Dedicated to Prof. Dr.-Ing. Dr. h.c. Wolfgang L. Wendland on the occasion of his 65th birthday

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1 ) MATHEMATICA BOHEMICA o. 1, 1 19 REMOVABILITY OF SIGULARITIES WITH AISOTROPIC GROWTH,, Praha Received May 15, 2001) Dedicated to Prof. Dr.-Ing. Dr. h.c. Wolfgang L. Wendland on the occasion of his 65th birthday Abstract. With help of suitable anisotropic Minkowski s contents and Hausdorff measures some results are obtained concerning removability of singularities for solutions of partial differential equations with anisotropic growth in the vicinity of the singular set. eywords: solutions of partial differential equations, removable singularities, anisotropic metric, Minkowski s contents MSC 2000 : 65Z05, 28A12 Let G be an open set. We consider differential operators of the form 1) P D) = α a α D α, α M, acting on distributions in G; M is a finite set of multiindices α = α 1, α 2,..., α ), where the components α j 1 j ) are nonnegative integers and a α are infinitely differentiable complex-valued functions on G. We write D α = D α1 1 Dα Dα, where D j = i j, j is the partial derivative with respect to the j-th variable and i is the imaginary unit. Let us fix m with components m 1, m 2,..., m is the set of all Support of the Research Project MSM/ of Ministry of Education of the Czech Republic is gratefully acknowledged. 1

2 positive integers) in such a way that After all, we can assume that α M = α : m M = { α k=1 α k m k 1. 0 ; α : m 1}, where 0 is the set of all nonnegative integers.) Put 2) m = max{m k ; 1 k } and define for x = x 1, x 2,..., x ) and y = y 1, y 2,..., y ) 3) ϱ m x, y) = max { x k y k m k/m ; 1 k }. Then ϱ m is a metric on. If λ is the Lebesgue measure in and 4) B r x, ϱ m ) = { y ; ϱ m x, y) r } is the closed ball centred at x of radius r 0, then and B r x, ϱ m ) = [x 1 r m/m1, x 1 + r m/m1 ] [x 2 r m/m2, x 2 + r m/m2 ] 5) λ Br x, ϱ m ) ) = 2 r mb, where b = In what follows b will always have this meaning.) For L we denote by 6) diaml, ϱ m ) = sup { ϱ m x, y); x, y L }... [x r m/m, x + r m/m ] k=1 1 m k. the diameter of the set L and for γ + + is the set of all nonnegative numbers in ) we define the outer anisotropic Hausdorff γ-dimensional measure of the set L by setting H γ, ϱ m ) = 0 2

3 and for L 7) H γ L, ϱ m ) = sup inf ε>0 { diam γ L n, ϱ m ); L n n } L n, 0 diaml n, ϱ m ) ε. The distance of x by from L with respect to the metric ϱ m will be denoted 8) ϱ m x, L) = inf { ϱ m x, y); y L }. For ε > 0 and L put 9) L ε = { x ; ϱ m x, L) < ε }. For compact denote by ε, ϱ m ) the minimal number of balls B ε x, ϱ m ) with x sufficient for covering. The lower γ-dimensional Minkowski s content of is given by 10) M γ, ϱ m ) = lim inf ε, ϱ m ) ε γ, the upper γ-dimensional Minkowski s content of is given by 11) M γ, ϱ m ) = lim sup ε, ϱ m ) ε γ. Further put 12) M ϱm -dim = inf { γ > 0; M γ, ϱ m ) = 0 }, which is the so-called Minkowski s dimension of corresponding to the metric ϱ m. "!$#&%' 1. There exist constants c > 0, d > 0 depending only on ) such that for each compact )*%+,+.- c M γ, ϱ m ) lim inf c M γ, ϱ m ) lim sup M ϱm -dim mb. λ ε ) ε mb γ d M γ, ϱ m ), λ ε ) ε mb γ d M γ, ϱ m ),. Given ε > 0, observe that ε, ϱ m ) closed balls of radius 2ε suffice to cover ε. Hence lim inf λ ε ) / ε mb γ M γ, ϱ m ) 2 mb+, lim sup λ ε ) / ε mb γ M γ, ϱ m ) 2 mb+ ; 3

