Quantitative uniqueness for the power of Laplacian with singular coefficients

Quantitative uniqueness for the power of Laplacian with singular coefficients

Ching-Lung Lin

Sei Nagayasu

Jenn-Nan Wang

Abstract

In this paper we study the local behavior of a solution to the lth power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.

1 Introduction

Assume that Ω is a connected open set containing 0 in Rⁿ for n ≥ 2. In this paper we are interested in the local behavior of u satisfying the following differential inequality:

|∆^lu| ≤ K₀P

|α|≤l−1|x|^−2l+|α||D^αu| + K₀P[3l/2]

|α|=l|x|^−2l+|α|+|D^αu|, (1.1) where 0 <  < 1/2 and [h] = k ∈ Z when k ≤ h < k + 1. For (1.1), a strong unique continuation was proved by the first author [8]. A similar result for the power of Laplacian with lower derivatives up to l-th order can be found in [5]. On the other hand, a unique continuation property for the l-th power

∗Department of Mathematics, National Chung Cheng University, Chia-Yi 62117, Tai- wan. Email:cllin@math.ccu.edu.tw

†Department of Mathematics, Taida Institute for Mathematical Sciences, National Tai- wan University, Taipei 106, Taiwan. Email:nagayasu@math.ntu.edu.tw

‡Department of Mathematics, Taida Institute of Mathematical Sci- ences, NCTS (Taipei), National Taiwan University, Taipei 106, Taiwan.

Email:jnwang@math.ntu.edu.tw

of Laplacian with the same order of lower derivatives as in (1.1) was given in [11]. Those results mentioned above concern only the qualitative behavior of the solution. In other words, they showed that if u vanishes at 0 in infinite order or u vanishes in an open subset of Ω, then u must vanishes identically in Ω. The aim of this paper is to study the strong unique continuation from a quantitative viewpoint. Namely, we are interested in the maximal vanishing order at 0 of any nontrivial solution to (1.1). It is worth mentioning that quantitative estimates of the strong unique continuation are useful in studying the nodal sets of eigenfunctions [2], or solutions of second order elliptic equations [6], [10], or the inverse problem [1].

Perhaps, for the quantitative uniqueness problem, the most popular tech- nique, introduced by Garofala and Lin [3], [4], is to use the frequency function related to the solution. This method works quite efficiently for second or- der strongly elliptic operators. However, this method can not be applied to (1.1). Another method to derive quantitative estimates of the strong unique continuation is based on Carleman estimates, which was first initiated by Donnelly and Fefferman [2] where they studied the maximal vanishing order of the eigenfunction with respect to the corresponding eigenvalue on a com- pact smooth Riemannian manifold. Their method does not work for (1.1) either.

Recently, the first and third authors and Nakamura [9] introduced a method based on appropriate Carleman estimates to prove a quantitative uniqueness for second order elliptic operators with sharp singular coefficients in lower order terms. A key strategy of our method is to derive three-sphere inequalities and then apply them to obtain doubling inequalities and the maximal vanishing order. Both steps require delicate choices of cut-off func- tions. Nevertheless, this method is quite versatile and can be adopted to treat many equations or even systems. The present work is an interesting application of the ideas of [9] to the lth power of Laplacian with singular coefficients. The power l = 2 is the most interesting and useful case. It cor- responds to the biharmonic operator with third order derivatives. Our work provides a quantitative estimate of the strong unique continuation for this equation. To our best knowledge, this quantitative estimate has not been derived before.

We now state main results of the paper. Assume that B_R⁰

0 ⊂ Ω for some R⁰₀ > 0.

Theorem 1.1 There exists a positive number ˜R₀ < e^−1/2 such that if 0 <

r₁ < r₂ < r₃ ≤ R⁰₀ and r₁/r₃ < r₂/r₃ < ˜R₀, then Z

|x|<r2

|u|²dx ≤ C

Z

|x|<r1

|u|²dx

τZ

|x|<r3

|u|²dx

1−τ

(1.2)

for u ∈ H^2l(B_R⁰

0) satisfying (1.1) in B_R⁰

0, where C and 0 < τ < 1 depend on r₁/r₃, r₂/r₃, n, l, and K₀.

