let D be an open subset of Rⁿ⁺¹ and M ⊂ D. Let v be a positive super- parabolic (resp. cosuperparabolic) function onD.

The superparabolic (resp. cosuperparabolic) reduction ofvonM is defined as R_v^M = inf{u;uis positive superparabolic function onD, u≧v onM}; (resp.^∗R_v^M = inf{u;uis positive cosuperparabolic function onD, u≧v onM}).

The smooth reductionkvk^M(x, t) (resp.^∗kvk^M(x, t)) is defined by kvk^M(x, t) = lim inf

(y,s)→(x,t)R_v^M(y, s);

(resp.^∗kvk^M(x, t) = lim inf

(y,s)→(x,t)

∗R^M_v (y, s)).

If the setD is not specified it is supposed thatD=Rⁿ⁺¹. Theorem. LetM ⊂Rⁿ⁺¹,Gbe the Green function. Then

∗kG(X, .)k^M(Y) =kG(., Y)k^M(X).

The common value will be denoted byG^M(X, Y).

Ifµis a measure onRⁿ⁺¹, then

kGµk^M =G^Mµ and kµGk^M =µG^M.

Proof: (See [4, p. 342].)

Parabolic and coparabolic thinness(see [4, p. 346]).

A set M ⊂Rⁿ⁺¹ is said to be parabolic (resp. coparabolic) thin at X (resp.

atY), if

G^M(X, .)6=G(X, .) on the set{G(X, .)>0}; (resp.G^M(., Y)6=G(., Y) on the set{G(., Y)>0}).

This definition is equivalent to the “usual” definition of parabolic and co- parabolic thinness. (See [4, p. 346].)

Parabolic (resp. coparabolic) thinness is a local property. It means: Letr ∈ R⁺. A setM is parabolic (resp. coparabolic) thin atX, if and only ifM∩B(X, r) is parabolic (resp. coparabolic) thin atX.

It is clear that the setM ⊂Rⁿ⁺¹is parabolic thin atX = (x, t), if and only if M^⊖ is coparabolic thin at (x,−t).

2. The case of the slab.

Let 0< T ≦∞. The Green function for a slabRⁿ×]0, T[ is the restriction of Gto (Rⁿ×]0, T[)×(Rⁿ×]0, T[).

The Weierstrass kernel forRⁿ×]0, T[ with the pole aty inRⁿis given by p(X, y) = (4πt)^−n/2exp(−kx−yk²

4t ), whereX = (x, t)∈Rⁿ×]0, T[.

Clearly,p(X, y) =b(x−y, t) =G(X,(y,0)).

Theorem 1. A functionuonRⁿ×]0, T[is a difference of two positive parabolic functions, if and only if there is a signed measureµ_u onRⁿfor which

Z

Rⁿ

exp(−kyk²

4t )d|µu|(y)<∞ for allt∈]0, T[and

u(X) = Z

Rⁿ

p(X, y)dµ_u(y), X ∈Rⁿ×]0, T[.

The map u→ µ_u is a one-to-one linear order-preserving map from the class of parabolic functions satisfying these conditions onto the vector lattice of charges onRⁿ satisfying the above inequality.

The functionuis bounded, if and only if there existsf_u ∈L_∞(Rⁿ)such that µ_u=f_uλ.

Proof: (See [4, p. 290].)

From this theorem inf

X∈Rⁿ×]0,T[u(X) = ess inf

y∈Rⁿ f_u(y) for any bounded parabolic functionuonRⁿ×]0, T[.

Coparabolic minimal thinness onRⁿ×]0, T[.

Let M ⊂ Rⁿ×]0, T[, Y ∈ Rⁿ× {0}. The set M is said to be coparabolic minimal thin atY, if

kp(., y)k^M 6=p(., y) on the set{p(., y)>0}. (Compare [4, p. 378].)