4 we see that it is sufficient to set d = 2 mb+. Consider the map π ε : given by π ε x 1, x 2,..., x ) = x 1 ε 1 m/m1, x 2 ε 1 m/m2,..., x ε 1 m/m ). For fixed z, π ε Bε z, ϱ m ) ) is the cube centred at π ε z) of side length 2 ε m/m1 ε 1 m/m1 = 2ε and for any compact π ε ) π ε ε ) π ε Bε z, ϱ m ) ). ext, we use the following consequence of the Morse-Besicovitch covering theorem cf. [2], 1, chap. 1, Th. 1.1): There exists a natural number γ) with the following property: If A is a bounded set in and if with each x A associate a closed euclidean cube Bx) centred at x, then there are finite sets such that z A 1, A 2,..., A γ) A A γ) x A i Bx), where for each fixed i {1, 2,..., γ)} we have Bx 1 ) Bx 2 ) = whenever x 1, x 2 A i, x 1 x 2. Put A = π ε ). There exist A 1, A 2,..., A γ) π ε ) such that π ε ) γ) π ε π εz) A i Bε z, ϱ m ) ) γ) [so B ε z, ϱ m ) for S i = πε 1A i)] and B ε z 1, ϱ m ) B ε z 2, ϱ m ) = z S i whenever z 1, z 2 S i, z 1 z 2. If cardm) means the number of elements in M, then 4 γ) ε, ϱ m ) card S i ).

5 If B denotes the interior of B, then we have Hence 2 ε mb ε, ϱ m ) 2 ε mb card = γ) γ) S i ) γ) ) λ Bε z, ϱ m) z S i z S i λ B ε z, ϱ m ) ) γ) λ ε ) = γ)λ ε ). ε γ ε, ϱ m ) ε γ γ) 2 ε mb λ ε ) = γ) 2 λ ε ) / ε mb γ. It is sufficient to put c = 2 /γ) to obtain c M γ, ϱ m ) lim inf λ ε ) / ε mb γ, c M γ, ϱ m ) lim sup λ ε ) / ε mb γ. We conclude that M γ, ϱ m ) = 0 for γ > mb, whence M ϱm -dim mb, which proves Remark 1. Lemma 1. Suppose that is compact and 0 q < mb. If then for each ε > 0. )*%+,+.- M ϱm -dim < mb q, ε ϱ m x, ) q dx < +. A similar reasoning occurs, e.g., in [5].) As M ϱm -dim < mb q, there exists a κ [0, mb q[ such that M κ, ϱ m ) = 0. For all sufficiently small ε > 0 we have λ ε ) c ε mb κ for a suitable c ]0, + [. Since mb κ > q 0, it follows that lim ε mb κ = 0 and so λ ) lim λ ε ) = 0. 5

6 { B rz z, ϱ m ) } z Q G\F F Hence λ ) = 0. Setting U j = { x ; 2 j 1 ε ϱ m x, ) < 2 j ε }, we get ε \ = Consequently, ϱ m x, ) q dx = ε + j=0 + j=0 U j ϱ m x, ) q dx, ϱ m x, ) q dx ε q 2 j+1)q λ 2 j ε) ε q 2 j+1)q c 2 j ε ) mb κ U j = c ε mb κ q 2 j+1)q jmb+jκ = c ε mb κ q 2 q 2 jq mb+κ). As κ mb + q < 0, we see that the series + j=0 U j. U j ϱ m x, ) q dx converges, whence ε ϱ m x, ) q dx < +. A simple compactness argument yields the next lemma. Lemma 2. Suppose that G is open, F G is relatively closed in G, g : ]0, + [ ]0, + [ and u: G \ F / are functions such that ux) = O gϱ m x, F )) ) as x z, x G \ F, for all z F G \ F. Then for each compact Q G there exist positive constants c, ε such that for all x Q \ F with ϱ m x, F ) < ε we have ux) cg ϱm x, F ) ). )*%+,+.-. With each z Q G \ F F we associate constants r z > 0 and c z > 0 such that ux) cz g ϱ m x, F ) ) for all x G\F with ϱ m x, z) < r z. We may clearly suppose that r z has been chosen small enough to get B rz z, ϱ m ) G. The open cover 6

