Let { Λ j } j = 1 ∞ be the nondecreasing sequence of eigenvalues of the Beltrami operator −δ on Ω (with the Dirichlet boundary condition) counted with their multiplicities, and let { ϕ j } j = 1 ∞ be an orthonormal (in L 2 ( Ω ) ) sequence of eigenfunctions corresponding to the eigenvalues  Λ j . Furthermore, we define

λ j ± = 2 − m 2 ± ( 1 − m / 2 ) 2 + Λ j and σ j = λ j + − 1 + m 2 .

This means that λ j ± are the solutions of the quadratic equation λ ( m − 2 + λ ) = Λ j . Obviously, λ j + > 0 and λ j − < 2 − m for j = 1 , 2 , …  .

By [8, Theorem 3],

| ∂ t k ∂ x ′ α ∂ y ′ γ G ( x ′ , y ′ , t ) | ≤ c t − k − ( m + | α | + | γ | ) / 2 ( | x ′ | | x ′ | + t ) λ 1 + − | α | − ε ( | y ′ | | y ′ | + t ) λ 1 + − | γ | − ε × ( d ( x ′ ) | x ′ | ) − ε α ( d ( y ′ ) | y ′ | ) − ε γ exp ( − κ | x ′ − y ′ | 2 t )

for | α | ≤ 2 , | γ | ≤ 2 . Here d(x′) denotes the distance of the point x′ from the boundary ∂K. Furthermore, ε α is defined as zero for | α | ≤ 1 , while εα is an arbitrarily small positive real number if | α | = 2 . Actually, the estimate (6) is proved in 8 only for k = 0 , but for a more general class of operators, parabolic operators with discontinuous in time coefficients. If the coefficients in 8 do not depend on t, then one can use the same argument as in the proof of [8, Theorem 3] when treating the derivatives along the edge of the domain D = K × R n − m . This argument shows that the kth derivative with respect to t will bring only an additional factor t − k to the right-hand side of (6).

The following lemma will be applied in the proof of Lemma 1.2. Here and in the sequel, we use the notation r = | x ′ | and ω x = x ′ / | x ′ | .

Lemma 1.1 Let G(x′,y′,t) be the Green function introduced above, and let G j ( r , ρ , t ) denote the Green function of the initial-boundary value problem

Then

∫ Ω G ( x ′ , y ′ , t ) ϕ j ( ω x ) d ω x = | y ′ | 1 − m G j ( | x ′ | , | y ′ | , t ) ϕ j ( ω y ) .

Proof The solution of the problem

is given by the formula

u ( x ′ , t ) = ∫ K G ( x ′ , y ′ , t ) ϕ ( y ′ ) d y ′ .

We define

U j ( r , t ) = ∫ Ω u ( x ′ , t ) ϕ j ( ω x ) d ω x .

Then it follows from (8) and (9) that

Furthermore,

U j ( r , 0 ) = Φ j ( r ) = def ∫ Ω ϕ ( x ′ ) ϕ j ( ω x ) d ω x .

Therefore,

U j ( r , t ) = ∫ 0 ∞ G j ( r , ρ , t ) Φ j ( ρ ) d ρ = ∫ 0 ∞ ∫ Ω G j ( r , ρ , t ) ϕ j ( ω y ) ϕ ( y ′ ) d ω y d ρ = ∫ K G j ( r , | y ′ | , t ) ϕ j ( ω y ) ϕ ( y ′ ) | y ′ | 1 − m d y ′ .

Comparing this with the formula

U j ( r , t ) = ∫ Ω u ( x ′ , t ) ϕ j ( ω x ) d ω x = ∫ K ∫ Ω G ( x ′ , y ′ , t ) ϕ j ( ω x ) d ω x ϕ ( y ′ ) d y ′ ,

we get (7). □

In the sequel, σ is an arbitrary real number satisfying the conditions

σ > λ 1 − , σ ≠ λ j + for all  j .