The reduction is, of course, taken with respect toD=Rⁿ×]0, T[. But:

The restriction of any function superparabolic onRⁿ⁺¹ toRⁿ×]0, T[ is super- parabolic onRⁿ×]0, T[.

Ifv is a positive function superparabolic onRⁿ×]0, T[, there exists a positive parabolic functionuonRⁿ×]0, T[ and a measureµonRⁿ×]0, T[ such that

v=G_Rⁿ_×]0,T[µ+uonRⁿ×]0, T[.

But knowing that G_Rn×]0,T[ = G and p(X, y) = G(X,(y,0) on Rⁿ×]0, T[ and using the representation ofuguaranteed by previous Theorem, we have

v=Gµ+Gµ_u onRⁿ×]0, T[.

The functionGµ+Gµ_u is a positive superparabolic function onRⁿ⁺¹. So the reduction can be taken with respect toRⁿ⁺¹.

Asp(., y) =G(., Y), we have for anyM ⊂Rⁿ×]0, T[:

kp(., y)k^M =kG(., Y)k^M =G^M(., Y).

It means that ifM ⊂Rⁿ×R⁺andY ∈Rⁿ× {0},M is coparabolic thin atY, if and only ifM ∩(Rⁿ×]0, T[) is coparabolic minimal thin atY. So we will write M is coparabolic (minimal) thin atY.

Theorem 2. LetM ⊂Rⁿ×R⁺. If the set of points ofRⁿ× {0}at whichM is coparabolic(minimal)thin has Lebesgue measure equal to zero, thenk1k^M = 1 onRⁿ×R⁺.

Proof: We have 1 = R

Rⁿ

p(X, y)dλ(y) onRⁿ×R⁺ and

k1k^M =k Z

Rⁿ

p(X, y)dλ(y)k^M =k Z

Rⁿ⁺¹

G(X, Y)dλ(Y)k^M =kGλk^M =G^Mλ

onRⁿ×R⁺.

BecauseG^M(., Y) =G(., Y) on Rⁿ×R⁺for λ-almost allY we haveG^Mλ=

Gλ= 1 onRⁿ×R⁺.

Theorem 3 (Harnack inequality). Letn∈N. Then there exists a constantc_H such that for anyT ∈]0,∞]and any(x₁, t₁),(x₂, t₂)belonging toRⁿ×]0, T[such thatt₂> t₁ and for any positive parabolic functionuonRⁿ×]0, T[

u(x₁, t₁)≦u(x₂, t₂).e^cH(

kx2−x1k t2−t1 +1)

t₂ t₁

cH

.

Proof: (See [6, p. 104].)

3. Parabolic capacity.

LetK⊂Rⁿ⁺¹ be a compact set. The parabolic capacity ofK is defined by γ(K) = sup{µ(Rⁿ⁺¹);µ∈ M⁺(K), Gµ≦1 inRⁿ⁺¹},

whereM⁺(K) is the set of Borel measures supported byK.

LetM ⊂Rⁿ⁺¹ be an arbitrary set. Then

γ_⋆(M) = sup{γ(K);K⊂M, K compact} is called the inner parabolic capacity ofM and

γ^⋆(M) = inf{γ_⋆(G);G⊃M, Gopen} the outer parabolic capacity ofM.

Lemma 1. LetF be a Borel subset of Rⁿ⁺¹,t₀∈R. Then γ(F× {t₀}) =λ_n(F).

Proof: See [8, p. 355].

Theorem. Let M ⊂Rⁿ⁺¹, Y ∈Rⁿ⁺¹. ThenM is parabolic thin at Y, if and only if

Z1

0

γ^⋆(M ∩B^p(Y, r))r⁻ⁿ²⁻¹dr <∞.

Proof: See [3, p. 99].

From this and from what was said about the relation between parabolic and coparabolic thinness and the relation between coparabolic minimal thinness on Rⁿ×]0, T[ and coparabolic thinness onRⁿ⁺¹ follows:

Theorem 4. Let M ⊂ Rⁿ×R⁺ and Y ∈ Rⁿ× {0}. Then M is coparabolic (minimal)thin atY, if and only if

Z1

0

γ^⋆(M^⊖∩B^p(Y, r))r⁻ⁿ²⁻¹dr <∞.