7 of the compact Q G \ F F contains a finite subcover {Br zi z i, ϱ m )} n : n F G \ F Q Br zi z i, ϱ m ). We assert that there is an ε > 0 such that 13) n x Q, 0 < ϱm x, F ) < ε) = x Br zi z i, ϱ m ). Admitting the contrary we would get, for each choice of ε = 1/k k ), a point x k G \ F with ϱ m x k, F ) < 1 n k, x k Q \ Br zi z i, ϱ m ). Passing to a convergent subsequence we would get lim x k = z. Obviously z Q k + F G \ F, consequently z Br zi z i, ϱ m ) for a suitable i {1, 2,..., n} and so x k Br zi z i, ϱ m ) for all sufficiently large k, which contradicts x k Q \ n Br zi z i, ϱ m ). Thus 13) is established for a suitable ε > 0. We may clearly suppose that ε min{r zi } n. Put c = max{c z i } n. If x Q \ F with ϱ m x, F ) < ε, then 13) tells us that, for a suitable i {1, 2,..., n}, ux) c zi g ϱ m x, F ) ) c g ϱ m x, F ) ), which proves the lemma. / Theorem 1. Let G be an open set, F G a relatively closed subset in G, u: G \ F a locally integrable function. Suppose that 14) M ϱm -dim Q < mb q for each compact Q F and ux) = O ϱ m x, F ) q) as x z, x G \ F, for each z F, where 0 q < mb [cf. 5)]. Then u is defined a.e. in G [i.e. λ F ) = 0] and it is locally integrable in G. If, moreover, γ mb 1) q 0 7

8 and M γ Q, ϱ m ) = 0 for each compact Q F, then the validity of the equation P D)u = 0 in G \ F in the sense of distributions) implies P D)u = 0 in the whole G. )*%+,+.-. In the proof of Lemma 1 we have seen that 14) implies λ Q) = 0. Since this is assumed for each compact Q F, we have λ F ) = 0 so that u is defined λ -a.e. in G. We have to verify that the growth estimate of u implies that u is integrable in a neighbourhood of any z G. It is sufficient to consider z F. According to Lemma 2, for each ball B r z, ϱ m ) G there exist constants c > 0, ε ]0, r] such that x Br z, ϱ m ) \ F, ϱ m x, F ) < ε ) = ux) c ϱm x, F ) q. We are going to show that Lemma 1 implies integrability of the function x ϱ m x, F ) q on a neighbourhood of z. Choosing δ ]0, r/2[, we have y Br z, ϱ m ) \ F, ϱ m y, z) δ ) = ϱ m y, B2δ z, ϱ m ) F ) = ϱ m y, F ). For these y and we get Q = B 2δ z, ϱ m ) F ϱ m y, Q) q = ϱ m y, F ) q. According to Lemma 1, the function y ϱ m y, Q) q is integrable on a neighbourhood of z. Thus the integrability of the function x ϱ m x, F ) q as well as of ux) on a neighbourhood of z is verified. Suppose now that P D)u = 0 on G \ F. DG) will denote the class of all infinitely differentiable functions with a compact support in G. Choose an arbitrary ϕ DG) and denote its support spt ϕ by. We are going to show that 15) P D)u, ϕ = 0. 8

9 0 This is obvious if F =. So let F. Fix an infinitely differentiable function ψ on such that spt ψ [ 1, 1], ψt) dt = 1. For L, χ L will denote the characteristic function of the set L, L ε for ε > 0 [as defined by 9)] is clearly an open set. Using the notation from 2), 5) we put for x and ε > 0 16) ψ ε x) = ε mb 0 χ F2ε y) ψ x j y j )ε ) m/mj dy 1... dy. j=1 Then ψ ε is an infinitely differentiable function on [3]). After substitution compare Chap. I, Section 5 in x j y j )ε m/mj = z j j = 1, 2,..., ) we get y j = x j ε m/mj z j, so that ψ ε x) = = 0 χ F2ε..., x j ε m/mj z j,...) ψz j ) dz 1... dz j=1 χ F2ε..., x j ε m/mj z j,...) ψz j ) dz 1... dz. {z 0 ; z j [ 1,1],1 j } j=1 Hence 17) ψε x) [ 1,1] j=1 ψzj ) dz1... dz n C 0, x. If x F ε then..., x j ε m/mj z j,...) F 2ε for z [ 1, 1] so that 18) ψ ε x) = ψz j ) dz 1... dz = [ 1,1] 1 j=1 j=1 1 1 ψz j ) dz j = 1, x F ε. If x \ F 3ε then..., x j ε m/mj z j,...) \ F 2ε so that χ F2ε..., x j ε m/mj z j,...) = 0, z [ 1, 1]. Hence 19) ψ ε x) = 0 for each x \ F 3ε. 9