Remark 1.1 From the proof, the constants C and τ can be explicitly written as C = max{C₀(r₂/r₁)ⁿ, exp(Bβ₀)} and τ = B/(A + B), where C₀ > 1 and β0 are constants depending on n, l, K0 and

A = A(r₁/r₃, r₂/r₃) = (log(r₁/r₃) − 1)²− (log(r₂/r₃))², B = B(r₂/r₃) = −1 − 2 log(r₂/r₃).

The explicit forms of these constants are important in the proof of Theo- rem 1.2.

Theorem 1.2 Let u ∈ H_loc^2l(Ω) be a nonzero solution to (1.1). Then we can find a constant R₂ (depending on n, l, , K₀) and a constant m₁ (depending on n, l, , K₀, kuk_L2(|x|<R₂²)/kuk_L2(|x|<R⁴₂)) satisfying

lim sup

R→0

1 R^m1

Z

|x|<R

|u|²dx > 0. (1.3)

Theorem 1.3 Let u ∈ H_loc^2l(Ω) be a nonzero solution to (1.1). Then there exist positive constants R₃ (depending on n, l, , K₀) and C₃ (depending on n, l, , K0, m1) such that if 0 < r ≤ R3, then

Z

|x|≤2r

|u|²dx ≤ C₃ Z

|x|≤r

|u|²dx, (1.4)

where m₁ is the constant obtained in Theorem 1.2.

The rest of the paper is devoted to the proofs of Theorem 1.1-1.3.

2 Three-sphere inequalities

We will prove Theorem 1.1 in this section. To begin, we recall a Carleman estimate with weight ϕ_β = ϕ_β(x) = exp(^β₂(log |x|)²) given in [8].

Lemma 2.1 [8, Corollary 3.3] There exist a sufficiently large number β₀ > 0 and a sufficiently small number r₀ > 0, depending on n and l, such that for all u ∈ U_r0 with 0 < r₀ < e⁻¹, β ≥ β₀, we have that

P

|α|≤2lβ^3l−2|α|R ϕ²_β|x|^2|α|−n(log |x|)^2l−2|α||D^αu|²dx

≤ ˜C₀R ϕ²_β|x|^4l−n|∆^lu|²dx, (2.1) where U_r0 = {u ∈ C₀^∞(Rⁿ \ {0}) : supp(u) ⊂ B_r0} and ˜C₀ is a positive constant depending on n and l. Here e = exp(1).

Remark 2.1 The estimate (2.1) in Lemma 2.1 remains valid if we assume u ∈ H_loc^2l(Rⁿ \ {0}) with compact support. This can be easily obtained by cutting off u for small |x| and regularizing.

We first consider the case where 0 < r₁ < r₂ < R < 1/e and B_R ⊂ Ω.

The constant R will be chosen later. To use the estimate (2.1), we need to cut-off u. So let ξ(x) ∈ C₀^∞(Rⁿ) satisfy 0 ≤ ξ(x) ≤ 1 and

ξ(x) =







0, |x| ≤ r₁/e, 1, r₁/2 < |x| < er₂, 0, |x| ≥ 3r₂.

It is easy to check that for all multiindex α

(|D^αξ| = O(r₁^−|α|) for all r1/e ≤ |x| ≤ r1/2

|D^αξ| = O(r₂^−|α|) for all er2 ≤ |x| ≤ 3r2. (2.2) On the other hand, repeating the proof of Corollary 17.1.4 in [7], we can show that

Z

a1r<|x|<a2r

||x|^|α|D^αu|²dx ≤ C⁰ Z

a3r<|x|<a4r

|u|²dx, |α| ≤ 2l, (2.3) for all 0 < a₃ < a₁ < a₂ < a₄ such that B_a4r ⊂ Ω, where the constant C⁰ is independent of r and u.