4. Geometrical properties of the heat ball, the coparabolic ball and the paraboloid.

Lemma 2. LetX ∈Rⁿ⁺¹,r∈R⁺. Then

B^p(X, r)⊂C(X, r2n

e

√r,[t, t−r]);

B^p(X, r)⊃D((x, t−r e),

r2n e

√r);

and

B^cp(X, r)⊂C(X, r2n

e

√r,[t, t+r]);

B^cp(X, r)⊃D((x, t+r e),

r2n e

√r).

Lemma 3. Leta∈R⁺, then there exists a numbera₁such that for allX∈Rⁿ⁺¹, r∈R⁺ andv≦a₁r isP(X, a, v)⊂B^cp(X, r).

Lemma 4. Leta₁, α, β∈R⁺.

Then there exists a number a₂ such that for any Y ∈ Rⁿ⁺¹, and for any X ∈P(Y, c₁),X 6=Y, the discD_X,α,β is a subset ofP(Y, a₂).

Proofs of these lemmas are elementary.

IV. Proof of Theorem

1. In this part it will be proved that (iii) ⇒(ii) for β >1.

Theorem 1. Let n ∈ N, α, β, γ ∈ R⁺ and γ ≧ β > 1. Then there exists a positive constant c such that for every T ∈]0,∞] and M ⊂ Rⁿ×]0, T[, and for every positive parabolic functionuonRⁿ×]0, T[,

X∈Minf_Cα,β,γu(X)≧c inf

X∈Mu(X).

Proof: Let M ⊂Rⁿ×]0, T[ and X = (x, t), X ∈ M. Sinceβ > 1, T ≧ γt≧ s ≧ βt > t whenever (y, s) ∈ C_X,α,β,γ∩(Rⁿ×]0, T[), we have by the Harnack inequality:

u(y, s).e^cH(ky−xk

2 s−t +1)

.s t

cH

≧u(x, t).

Using the fact thatγt≧s≥βtandky−xk≦α√

t we arrive at u(y, s).e^cH(α

2t βt−t+1)

. γt

t cH

≧u(x, t) or

u(y, s).e^cH(α

2 β−1+1)

.γ^cH ≧u(x, t), thus

u(y, s)≧e^−cH(α

2 β−1+1)

.γ^−cHu(x, t).

From here the theorem immediately follows.

Theorem 2. Let0< T ≦∞andM ⊂Rⁿ×]0, T[for which there existα, β, γ∈ R⁺, γ ≧ β > 1 such that the set of points of Rⁿ× {0} at which M_Cα,β,γ is (minimal)coparabolic thin has Lebesgue measure zero.

Then there exists a constantcdepending only on α, β, γ, nsuch that

X∈Rinfⁿ×]0,T[u(X)≧c inf

X∈Mu(X) for all positive parabolic functionsuonRⁿ×]0, T[.

Proof: This theorem is obtained by combining the previous Theorem and The- orem III.2.

Theorem 3. Let0< T ≦∞andM ⊂Rⁿ×]0, T[. Then the following statements are equivalent:

(i)

X∈Rinfⁿ×]0,T[u(X) = inf

X∈Mu(X) for all positive parabolic functionsuonRⁿ×]0, T[;

(ii)there existsc >0 such that

X∈Rinfⁿ×]0,T[u(X)≧c inf

X∈Mu(X) for all positive parabolic functionsuonRⁿ×]0, T[.

Proof: (i)⇒(ii) is clear, putc= 1.

(ii) ⇒ (i) Let us suppose that there exists a set M satisfying (ii), but not (i).

Thencin (ii) belongs to (0,1).

Letube a positive parabolic function for which (i) is not true.