We define G σ ( x ′ , y ′ , t ) = 0 for σ < λ 1 + , while

G σ ( x ′ , y ′ , t ) = ∑ λ j + < σ u j ( m j ) ( x ′ , ∂ t ) c j ( y ′ , t ) for  σ > λ 1 + ,

where

and m j = [ σ − λ j + 2 ] . Here, we used the notation

σ ( μ ) = σ ( σ − 1 ) ⋯ ( σ − μ + 1 ) for  μ = 1 , 2 , … and σ ( 0 ) = 1 .

We define V p , β l ( K ) as the weighted Sobolev space with the norm

∥ u ∥ V p , β l ( K ) = ( ∫ K ∑ | α | ≤ l r p ( β − l + | α | ) | ∂ x α u ( x ) | p d x ) 1 / p

for 1 < p < ∞ and integer l≥0.

Lemma 1.2 Suppose that σ is a real number such that σ > λ 1 − and ( σ − λ j + ) / 2 is not integer for λ j + ≤ σ . Furthermore, let 1<p<∞ and β = 2 − σ − m / p . Then

G ( x ′ , y ′ , t ) = G σ ( x ′ , y ′ , t ) + R σ ( x ′ , y ′ , t ) ,

where ∂ t k ∂ y ′ γ R σ ( ⋅ , y ′ , t ) ∈ V p , β 2 ( K ) for y′∈K, t > 0 , |γ|≤2.

Proof We prove the lemma by induction in m 1 = [ ( σ − λ 1 + ) / 2 ] .

First, let λ 1 − < σ < λ 1 + . Then it follows from [7, Corollary 4.1 and Theorem 4.2] (see also [6, Theorem 3.2]) that ∂ t k ∂ y ′ γ G ( ⋅ , y ′ , t ) ∈ V p , β 2 ( K ) for all y′∈K, t>0, |γ|≤2, where β = 2 − σ − m / p . Thus, the assertion of the lemma is true for σ<λ1+.

Suppose the assertion is proved for σ < λ 1 + + 2 l . Now let λ 1 + + 2 l < σ < λ 1 + + 2 ( l + 1 ) . We set σ ′ = σ − 2 if l > 0 and σ ′ = λ 1 + − ε if l=0, where ε is a sufficiently small positive number. Then

[ σ ′ − λ j + 2 ] = [ σ − λ j + 2 ] − 1 = m j − 1 for  λ j + < σ ′ .

By the induction hypothesis, we have

G ( x ′ , y ′ , t ) = G σ ′ ( x ′ , y ′ , t ) + R σ ′ ( x ′ , y ′ , t ) ,

where G σ ′ is given by (11) (with σ ′ instead of σ and m j − 1 instead of mj), ∂ t k ∂ y ′ γ R σ ′ ( ⋅ , y ′ , t ) ∈ V p , β ′ 2 ( K ) , β ′ = 2 − σ ′ − m / p . The coefficients c j ( y ′ , t ) in Gσ′ are given by (13) and satisfy the equation ( ∂ t − Δ y ′ ) c j ( y ′ , t ) = 0 . Therefore,

( ∂ t − Δ y ′ ) R σ ′ ( x ′ , y ′ , t ) = 0

for x ′ , y ′ ∈ K , t>0. Obviously, G σ ′ ( a x ′ , a y ′ , a 2 t ) = a − m G σ ′ ( x ′ , y ′ , t ) for a > 0 . Using the same equality for the Green function G(x′,y′,t), we obtain

R σ ′ ( a x ′ , a y ′ , a 2 t ) = a − m R σ ′ ( x ′ , y ′ , t ) for  a > 0 .

Furthermore,

Δ x ′ R σ ′ ( x ′ , y ′ , t ) = Δ x ′ G ( x ′ , y ′ , t ) − Δ x ′ G σ ′ ( x ′ , y ′ , t ) = ( ∂ t − Δ x ′ ) G σ ′ ( x ′ , y ′ , t ) + ∂ t R σ ′ ( x ′ , y ′ , t ) = ( ∂ t − Δ x ′ ) ∑ λ j + < σ ′ ∑ k = 0 m j − 1 ∂ t k c j ( y ′ , t ) 4 k k ! ( σ j + k ) ( k ) r λ j + + 2 k ϕ j ( ω x ) + ∂ t R σ ′ ( x ′ , y ′ , t ) .