Noting that the commutator [∆^l, ξ] is a 2l − 1 order differential operator.

Applying (2.1) to ξu and using (1.1), (2.2), (2.3) implies X

|α|≤2l

β^3l−2|α|

Z

r1/2<|x|<er2

ϕ²_β|x|^2|α|−n(log |x|)^2l−2|α||D^αu|²dx

≤ X

|α|≤2l

β^3l−2|α|

Z

ϕ²_β|x|^2|α|−n(log |x|)^2l−2|α||D^α(ξu)|²dx

≤ C˜₀ Z

ϕ²_β|x|^4l−n|∆^l(ξu)|²dx

≤ 2 ˜C₀ Z

ϕ²_β|x|^4l−nξ²(K₀ X

|α|≤l−1

|x|^−2l+|α||D^αu| + K₀

[3l/2]

X

|α|=l

|x|^−2l+|α|+|D^αu|)²dx

+2 ˜C₀ Z

ϕ²_β|x|^4l−n

[∆^l, ξ]u 

2dx

≤ C˜₁nZ

r1/2<|x|<er2

ϕ²_β( X

|α|≤l−1

|x|^2|α|−n|D^αu|²+

[3l/2]

X

|α|=l

|x|^2|α|−n+2|D^αu|²)dx

+ Z

r1/e<|x|<r1/2

ϕ²_β X

|α|≤2l−1

|x|^2|α|−n|D^αu|²dx

+ Z

er2<|x|<3r2

ϕ²_β X

|α|≤2l−1

|x|^2|α|−n|D^αu|²dxo

≤ C˜₂nZ

r1/2<|x|<er2

ϕ²_β( X

|α|≤l−1

|x|^2|α|−n|D^αu|²+

[3l/2]

X

|α|=l

|x|^2|α|−n+2|D^αu|²)dx

+r⁻ⁿ₁ ϕ²_β(r₁/e) Z

r1/e<|x|<r1/2

X

|α|≤2l−1

||x|^|α|D^αu|²dx

+r⁻ⁿ₂ ϕ²_β(er₂) Z

er2<|x|<3r2

X

|α|≤2l−1

||x|^|α|D^αu|²dxo

≤ C˜₃nZ

r1/2<|x|<er2

ϕ²_β( X

|α|≤l−1

|x|^2|α|−n|D^αu|²+

[3l/2]

X

|α|=l

|x|^2|α|−n+2|D^αu|²)dx

+r⁻ⁿ₁ ϕ²_β(r₁/e) Z

r1/4<|x|<r1

|u|²dx + r₂⁻ⁿϕ²_β(er₂) Z

2r2<|x|<4r2

|u|²dxo ,

(2.4)

where ˜C₁, ˜C₂, and ˜C₃ are independent of r₁, r₂, and u.

We now choose r₀ < e^−−1([3l/2]−l)−1 small enough such that ((log(er₀))⁻² ≤ _{2 ˜C}¹

3

(er₀)^2(log(er₀))2([3l/2]−l) ≤ ¹

2 ˜C3.

Letting R ≤ r₀ and β ≥ β₀ ≥ max{2 ˜C₃, 1}, we can absorb the integral over r₁/2 < |x| < er₂ on the right side of (2.4) into its left side to obtain

Z

r1/2<|x|<er2

ϕ²_β( X

|α|≤l−1

|x|^2|α|−n|D^αu|²+

[3l/2]