10 For any multiindex α = α 1, α 2,..., α ) we get by differentiating under the sign of the integral D α ψ ε x) = 1 ε mα:m) i α [ 1,1] χ F2ε..., x j ε m/mj z j,...) ψ αj) z j ) dz 1... dz, j=1 20) D α ψ ε x) ε mα:m) [ 1,1] j=1 = C α ε mα:m), x ψ αj ) z j ) dz 1... dz. If T is a distribution on G we denote by spt T its support. As ψ ε = 1 in a neighbourhood of spt P D)u F we have P D)u, ϕ = P D)u, ψε ϕ = u, t P D)ψ ε ϕ), where t P D) is the formal adjoint of P D) cf. p. 40 in [4]); t P D)ψ ε ϕ) = α:m 1 ϕ α D α ψ ε, where ϕ α are certain functions in DG) independent of ε which depend on ϕ and the coefficients a ω of the operator P D) only; they have the form of a finite sum of the type ϕ α = c α βδω Dβ a ω D δ ϕ, where c α βδω are constants so that spt ϕ α spt ϕ. We have 21) P D)u, ϕ = α:m 1 u, ϕ α D α ψ ε. Fix an ε 0 ]0, 1[ small enough to have ε0 G. We will consider ε ]0, ε 0 ]. For α = 0 ) we have 22) u, ϕ0 D 0 ψ ε F 3ε G sup ϕ 0 u ϕ 0 ψ ε dλ F 3ε u dλ 0 as ε 0, because u is locally integrable in G, F 3ε F as ε 0 and λ F ) = 0. 10

11 For α > 0 we make use of the equality D α ψ ε = 0 on F ε to get the estimate u, ϕα D α ψ ε sup ϕ α ux) dx Cα ε mα:m) ; \F ε) F 3ε according to Lemma 2 there are constants c and ε 1 > 0 such that ux) c ϱ m x, F ) q c ε q, x \ F ε ) F ε1, so that for 3ε < ε 1 we get u, ϕα D α ψ ε sup ϕ α C α ε mα:m) c ϱ m x, F ) F q dx 3ε \F ε sup ϕ α C α ε mα:m) c ε q λ F 3ε ). For ε < ε 0 /3 we have F 3ε F 3ε ) 3ε F 3ε ) 3ε. Indeed, if y F 3ε, then there exists a z F such that ϱ m y, z) < 3ε, whence z F 3ε and, consequently, Assuming we arrive at y F 3ε ) 3ε F ε0 )3ε. 0 < ε < minε 0, ε 1 ) 3 < 1 u, ϕ α D α ψ ε sup ϕ α C α c ε mα:m) q ) λ F ε0 ) 3ε Lα) ε m q λf ε0 ) 3ε ) 3ε) mb γ ε mb γ where Q = F ε0 = Lα) ε mb 1) q mb 1)+q λq 3ε ) 3ε) mb γ, F is compact and Lα) = C α c sup ϕ α 3 mb γ. Hence u, ϕα D α ψ ε Lα) λ Q 3ε ) 3ε) mb γ for 0 < α : m 1. 11