Using the formula

Δ x ′ r λ j + + 2 k ϕ j ( ω ) = 4 k ( σ j + k ) r λ j + + 2 k − 2 ϕ j ( ω x ) ,

we get

Δ x ′ R σ ′ ( x ′ , y ′ , t ) = ∑ λ j + < σ ′ ∂ t m j c j ( y ′ , t ) r λ j + + 2 m j − 2 ϕ j ( ω x ) 4 m j − 1 ( m j − 1 ) ! ( σ j + m j − 1 ) ( m j − 1 ) + ∂ t R σ ′ ( x ′ , y ′ , t ) = Δ x ′ Σ ′ + ∂ t R σ ′ ( x ′ , y ′ , t ) ,

where

Σ ′ = ∑ λ j + < σ ′ ∂ t m j c j ( y ′ , t ) 4 m j m j ! ( σ j + m j ) ( m j ) r λ j + + 2 m j ϕ j ( ω x )

( Σ ′ = 0 for l=0). Let χ be a smooth function with compact support on [ 0 , ∞ ) such that χ ( r ) = 1 for r < 1 . Using the notation r=|x′|, the function χ can be also considered as a function in K. Since σ ′ < λ j + + 2 m j < σ for λ j + < σ ′ , we have χ ∂ t k ∂ y ′ γ Σ ′ ( ⋅ , y ′ , t ) ∈ V p , β ′ 2 ( K ) and ( 1 − χ ) ∂ t k ∂ y γ Σ ′ ( ⋅ , y ′ , t ) ∈ V p , β 2 ( K ) for all y′∈K, t>0. Consequently, ∂ t k ∂ y ′ γ ( R σ ′ ( ⋅ , y ′ , t ) − χ Σ ′ ( ⋅ , y ′ , t ) ) ∈ V p , β ′ 2 ( K ) and

Applying [7, Theorem 4.2], we obtain

where v k , γ ( ⋅ , y ′ , t ) ∈ V p , β 2 ( K ) . The coefficients c μ , k , γ are given by the formula

c μ , k , γ ( y ′ , t ) = ∫ K ∂ t k ∂ y ′ γ ( ∂ t R σ ′ ( x ′ , y ′ , t ) + Δ x ′ ( 1 − χ ) Σ ′ ( x ′ , y ′ , t ) ) v μ ( x ′ ) d x ′ ,

where v μ ( x ′ ) = − 1 2 σ μ r λ μ − ϕ μ ( ω x ) . The integral in (16) is well defined, since

∂ t k ∂ y ′ γ ( ∂ t R σ ′ ( ⋅ , y ′ , t ) + Δ x ′ ( 1 − χ ) Σ ′ ( ⋅ , y ′ , t ) ) ∈ V p , β 0 ( K ) ∩ V p , β ′ 0 ( K )

and v μ ∈ V p ′ , − β 0 ( K ) + V p ′ , − β ′ 0 ( K ) , p ′ = p / ( p − 1 ) , for σ ′ < λ μ + < σ . The remainder v k , γ and the coefficients cμ,k,γ in (15) satisfy the estimate

Obviously, c μ , k , γ ( y ′ , t ) = ∂ t k ∂ y ′ γ c μ ( y ′ , t ) = ∂ t k ∂ y ′ γ c μ , 0 , 0 ( y ′ , t ) . This means that

R σ ′ ( x ′ , y ′ , t ) − χ ( r ) Σ ′ ( x ′ , y ′ , t ) = ∑ σ ′ < λ μ + < σ c μ ( y ′ , t ) r λ μ + ϕ μ ( ω x ) + v ( x ′ , y ′ , t ) ,

where ∂ t k ∂ y ′ γ v ( ⋅ , y ′ , t ) = v k , γ ( ⋅ , y ′ , t ) ∈ V p , β 2 ( K ) . Consequently,