X

|α|=l

|x|^2|α|−n+2|D^αu|²)dx

≤ C˜₄n

r₁⁻ⁿϕ²_β(r₁/e) Z

r1/4<|x|<r1

|u|²dx + r⁻ⁿ₂ ϕ²_β(er₂) Z

2r2<|x|<4r2

|u|²dxo , (2.5) where ˜C₄ = 1/ ˜C₃. Using (2.5) we have that

r₂⁻ⁿϕ²_β(r₂) Z

r1/2<|x|<r2

|u|²dx

≤ Z

r1/2<|x|<er2

ϕ²_β|x|⁻ⁿ|u|²dx

≤ C˜₄n

r⁻ⁿ₁ ϕ²_β(r₁/e) Z

r1/4<|x|<r1

|u|²dx + r₂⁻ⁿϕ²_β(er₂) Z

2r2<|x|<4r2

|u|²dxo . (2.6) Dividing r⁻ⁿ₂ ϕ²_β(r₂) on the both sides of (2.6) implies

Z

r1/2<|x|<r2

|u|²dx

≤ C˜₄n

(r₂/r₁)ⁿ[ϕ²_β(r₁/e)/ϕ²_β(r₂)]

Z

r1/4<|x|<r1

|u|²dx +[ϕ²_β(er2)/ϕ²_β(r2)]

Z

2r2<|x|<4r2

|u|²dx o

≤ C˜₅n

(r₂/r₁)ⁿ[ϕ²_β(r₁/e)/ϕ²_β(r₂)]

Z

|x|<r1

|u|²dx +(r₂/r₁)ⁿ[ϕ²_β(er₂)/ϕ²_β(r₂)]

Z

|x|<4r₂

|u|²dxo

, (2.7)

where ˜C₅ = max{ ˜C₄, 1}. With such choice of ˜C₅, we can see that C˜₅(r₂/r₁)ⁿ[ϕ²_β(r₁/e)/ϕ²_β(r₂)] > 1

for all 0 < r₁ < r₂. AddingR

|x|<r1/2|u|²dx to both sides of (2.7) and choosing r2 ≤ R = min{r0, 1/4}, we get that

Z

|x|<r2

|u|²dx

≤ 2 ˜C₅(r₂/r₁)ⁿ[ϕ²_β(r₁/e)/ϕ²_β(r₂)]

Z

|x|<r₁

|u|²dx +2 ˜C₅(r₂/r₁)ⁿ[ϕ²_β(er₂)/ϕ²_β(r₂)]

Z

|x|<1

|u|²dx. (2.8)

By denoting

A = β⁻¹ log[ϕ²_β(r1/e)/ϕ²_β(r2)] = (log r1− 1)²− (log r2)² > 0, B = −β⁻¹ log[ϕ²_β(er2)/ϕ²_β(r2)] = −1 − 2 log r2 > 0,

(2.8) becomes Z

|x|<r2

|u|²dx

≤ 2 ˜C₅(r₂/r₁)ⁿn

exp(Aβ) Z

|x|<r1

|u|²dx + exp(−Bβ) Z

|x|<1

|u|²dxo . (2.9) To further simplify the terms on the right hand side of (2.9), we consider two cases. If

Z

|x|<r₁

|u|²dx 6= 0 and

exp (Aβ0) Z

|x|<r₁

|u|²dx < exp (−Bβ0) Z

|x|<1

|u|²dx, then we can pick a β > β₀ such that

exp (Aβ) Z

|x|<r₁

|u|²dx = exp (−Bβ) Z

|x|<1

|u|²dx.

Using such β, we obtain from (2.9) that Z

|x|<r2

|u|²dx

≤ 4 ˜C₅(r₂/r₁)ⁿexp (Aβ) Z

|x|<r₁

|u|²dx

= 4 ˜C₅(r₂/r₁)ⁿ

Z

|x|<r1

|u|²dx

_A+B^B Z

|x|<1

|u|²dx

_A+B^A

. (2.10)

If Z

|x|<r₁

|u|²dx = 0, then it follows from (2.9) that

Z

|x|<r2

|u|²dx = 0

since we can take β arbitrarily large. The three-sphere inequality obviously holds.