12 By virtue of 21) we get P D)u, ϕ u, ϕ0 D 0 ψ ε + λ Q 3ε ) 3ε) mb γ 0<α:m 1 Lα). Passing to lim inf we obtain in view of 22) P D)u, ϕ 0 + Mγ Q) 0<α:m 1 Lα) = 0. Thus 15) is verified and the proof is complete. The following simple lemma plays the role similar to that of Lemma 2. Lemma 2. Let G be open and F G relatively closed in G. Suppose that u: G \ F / and g : ]0, + [ ]0, + [ are functions satisfying 23) ux) = o gϱ m x, F )) ) as x z, x G \ F for each z F G \ F. Then for each compact H F G \ F there exist δ > 0 and a nondecreasing function f : ]0, + [ ]0, + [ such that, in the notation of 9), 24) H δ G, x H δ \ F = ux) f ϱm x, F ) ) g ϱ m x, F ) ), f0+) lim t 0 ft) = 0. )*%+,+.-. Choose an ε 0 > 0 small enough to have H ε0 G. According to Lemma 2, given a compact Q = H ε0, there exist c > 0 and ε > 0 such that 25) x Q \ F, ϱm x, F ) < ε ) = ux) c. gϱ m x, F )) Fix a δ ]0, ε[ and define for t > 0 { ux) } 26) ft) = sup gϱ m x, F )) ; x Q \ F, ϱ mx, F ) mint, δ). Then ft) c for t ]0, + [ by virtue of 25). Clearly, f is nondecreasing, for x H δ \ F Q \ F we have setting τ = ϱ m x, F ) 12 ux) gϱ m x, F )) fτ) = f ϱ m x, F ) )

13 so that ux) f ϱm x, F ) ) g ϱ m x, F ) ). We are going to verify that lim ft) = 0. Choose a decreasing sequence t k > 0 with t 0 lim t k = 0. Then k lim ft k) = inf ft) α. k t>0 If α > 0 then ft k ) > α/2 for each k and the definition of f would imply the existence of x k Q \ F with ϱ m x k, F ) t k such that 27) ux k ) gϱ m x k, F )) > α 2. Passing to a convergent subsequence we could achieve the existence of the limit lim k x k = z; obviously, z F G \ F and, in view of the assumption 23), ux k ) gϱ m x k, F )) 0, which contradicts 27). Thus 24) is verified. The following theorem is similar to Theorem 1. Theorem 2. Let G be an open set and let F G be relatively closed in G. Suppose that u: G \ F / is a locally integrable function satisfying ux) = o ϱ m x, F ) q) as x z, x G \ F for each z F, where 0 q < mb [cf. 5)]. If M ϱm -dim Q < mb q for each compact Q F, then u is defined a.e. in G and it is locally integrable in G. If, moreover, γ mb 1) q 0 and M γ Q, ϱ m ) < + for each compact Q F, then the validity of the equation P D)u = 0 in G \ F in the sense of distributions) implies P D)u = 0 in the whole G. 13

14 )*%+,+.-. We have seen in the proof of Theorem 1 that u is locally integrable in G. Choose an arbitrary ϕ DG). We are going to verify 15). Put = spt ϕ, which is a compact set. It is again sufficient to consider the case when F. As in the proof of Theorem 1 we construct for each ε > 0 an auxiliary function ψ ε. We have D α ψ ε x) C α ε mα:m) for each multiindex α. Besides that, ψ ε x) = 1, x F ε, ψ ε x) = 0, x \ F 3ε. As ψ ε = 1 on a neighbourhood of F spt ϕ spt P D)u we get P D)u, ϕ = P D)u, ψε ϕ = u, t P D)ψ ε ϕ), which implies 21) with ϕ α DG) independent of ε with spt ϕ α. For α = 0 ) we obtain as in the proof of Theorem 1 28) u, ϕ0 D 0 ψ ε sup ϕ 0 u dλ u dλ = 0 F 3ε F as ε 0. We will consider ε ]0, ε 0 ], where ε 0 ]0, 1[ has been chosen small enough to satisfy G. We have seen that ε0 29) F 3ε F ε0 ) For 0 < α : m 1 we get from 18) 3ε for 0 < ε < ε 0 /3. D α ψ ε = 0 on F ε and, in view of 19), 20), u, ϕα D α ψ ε sup ϕ α C α ε mα:m) u dλ. F 3ε \F ε Putting H = F ε0 we get from 29) for ε ]0, ε 0 /3[ u dλ u dλ. F 3ε \F ε H 3ε\F ε According to Lemma 3 there exist a δ > 0 and a nondecreasing function 14 f : ]0, + [ ]0, + [