R σ ′ ( x ′ , y ′ , t ) = Σ ( x ′ , y ′ , t ) + R σ ( x ′ , y ′ , t ) ,

where

Σ ( x ′ , y ′ , t ) = Σ ′ ( x ′ , y ′ , t ) + ∑ σ ′ < λ μ + < σ c μ ( y ′ , t ) r λ μ + ϕ μ ( ω x ) = ∑ λ j + < σ ∂ t m j c j ( y ′ , t ) r λ j + + 2 m j ϕ j ( ω x ) 4 m j m j ! ( σ j + m j ) ( m j )

and R σ ( x ′ , y ′ , t ) = v ( x ′ , y ′ , t ) + ( χ − 1 ) Σ ′ ( x ′ , y ′ , t ) . Obviously, ∂ t k ∂ y ′ γ R σ ( ⋅ , y ′ , t ) ∈ V p , β 2 ( K ) for |γ|≤2. Using (18) and the equality

G σ ′ ( x ′ , y ′ , t ) + Σ ( x ′ , y ′ , t ) = G σ ( x ′ , y ′ , t ) ,

we conclude that

G ( x ′ , y ′ , t ) = G σ ′ ( x ′ , y ′ , t ) + R σ ′ ( x ′ , y ′ , t ) = G σ ( x ′ , y ′ , t ) + R σ ( x ′ , y ′ , t ) .

It remains to show that the coefficients

in (15) have the form (13) for σ ′ < λ μ + < σ . First, note that

( ∂ t − Δ y ′ ) c μ ( y ′ , t ) = 0 for  y ′ ∈ K , t > 0 ,

since ( ∂ t − Δ y ′ ) R σ ′ ( x , y , t ) = 0 and ( ∂ t − Δ y ′ ) Σ ′ ( x , y , t ) = 0 .

Obviously, the functions ∂ t G σ ′ ( x ′ , y ′ , t ) and

contain only functions ϕ j ( ω x ) with λ j + < σ ′ . Thus, the orthogonality of the functions ϕj implies

for λ μ + > σ ′ . Applying Lemma 1.1, we conclude that c μ ( y ′ , t ) has the form

c μ ( y ′ , t ) = ρ 1 − m ϕ μ ( ω y ) f μ ( ρ , t ) ,

where ρ = | y ′ | . Since R σ ′ ( a x ′ , a y ′ , a 2 t ) = a − m R σ ′ ( x ′ , y ′ , t ) and Σ ′ ( a x ′ , a y ′ , a 2 t ) = a − m Σ ′ ( x ′ , y ′ , t ) for all a>0, it follows from (18) that

∑ σ ′ < λ μ + < σ ( a λ μ + c μ ( a y ′ , a 2 t ) − a − m c μ ( y ′ , t ) ) r λ μ + ϕ μ ( ω x ) = a − m R σ ( x ′ , y ′ , t ) − R σ ( a x ′ , a y ′ , a 2 t ) .

The function on the right-hand side belongs to V p , β 2 ( K ) for all y′∈K, t>0, a>0, while the left-hand side belongs only to Vp,β2(K) if

c μ ( a y ′ , a 2 t ) = a − m − λ μ + c μ ( y ′ , t ) .

Combining the last equality with (21), we get the representation

c μ ( y ′ , t ) = ρ − m − λ μ + ϕ μ ( ω y ) h μ ( ρ 2 4 t ) = ρ λ μ − − 2 ϕ μ ( ω y ) h μ ( ρ 2 4 t ) .

Inserting this into the equation ( ∂ t − Δ y ′ ) c μ ( y ′ , t ) = 0 , we obtain

r 2 h μ ″ ( r ) + ( r − σ μ − 1 ) r h μ ′ ( r ) + ( σ μ + 1 ) h μ ( r ) = 0 .