On the other hand, if exp (−Bβ₀) Z

|x|<1

|u|²dx ≤ exp (Aβ₀) Z

|x|<r1

|u|²dx, then we have

Z

|x|<r2

|u|²dx

≤

Z

|x|<1

|u|²dx

_A+B^B Z

|x|<1

|u|²dx

_A+B^A

≤ exp (Bβ₀)

Z

|x|<r1

|u|²dx

_A+B^B Z

|x|<1

|u|²dx

_A+B^A

. (2.11)

Putting together (2.10), (2.11), and setting ˜C₆ = max{4 ˜C₅(r₂/r₁)ⁿ, exp (Bβ₀)}, we arrive at

Z

|x|<r₂

|u|²dx ≤ ˜C₆

Z

|x|<r₁

|u|²dx

_A+B^B Z

|x|<1

|u|²dx

_A+B^A

. (2.12)

Now for the general case, we take ˜R₀ = R and consider 0 < r₁ < r₂ < r₃ with r₁/r₃ < r₂/r₃ ≤ ˜R₀. By scaling, i.e. defining bu(y) := u(r₃y), we derive from (2.12) that

Z

|y|<r2/r3

|u|b²dy ≤ C(

Z

|y|<r1/r3

|u|b²dy)^τ( Z

|y|<1

|bu|²dy)^1−τ, (2.13) where τ = B/(A + B) with

A = A(r1/r3, r2/r3) = (log(r1/r3) − 1)²− (log(r2/r3))², B = B(r₂/r₃) = −1 − 2 log(r₂/r₃),

and C = max{4 ˜C₅(r₂/r₁)ⁿ, exp(Bβ₀)}. Note that that ˜C₅ can be chosen independent of the scaling factor r₃ provided r₃ < 1. Replacing the variable y = x/r₃ in (2.13) gives

Z

|x|<r2

|u|²dx ≤ C(

Z

|x|<r1

|u|²dx)^τ( Z

|x|<r3

|u|²dx)^1−τ.

This ends the proof.

2

3 Doubling inequalities and maximal vanish- ing order

In this section, we prove Theorem 1.2 and Theorem 1.3. We begin with an- other Carleman estimate derived in [8, Lemma 2.1]: for any u ∈ C₀^∞(Rⁿ\{0}) and for any m ∈ {k + 1/2, k ∈ N}, we have the following estimate

X

|α|≤2l

Z

m^2l−2|α||x|^{−2m+2|α|−n}|D^αu|²dx ≤ C Z

|x|^−2m+4l−n|4^lu|²dx, (3.1) where C depends only on the dimension n and the power l.

Remark 3.1 Using the cut-off function and regularization, estimate (3.1) remains valid for any fixed m if u ∈ H_loc^2l(Rⁿ\{0}) with compact support.

In view of Remark 3.1, we can apply (3.1) to the function χu with χ(x) ∈ C₀^∞(Rⁿ\{0}). Thus, we define χ(x) ∈ C₀^∞(Rⁿ\{0}) as

χ(x) =







0 if |x| ≤ δ/3,

1 in δ/2 ≤ |x| ≤ (R₀+ 1)R₀R/4 = r₄R, 0 if 2r₄R ≤ |x|,

where δ ≤ R²₀R/4, R₀ > 0 is a small number which will be chosen later and R < 1 is sufficiently small. Here the number R is not yet fixed and is given by R = (γm)^−l/2, where γ > 0 is a large constant which will be determined later. Using the estimate (3.1) and the equation (1.1), we can derive that

X

|α|≤2l

Z

δ/2≤|x|≤r4R

m^2l−2|α||x|^{−2m+2|α|−n}|D^αu|²dx

≤ X

|α|≤2l

Z

m^2l−2|α||x|^{−2m+2|α|−n}|D^α(χu)|²dx

≤ C Z

|x|^−2m+4l−n|∆^l(χu)|²dx

= C Z

δ/2≤|x|≤r4R

|x|^−2m+4l−n|∆^lu|²dx + C Z

|x|>r4R

|x|^−2m+4l−n|∆^l(χu)|²dx +C

Z

δ/3≤|x|≤δ/2

|x|^−2m+4l−n|∆^l(χu)|²dx

≤ C⁰K₀² Z

δ/2≤|x|≤r4R

X

|α|≤l−1

|x|^{2|α|−n−2m}|D^αu|²+

[3l/2]