15 such that H δ G and 24) holds together with x H δ \ F = ux) f ϱm x, F ) ) ϱ m x, F ) q. Assuming 3ε < δ we have for x H 3ε estimates ϱ m x, F ) 3ε and u dλ f3ε) ϱ m x, F ) q dλ x). H 3ε\F ε H 3ε\F ε We see that for sufficiently small ε ]0, 1[ where so that u, ϕα D α ψ ε sup ϕ α C α ε m f3ε) ϱ m x, F ) q dλ x) H 3ε\F ε 0<α:m 1 sup ϕ α C α ε m q f3ε)λ H 3ε ) = λ H 3ε ) 3ε) mb γ f3ε)e α, E α = C α sup ϕ α 3 mb γ, u, ϕα D α ψ ε f3ε) λ F 3ε ) As M γ H, ϱ m ) < + we obtain from 24), 28) whence lim inf α:m 1 and the proof is complete. "!$#&%' 3ε) mb γ 0<α:m 1 u, ϕα D α ψ ε f0+)mγ H, ϱ m ) P D)u, ϕ lim inf α:m 1 0<α:m 1 u, ϕ α D α ψ ε = 0 E α. E α = 0, 2. Let G be an open set, F G a relatively closed subset in G. We have seen that the assumption M ϱm -dim Q < mb q 0 q < mb) for each compact Q F guarantees that λ F ) = 0 and each function u locally integrable in G \ F satisfying 30) ux) = O ϱ m x, F ) q) as x z, x G \ F, z F, 15

16 is locally integrable in G. We say that F is P D)-removable for functions u locally integrable in G \ F satisfying 30), if each such function u fulfilling P D)u = 0 in G \ F in the sense of distributions) fulfils P D)u = 0 in the whole G. The previous theorems describe in terms of Minkowski s contents sufficient conditions for the P D)-removability of F for functions u satisfying 30) or the growth estimate 31) ux) = o ϱ m x, F ) q) as x z, x G \ F, z F. It turns out that for certain operators P D) it is possible to use Hausdorff measures for obtaining necessary conditions for the P D)-removability of F. ow we are going to consider operators of the form 32) P D) = α:m 1 a α D α, whose coefficients a α are complex constants. We associate with such an operator 32) the polynomial P m ξ) = a α ξ α in real variables ξ 1, ξ 2,..., ξ. P D) is termed semielliptic, if α:m=1 ξ, P m ξ) = 0 ) = ξ = 0 ), which means that P m ξ) has no nontrivial zeros in then uniquely determined cf. [14].). The corresponding m is Theorem 3. Suppose that G, F have the meaning described above. Let P D) be a semielliptic operator. Assume that [cf. 5)] b > 1, q mb 1), γ mb 1) q 0. For F to be P D)-removable for all locally integrable functions u in G \ F satisfying 30) it is necessary that 33H ) H γ F, ϱ m ) = 0 and sufficient that 33 M ) M γ Q, ϱ m ) = 0 for each compact Q F. 16

17 1 1 For the P D)-removability of F for all locally integrable functions u satisfying the growth condition 31) it is necessary that F have a σ-finite H γ -measure, and sufficient that M γ Q, ϱ m ) < + for each compact Q F. The proof follows from a combination of Theorem 1, Theorem 2 above and Theorem 5 in [6]. We should point out that M γ denotes the upper Minkowski s content in [6] so that this Theorem 3 is more general than the corresponding result in [6]. An important example is given by the heat conduction operator = n n+1 in n+1. This operator is semielliptic with the choice m j = 2 for 1 j n, m n+1 = 1 so that m = 2,..., 2, 1) n+1, m = 2 and the corresponding metric ϱ ϱ m is given by ϱx, y) = max { x 1 y 1,..., x n y n, x n+1 y n+1 1/2} for x = x 1,..., x n+1 ), y = y 1,..., y n+1 ). ow we will consider an open set G n+1 and a relatively closed subset F G, where 34) M ϱ -dim Q < n + 2 q, γ = n q for each compact Q F. A function u, which is of the class C 2) on an open set U on U, if it satisfies u = 0 on U. n+1, is termed caloric Theorem 4. In order that each caloric function u on G \ F satisfying 35) ux) = O ϱx, F ) q) as x z, x G \ F, z F be extendable to a caloric function on the whole G it is necessary that 36) H γ F, ϱ) = 0 and it is sufficient that 37) M γ Q, ϱ) = 0 for each compact Q F. 17