The substitution h μ ( r ) = e − r r σ μ + 1 u ( r ) leads to the differential equation

r 2 u ″ ( r ) + ( σ μ + 1 − r ) r u ′ ( r ) = 0

which has the solution

u ( r ) = d 1 + d 2 ∫ r 1 s − σ μ − 1 e s d s

with arbitrary constants d 1 and d 2 . Consequently,

c μ ( y ′ , t ) = ρ λ μ − − 2 ϕ μ ( ω y ) ( ρ 2 4 t ) σ μ + 1 exp ( − ρ 2 4 t ) ( d 1 + d 2 ∫ ρ 2 / ( 4 t ) 1 s − σ μ − 1 e s d s ) .

Using (6) and (17), one gets the estimate

| ∂ t k c μ ( y ′ , t ) | ≤ C k ( t ) ρ λ 1 + − ε

with certain functions C k for ρ = | y | < t . Thus, the constant d2 in (22) must be zero. Integrating (19), we get

∫ 0 ∞ c μ ( y ′ , t ) d t = − v μ ( y ′ ) = 1 2 σ μ ρ λ μ − ϕ μ ( ω y )

by means of (20). Hence,

d 1 ρ λ μ − − 2 ϕ μ ( ω y ) ∫ 0 ∞ ( ρ 2 4 t ) σ μ + 1 exp ( − ρ 2 4 t ) d t = 1 2 σ μ ρ λ μ − ϕ μ ( ω y ) .

The integral on the left-hand side is equal to 1 4 ρ 2 Γ ( σ μ ) . Thus, we get u ( r ) = d 1 = 2 / Γ ( σ μ + 1 ) and

h μ ( r ) = 2 Γ ( σ μ + 1 ) r σ μ + 1 e − r .

This means that the formula (13) is valid for the coefficients cj if σ ′ < λ j + < σ . The proof of the lemma is complete. □

We are interested in point estimates for the remainder R σ ( x ′ , y ′ , t ) in Lemma 1.2 in the case |x′|<t. For this, we need the following lemma.

Lemma 1.3 Suppose that u ∈ L p , β ( K ) and d ∇ u ∈ L p , β ( K ) , where p > m . Then

sup x ∈ K d ( x ′ ) m / p r ( x ) β | u ( x ′ ) | ≤ c ( ∫ K r p β ( | d ( x ′ ) ∇ u ( x ′ ) | p + | u ( x ′ ) | p ) d x ′ ) 1 / p

with a constant c independent of u.

Proof Let x 0 ′ be a point int K, and let B 0 be a ball centered at x0′ with radius d 0 / 2 = d ( x 0 ′ ) / 2 . We introduce the new coordinates y ′ = d 0 − 1 x ′ and set v ( y ′ ) = u ( d 0 y ′ ) = u ( x ′ ) . Obviously, the point y 0 ′ = d 0 − 1 x 0 ′ has the distance 1 from ∂K. Hence,

| v ( y 0 ′ ) | p ≤ c ∫ | y ′ − y 0 ′ | < 1 / 2 ( | ∇ y ′ v ( y ′ ) | p + | v ( y ′ ) | p ) d y ′ .

This implies

| u ( x 0 ′ ) | p ≤ c d 0 − m ∫ B 0 ( | d 0 ∇ x ′ u ( x ′ ) | p + | u ( x ′ ) | p ) d x ′ .

Since d 0 / 2 < d ( x ′ ) < 3 d 0 / 2 and r ( x 0 ′ ) / 2 < r ( x ′ ) < 3 r ( x 0 ′ ) / 2 for x ′ ∈ B 0 , we obtain

d 0 m r ( x 0 ′ ) p β | u ( x 0 ′ ) | p ≤ c ∫ B 0 r p β ( | d ( x ′ ) ∇ x ′ u ( x ′ ) | p + | u ( x ′ ) | p ) d x ′ .

The result follows. □

Using the last two lemmas, we can prove the following theorem.

Theorem 1.1 Suppose that σ is a real number satisfying (10). Then

G ( x ′ , y ′ , t ) = G σ ( x ′ , y ′ , t ) + R σ ( x ′ , y ′ , t ) ,

where

| ∂ t k ∂ x ′ α ∂ y ′ γ R σ ( x ′ , y ′ , t ) | ≤ c t − k − ( m + | α | + | γ | ) / 2 ( | x ′ | t ) σ − | α | ( | y ′ | | y ′ | + t ) λ 1 + − | γ | − ε × ( d ( x ′ ) | x ′ | ) − ε α ( d ( y ′ ) | y ′ | ) − ε γ exp ( − κ | y ′ | 2 t )

for |x′|<t, |α|≤2, |γ|≤2. Here ε α = 0 for |α|≤1, while εα is an arbitrarily small positive real number if |α|=2.