X

|α|=l

|x|2|α|−n−2m+2|D^αu|²
dx

+C Z

|x|>r₄R

|x|^−2m+4l−n|∆^l(χu)|²dx + C Z

δ/3≤|x|≤δ/2

|x|^−2m+4l−n|∆^l(χu)|²dx

≤ C⁰K₀²(r₄R)^2

Z

δ/2≤|x|≤r4R [3l/2]

X

|α|=l

|x|^{2|α|−n−2m}|D^αu|²dx

+C⁰K₀² Z

δ/2≤|x|≤r4R

X

|α|≤l−1

|x|^{2|α|−n−2m}|D^αu|²dx

+C Z

|x|>r₄R

|x|^−2m+4l−n|∆^l(χu)|²dx + C Z

δ/3≤|x|≤δ/2

|x|^−2m+4l−n|∆^l(χu)|²dx, (3.2) where the constant C⁰ depends on n and l.

By carefully checking terms on both sides of (3.2), we now choose γ ≥ (2C⁰K₀²)^1/l and thus

R^2 = (γm)^−l ≤ m^−l 2C⁰K₀².

Hence, choosing R₀ < 1 (suffices to guarantee r^2/₄ = R^2₀ (R₀+ 1)^2/4^2 < 1) and m such that m² > 2C⁰K₀², we can remove the first two terms on the right hand side of the last inequality in (3.2) and obtain

X

|α|≤2l

Z

δ/2≤|x|≤r4R

m^2l−2|α||x|^{−2m+2|α|−n}|D^αu|²dx

≤ 2C Z

δ/3<|x|<δ/2

|x|^−2m+4l−n|∆^l(χu)|²dx +2C

Z

r4R<|x|<2r4R

|x|^−2m+4l−n|∆^l(χu)|²dx. (3.3) In view of the definition of χ, it is easy to see that for all multiindex α

(|D^αχ| = O(δ^−|α|) for all δ/3 < |x| < δ/2,

|D^αχ| = O((r₄R)^−|α|) for all r₄R < |x| < 2r₄R. (3.4) Note that R²₀ ≤ r₄ provided R₀ ≤ 1/3. Therefore, using (3.4) and (2.3) in (3.3), we derive

m²(2δ)^−2m−n Z

δ/2<|x|≤2δ

|u|²dx + m²(R²₀R)^−2m−n Z

2δ<|x|≤R²₀R

|u|²dx

≤ X

|α|≤2l

Z

δ/2≤|x|≤r4R

m^2l−2|α||x|^{−2m+2|α|−n}|D^αu|²dx

≤ C⁰⁰ X

|α|≤2l

δ^−4l+2|α|

Z

δ/3<|x|<δ/2

|x|^−2m+4l−n|D^αu|²dx

+C⁰⁰ X

|α|≤2l

(r₄R)^−4l+2|α|

Z

r4R<|x|<2r4R

|x|^−2m+4l−n|D^αu|²dx

≤ C˜⁰δ^−2m−n Z

|x|≤δ

|u|²dx + C⁰⁰(r₄R)^−2m−n Z

|x|≤R0R

|u|²dx, (3.5)

where ˜C⁰ = C⁰⁰3^2m+n and C⁰⁰ is independent of R₀, R, and m.