Denote inf

X∈Rⁿ×]0,T[u(X) =c₁ and inf

X∈Mu(X) =c₂.

We suppose that c₂ > c₁ ≧ c.c₂. Let ε be a positive number and v(X) = u(X)−c₁+εforX ∈Rⁿ×]0, T[.

Thenv is a positive parabolic function and

X∈Rinfⁿ×]0,T[v(X) =c₁−c₁+ε=ε, and inf

X∈Mv(X) =c₂−c₁+ε.

It follows from (ii) thatε≧c(c2−c₁+ε) for everyε >0, which is a contradiction.

Theorem 4. Let0< T ≦∞andM ⊂Rⁿ×]0, T[for which there existα, β, γ∈ R⁺, γ ≧ β > 1 such that the set of points of Rⁿ× {0} at which M_Cα,β,γ is (minimal)coparabolic thin has Lebesgue measure zero.

Then

X∈Rinfⁿ×]0,T[u(X) = inf

X∈Mu(X) for all positive parabolic functionsuonRⁿ×]0, T[.

Proof: The result is obtained by combining two previous Theorems.

The implication (iii)⇒(ii) is proved.

2. In this part the implication (v) ⇒ (iv) will be proved.

Lemma 1. Let {α_k}^∞_k=0 be a decreasing sequence of strictly positive numbers with limit zero.

Then ∞

X

k=1

(1− α_k

α_k−1) =∞. Proof: The infinite productQ_∞

k=1 αk

α_k−1 obviously diverges to 0. Consequently,

the above sum diverges.

Theorem 5. Let0< T ≦∞andM ⊂Rⁿ×]0, T[,Y ∈Rⁿ× {0}andα, β∈R⁺. If Y is a parabolic limit point of M, then M_Dα,β is not coparabolic (minimal) thin atY.

Proof: Theorem III.4 will be used and so we are interested in the setM_D^⊖

α,β∩ B^p(Y, r), which is clearly equal to the set (M_Dα,β∩B^cp(Y, r))^⊖.

Leta₁∈R⁺and{X_k}^∞_k=1 be a sequence of points ofM for which X_k= (x_k, t_k), lim

k→∞x_k=y, lim

k→∞t_k= 0, kx_k−yk≦a₁t_k. We can suppose thatt_k↓0.

By Lemma III.4, there exists a constanta₂ ∈ R⁺ depending on a₁, α, β such thatD_Xk,α,β is a subset ofP(Y, a₂) for allX_k.

Havinga₂, there exists, by Lemma III.3, a positive constanta₃, such that P(Y, a₂, v)⊂B^cp(Y, r) for allv≦a₃r.

Now it is clear thatD_Xk,α,β ⊂B^cp(Y, r) for allX_k= (x_k, t_k) satisfyingβt_k≦ a₃r. Let us denoter_k = _a^β

3t_k. Without loss of generality we can suppose that r₁ ≦1 and putr₀ = 1. Of course,r_k↓0.

Now we have thatD_Xk,α,β ⊂B^cp(y, r) for allk∈Nsuch thatr≧r_k. Forr∈]r_k, r_k−1] let us take instead ofM_Dα,β∩B^cp(Y, r) its subsetD_Xk,α,β. Then

Z1

0

γ^⋆(M_D^⊖

α,β∩B^p(Y, r))r⁻ⁿ²⁻¹dr= Z1

0

γ^⋆((M_Dα,β∩B^cp(Y, r))^⊖)r⁻ⁿ²⁻¹dr=

X∞ k=1

r_k−1

Z

rk

γ^⋆((M_Dα,β∩B^cp(Y, r))^⊖)r⁻ⁿ²⁻¹dr≧ X∞ k=1

r_k−1

Z

rk

γ^⋆(D^⊖_X

k,α,β)r⁻ⁿ²⁻¹dr = X∞

k=1

γ^⋆(D^⊖_X

k,α,β)

−2 nr⁻ⁿ²

r_k−1 rk

= 2 n

X∞ k=1

γ^⋆(D^⊖_X

k,α,β)(r⁻

n 2

k −r⁻

n 2

k−1).