18 If we replace 35) by 38) ux) = o ϱx, F ) q) as x z, x G \ F, z F, then the assertion remains valid with 36) replaced by the requirement of σ-finiteness of H γ F, ϱ) and the assumption 37) replaced by the requirement M γ Q, ϱ) < + for each compact Q F. The proof follows from Theorem 3. Indeed, if u is a caloric function on G \ F satisfying 35) then, according to Theorem 3, under the condition 34), u is defined a.e. in G, it is locally integrable and satisfies 1 u = 0 in the sense of distributions. Since the operator 1 is semielliptic, there exists ũ C G) satisfying 1 ũ = 0 on G in the classical sense such that ũ = u a.e. on G cf. [14]), whence ũ = u on G\F and ũ is the required extension. "!$#&%'32. Sufficient conditions for removability of singularities of caloric functions with anisotropic growth were described in terms of the upper anisotropic Minkowski s contents by H. Zlonická in [15]. Another proof in [7] based on anisotropic Whitney s decomposition) showed that the lower Minkowski s content is sufficient for this purpose. Application of the upper Minkowski s content derived from the euclidean metric) in this context goes back to Bochner compare [1], [4]). Later Polking pointed out in [11] that a modification of Bochner s proof makes it possible to replace the upper Minkowski s content by the lower Minkowski s content; his method is also described in [13]. Theorems of the Bochner type for operators on manifolds, where the singular set is formed by a submanifold of a suitable dimension, have been studied by Sugimoto and Uchida in [12]. Various applications of anisotropic metrics and anisotropic Minkowski s contents in connection with removability of singularities are described by Littman in [8]. In his thesis [10] M. Píštěk described in terms of the upper anisotropic Minkowski s content) removable singularities of caloric functions which are locally integrable in power p with a weight depending on the distance from the singular set measured by the caloric metric. It is natural to ask whether in his investigation the upper Minkowski s content could also be replaced by the lower content of the same dimension. 18

19 References [1] S. Bochner: Weak solutions of linear partial differential equations. J. Math. Pures Appl ), [2] M. de Guzmán: Differentitation of Integrals in 4 n. Springer, Berlin, [3] S. Fučík, O. John, A. ufner: Function Spaces I. SP, Praha, In Czech.) [4] R. Harvey, J. Polking: Removable singularities of solutions of linear partial differential equations. Acta Mathematica ), [5] P. oskela: Removable singularities for analytic functions. Michigan Math. J ), [6] J. rál: Removability of singularities in potential theory. Potential Analysis ), [7] J. rál, Jr.: Removable singularities for harmonic and caloric functions. Thesis, FJFI ČVUT, Praha, In Czech.) [8] W. Littman: Polar sets and removable singularities of partial differential equations. Ark. Mat ), 1 9. [9] O. Martio, M. Vuorinen: Whitney cubes, p-capacity and Minkowski content. Expo. Math ), [10] M. Píštěk: Removable singularities of solutions of partial differential equations. Thesis, MFF U, Praha, In Czech.) [11] J. C. Polking: A survey of removable singularities. Seminar onlinear Partial Differential Equations, ew York, 1987, pp [12] M. Sugimoto, M. Uchida: A generalization of Bochner s extension theorem and its application. Ark. Mat ), [13].. Tarkhanov: The Analysis of Solutions of Elliptic Equations. luver Academic Publishers, [14] J. F. Trèves: Lectures on Linear Partial Differential Equations with Constant Coefficients. Rio de Janeiro, [15] H. Zlonická: Singularity of solutions of partial differential equations. Thesis, MFF U, Praha, In Czech.) Authors addresses: Miroslav Dont, Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, Technická 2, Praha, Czech Republic, dont@math.feld.cvut.cz; Josef rál, Palackého 500, Pečky, Czech Republic. 19

NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES

NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES JUHA LEHRBÄCK Abstract. We establish necessary conditions for domains Ω R n which admit the pointwise (p, β)-hardy inequality u(x) Cd Ω(x)