We then add m²(2δ)^−2m−nR

|x|≤δ/2|u|²dx to both sides of (3.5) and obtain 1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx + m²(R²₀R)^−2m−n Z

|x|≤R²₀R

|u|²dx

= 1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx + m²(R²₀R)^−2m−n Z

|x|≤2δ

|u|²dx +m²(R₀²R)^−2m−n

Z

2δ<|x|≤R²₀R

|u|²dx

≤ 1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx +1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx +m²(R₀²R)^−2m−n

Z

2δ<|x|≤R²₀R

|u|²dx

≤ C˜⁰⁰δ^−2m−n Z

|x|≤δ

|u|²dx + C⁰⁰(r₄R)^−2m−n Z

|x|≤R₀R

|u|²dx

= C˜⁰⁰δ^−2m−n Z

|x|≤δ

|u|²dx

+m²(R₀²R)^−2m−nC⁰⁰m⁻²(R²₀ r₄ )^2m+n

Z

|x|≤R₀R

|u|²dx (3.6)

with ˜C⁰⁰ = ˜C⁰+ 2^2m+nm². We first observe that

C⁰⁰m⁻²(R²₀ r4

)^2m+n = C⁰⁰m⁻²

 4R₀ R0+ 1

2m+n

≤ C⁰⁰m⁻²(4R₀)^2m+n ≤ exp(−2m) for all R₀ ≤ 1/16 and m² ≥ C⁰⁰. Thus, we obtain that

1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx + m²(R²₀R)^−2m−n Z

|x|≤R²₀R

|u|²dx

≤ C˜⁰⁰δ^−2m−n Z

|x|≤δ

|u|²dx +m²(R₀²R)^−2m−nexp(−2m)

Z

|x|≤R₀R

|u|²dx. (3.7)

It should be noted that (3.7) is valid for all m = j + ¹₂ with j ∈ N and j ≥ j₀, where j₀ depends on n, l, , and K₀. Setting R_j = (γ(j +¹₂))^−l/2 and

using the relation m = (γ)⁻¹(R)^−2/l, we get from (3.7) that 1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx + m²(R²₀R_j)^−2m−n Z

|x|≤R²₀Rj

|u|²dx

≤ C˜⁰⁰δ^−2m−n Z

|x|≤δ

|u|²dx

+m²(R²₀R_j)^−2m−nexp(−2cR_j^−2/l) Z

|x|≤R₀Rj

|u|²dx (3.8)

for all j ≥ j₀ and c = γ⁻¹. We now let j₀ be large enough such that R_j+1 < R_j < 2R_j+1 for all j ≥ j₀.

Thus, if R_j+1 < R ≤ R_j for j ≥ j₀, we can conclude that

( R

|x|≤R²₀R|u|²dx ≤R

|x|≤R²₀Rj|u|²dx, exp(−2cR^−2/l_j )R

|x|≤R0Rj|u|²dx ≤ exp(−cR^−2/l)R

|x|≤R|u|²dx, (3.9) where we have used the inequality R₀R_j ≤ R_j/16 < R_j+1to derive the second inequality above. Namely, we have from (3.8) and (3.9) that

1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx + m²(R²₀R_j)^−2m−n Z

|x|≤R₀²R

|u|²dx

≤ C˜⁰⁰δ^−2m−n Z

|x|≤δ

|u|²dx

+m²(R²₀R_j)^−2m−nexp(−cR^−2/l) Z

|x|≤R

|u|²dx. (3.10)

If there exists s ∈ N such that

Rj+1 < R^2s₀ ≤ Rj for some j ≥ j0, (3.11) then replacing R by R^2s₀ in (3.10) leads to

1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx + m²(R²₀R_j)^−2m−n Z

|x|≤R^2s+2₀

|u|²dx

≤ C˜⁰⁰δ^−2m−n Z

|x|≤δ

|u|²dx

+m²(R₀²R_j)^−2m−nexp(−cR^−4s/l₀ ) Z

|x|≤R^2s₀

|u|²dx. (3.12)

Here s and R₀ are yet to be determined. The trick now is to find suitable s and R₀ satisfying (3.11) and the inequality

exp(−cR^−4s/l₀ ) Z

|x|≤R^2s₀

|u|²dx ≤ 1 2

Z

|x|≤R^2s+2₀

|u|²dx (3.13)

holds with such choices of s and R₀.