Letκdenote the volume of the unit ball inRⁿ. SinceD^⊖_X

k,α,β =B(x_k, α√ t_k)× {−βt_k}, Lemma III.1. yields

γ^⋆(D_Xk,α,β) =λ_n(B(x_k, αp

t_k)) =καⁿt

n

k2

and this is equal toκαⁿ(^a_β³)ⁿ²r_kⁿ². Denotinga₄ =καⁿ(^a_β²)ⁿ² we arrive at

γ^⋆(D^⊖_X

k,α,β) =a₄r_kⁿ². So the series is equal to

2 na₄

X∞ k=1

r_kⁿ²( 1

r_kⁿ² − 1

r_k−1ⁿ² ) = 2 na₄

X∞ k=1

(1−( r_k r_k−1)ⁿ²)

and by Lemma 1 its sum is equal to∞, finishing the proof.

The implication (v)⇒(iv) immediately follows from this theorem.

3. So far we have proved the equivalence of (i), (ii), (iii), (iv), (v) for β >1. Now this condition will be removed.

Let 0< T ≦∞andM ⊂Rⁿ×]0, T[,d∈R⁺and

M(d) ={(y, s)∈Rⁿ×]0, T[; ex. (x, t)∈M, x=y, s=d.t}. LetX= (x, t), X∈Rⁿ×R⁺. We denoteX(d) the point (x, d.t).

Lemma 2. Let0 < T ≦∞ and M ⊂Rⁿ×]0, T[, d∈R⁺,Y ∈Rⁿ× {0}. The point Y is a parabolic limit point of the set M if and only if Y is a parabolic limit point of M(d).

Proof: Let Y be a parabolic limit point of M. Then there existc ∈ R⁺ and {X_k}^∞_k=1 such that

X_k= (x_k, t_k)∈M, lim

k→∞t_k= 0, andkx_k−yk≦ct_k. There existk₀ such that for allk > k₀ isdt_k< T and soX_k(d)∈M(d).

And lim

k→∞dt_k=d lim

k→∞t_k= 0, so lim

k→∞X_k(d) =Y andkx_k−yk≦ c d.d.t_k. Thus Y is a parabolic limit of {X_k(d)}^∞_k=k0 and thus a parabolic limit point of M(d).

AsM(d)(¹_d)⊂M, the opposite is true.

Remark. From this lemma and Theorem (v) it follows thatM is a set of deter- mination if and only ifM(d) is a set of determination.

Lemma 3. LetX ∈Rⁿ×R⁺,X ∈Rⁿ×R⁺,α, β, γ, d∈R⁺. Then C_X,α,β,γ=C_X(d),α,β

d,^γ_d and M_Cα,β,γ=M(d)_C

α,β d,γ

d

.

Proof: A straightforward calculation.

Now we will remove the conditionβ >1:

Letα, β, γ∈R⁺, γ≧β >0.

Using Remark to Lemma 2 we have the equivalence of these conditions:

(i)M is a set of determination;

(i₁) there existsβ ∈R⁺ such thatM(^β₂) is a set of determination;

(i₂) for anyβ∈R⁺,M(^β₂) is a set of determination.

Now we will use the equivalence of (i), (iii) and (iv) of Theorem for M(^β₂)_C

α,2,2γ β

. (2 > 1.) From the equivalence of (i) and (iii) it follows that (i₁) is equivalent with (ii₁) and from the equivalence of (i) and (iv) it follows that (i₂) is equivalent with (ii₂):

(ii1) There existα, β, γ∈R⁺, γ≧β such that the set of points ofRⁿ× {0}at whichM(^β₂)_C

α,2,2γ β

is (minimal) coparabolic thin has Lebesgue measure zero;

(ii2) for anyα, β, γ∈R⁺, γ ≧β

the set of points ofRⁿ× {0}at whichM(^β₂)_C

α,2,2γ β

is (minimal) coparabolic thin has Lebesgue measure zero.