Proof Since G σ = G σ + ε for small positive ε, we may assume, without loss of generality, that (σ−λj+)/2 is not integer for λ j + < σ . We prove the theorem by induction in m1=[(σ−λ1+)/2].

If λ1−<σ<λ1+, then the assertion of the theorem follows from [8, Theorem 3]. Suppose that λ1++2l<σ<λ1++2(l+1), l ≥ 0 , and that the theorem is proved for σ<λ1++2l. We set σ′=σ−2 if l>0. In the case l=0, let σ′ be an arbitrary real number satisfying the inequalities λ 1 − < σ ′ < λ 1 + and σ ′ ≥ σ − 2 . By the induction hypothesis, we have

G ( x ′ , y ′ , t ) = G σ ′ ( x ′ , y ′ , t ) + R σ ′ ( x ′ , y ′ , t ) ,

where Gσ′ is given by (11) (with σ′ instead of σ and mj−1 instead of mj). Since G σ ′ = G σ ′ + δ for sufficiently small δ, it follows from the induction hypothesis that

| ∂ t k ∂ x ′ α ∂ y ′ γ R σ ′ ( x ′ , y ′ , t ) | ≤ c t − k − ( m + | α | + | γ | ) / 2 ( | x ′ | t ) σ ′ + δ − | α | ( | y ′ | | y ′ | + t ) λ 1 + − | γ | − ε × ( d ( x ′ ) | x ′ | ) − ε α ( d ( y ′ ) | y ′ | ) − ε γ exp ( − κ | y ′ | 2 t )

for | x ′ | < 2 t , |α|≤2, |γ|≤2. As was shown in the proof of Lemma 1.2, the remainder R σ ′ admits the decomposition

R σ ′ ( x ′ , y ′ , t ) = Σ ( x ′ , y ′ , t ) + R σ ( x ′ , y ′ , t ) ,

where

Σ ( x ′ , y ′ , t ) = ∑ λ j + < σ r λ j + + 2 m j ϕ j ( ω x ) ∂ t m j c j ( y ′ , t ) 4 m j m j ! ( σ j + m j ) ( m j )

and ∂ t k ∂ y ′ γ R σ ( ⋅ , y ′ , t ) ∈ V p , β 2 ( K ) for t>0, y′∈K, |γ|≤2. Here β=2−σ−m/p. Furthermore (cf. (14)),

Δ x ′ R σ ( x ′ , y ′ , t ) = Δ x ′ ( R σ ′ ( x ′ , y ′ , t ) − Σ ( x ′ , y ′ , t ) ) = Δ x ′ ( R σ ′ ( x ′ , y ′ , t ) − Σ ′ ( x ′ , y ′ , t ) ) = ∂ t R σ ′ ( x ′ , y ′ , t ) .

Let χ be a smooth cut-off function on the interval [0,∞), χ = 1 in [ 0 , 1 ) and χ = 0 on ( 2 , ∞ ) . We define χ 1 ( x ′ , t ) = χ ( t − 1 / 2 | x ′ | ) for x ′ ∈ K , t>0. Then

Δ x ′ ( χ 1 ( x ′ , t ) ∂ y ′ γ ∂ t k R σ ( x ′ , y ′ , t ) ) = f ( x ′ , y ′ , t ) ,

where

f = χ 1 ∂ y ′ γ ∂ t k + 1 R σ ′ + 2 ∇ x ′ χ 1 ⋅ ∇ x ′ ∂ y ′ γ ∂ t k ( R σ ′ − Σ ) + ( Δ x ′ χ 1 ) ∂ y ′ γ ∂ t k ( R σ ′ − Σ ) .