From Lemma 3 we have

M_Cα,β,γ=M(β 2)_C

α,2,2γ β

.

Using this equality, (ii1) and (ii2) can be formulated in this way:

(iii) There existα, β, γ∈R⁺,γ≧β such that

the set of points ofRⁿ× {0}at whichM_Cα,β,γ is (minimal) coparabolic thin has Lebesgue measure zero;

(iv) for anyα, β, γ∈R⁺,γ≧β

the set of points ofRⁿ× {0}at whichM_Cα,β,γ is (minimal) coparabolic thin has Lebesgue measure zero.

The conditionβ >1 was removed. The equivalence of (i), (ii), (iii), (iv) and (v) is proved.

4. In this part the rest of Theorem and Corollary will be proved.

Lemma 4. Let0< T ≦∞and M ⊂Rⁿ×]0, T[andα₁, β₁, γ₁, α₂, β₂∈R⁺ and let with every point ofM a setA_X be associated such that

D_X,α2,β2 ⊂A_X ⊂C_{X,α1,β1,γ1}.

ThenM is a set of determination if and only if the set of points ofRⁿ× {0} at whichM_Ais(minimal)coparabolic thin has Lebesgue measure zero.

Proof: Let M be a set of determination. Then by Theorem (iv), the set of points ofRⁿ× {0}at whichM_Dα

2,β2 is (minimal) coparabolic thin has Lebesgue measure zero. FromM_A⊃M_Dα

2,β2 the assertion of the lemma follows.

If the set of points of Rⁿ× {0} at which M_A is (minimal) coparabolic thin has Lebesgue measure zero, then the same is true forM_Cα

1,β1,γ1 (becauseM_A⊂ M_Cα

1,β1,γ1) and by Theorem (iii) the converse implication of this lemma follows.

Proof of Theorem (v), (vi) and Corollary:

Using Lemma III.2 we have DX,

q

2n

e δ,1−^δ_e ⊂B_X,δ^p ⊂C

X,

q

2n e δ,1−δ,1;

and D

X,

q

2n

e δ,1+^δ_e ⊂B_X,δ^cp ⊂C

X,

q

2n e δ,1,1+δ. Similar properties are true for the paraboloid and the interval:

D_X,^√_δ,1+δ⊂P_X,a,δ⊂C_X,^√_δ,1,1+δ,; and D_X,δ,1⊂I_X,δ⊂C_X,δ,1−δ2,1.

From here and from previous lemma, Theorem (v), (vi) and Corollary imme- diately follow.

Added in the proof. After having submitted the paper I found out that Theo- rem 5 is a known result, see Proposition 3.1 in Mair B.: Fine and parabolic limits for solutions of second order linear parabolic equations on an infinite slab, Trans.

Amer. Math. Soc.284(1984), 583–599.

Theorem 5 is a consequence of Corollary 2 in Netuka I.: Thinnesss and the heat equation, ˇCasopis Pˇest. Mat.99(1974), 293–299, as well.

References

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

Soc. Japan, to appear.

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

[3] Brzezina M.,On the base and the essential base in parabolic potential theory, Czechoslovak Math. J.40 (115)(1990), 87–103.

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

[5] Gardiner S.J.,Sets of determination for harmonic function, Trans. Amer. Math. Soc.338.1 (1993), 233–243.

[6] Moser J., A Harnack inequality for parabolic differential equations, Comm. Pure Appl.

Math.XVII(1964), 101–134.

[7] Ranoˇsov´a J., Sets of determination for parabolic functions on a half-space, Comment.

Math. Univ. Carolinae35(1994), 497–513.

[8] Watson A.N.,Thermal capacity, Proc. London Math. Soc.37.3(1987), 342–362.

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

E-mail: [email protected]

(Received October 12, 1995)