It is time to use the three-sphere inequality (1.2). To this end, we choose r₁ = R^2k+2₀ , r₂ = R^2k₀ and r₃ = R₀^2k−2 for k ≥ 1. Note that r₁/r₃ < r₂/r₃ ≤ R²₀ ≤ ˜R₀. Thus (1.2) implies

Z

|x|<R^2k₀

|u|²dx/

Z

|x|<R^2k+2₀

|u|²dx ≤ C^1/τ( Z

|x|<R^2k−2₀

|u|²dx/

Z

|x|<R^2k₀

|u|²dx)^a, (3.14) where

C = max{C₀R⁻²ⁿ₀ , exp(β₀(−1 − 4 log R₀))}

and

a = 1 − τ

τ = A

B = (log(r₁/r₃) − 1)²− (log(r₂/r₃))²

−1 − 2 log(r₂/r₃)

= (4 log R₀− 1)²− (2 log R₀)²

−1 − 4 log R₀ . It is not hard to see that

(1 < C ≤ C₀R^−β₀ ¹,

2 < a ≤ −4 log R₀, (3.15) where β₁ = max{2n, 4β₀} and if R₀ is sufficiently small, e.g., R₀ ≤ e⁻⁴. Combining (3.15) and using (3.14) recursively, we have that

Z

|x|≤R^2s₀

|u|²dx/

Z

|x|≤R₀^2s+2

|u|²dx

≤ C^1/τ( Z

|x|<R^2s−2₀

|u|²dx/

Z

|x|<R^2s₀

|u|²dx)^a

≤ C^{as−1−1τ (a−1)} ( Z

|x|<R²₀

|u|²dx/

Z

|x|<R⁴₀

|u|²dx)^as−1 (3.16)

for all s ≥ 1. Now from the definition of a, we have τ = 1/(a + 1) and thus a^s−1− 1

τ (a − 1) = a + 1

a − 1(a^s−1− 1) ≤ 3a^s−1. Then it follows from (3.16) that

Z

|x|≤R^2s₀

|u|²dx/

Z

|x|≤R^2s+2₀

|u|²dx

≤ C^{3(−4 log R0)s−1}( Z

|x|<R²₀

|u|²dx/

Z

|x|<R⁴₀

|u|²dx)^as−1

≤ (C₀³(R₀)^−3β1)^{(−4 log R0)s−1}( Z

|x|<R²₀

|u|²dx/

Z

|x|<R⁴₀

|u|²dx)^as−1. (3.17) Thus, by (3.17), we can get that

exp(−cR₀^−4s/l) Z

|x|≤R^2s₀

|u|²dx

≤ exp(−cR₀^−4s/l)(C₀³(R0)^−3β1)^{(−4 log R0)s−1} (

Z

|x|<R²₀

|u|²dx/

Z

|x|<R⁴₀

|u|²dx)^as−1 Z

|x|≤R^2s+2₀

|u|²dx.

(3.18) Let µ = − log R₀, then if R₀ (≤ min{e⁻⁴,p ˜R₀}) is sufficiently small, i.e., µ is sufficiently large, we can see that

4tµ/l > (t − 1) log(4µ) + log(log C₀³+ 3β₁µ) − log(c/4), for all t ∈ N. In other words, we have that for R0 small

(C₀³R^−3β₀ ¹)^{(−4 log R0)t−1} < exp(cR^−4t/l₀ /4) < (1/2) exp(cR^−4t/l₀ /2), (3.19) for all t ∈ N. We now fix such R0 so that (3.19) holds and

−4ε

l log R₀ − 2 log a > 0.

It is a key step in our proof that we can find a universal constant R₀. After fixing R0, we then define a number t0, depending on R0 and u, as

t₀ = (log 2 − log(ac) + log log(

Z

|x|<R²₀

|u|²dx/

Z

|x|<R⁴₀

|u|²dx))

×(−4ε

l log R0− log a)⁻¹.

With the choice of t₀, we can see that (

Z

|x|<R²₀

|u|²dx/

Z

|x|<R⁴₀

|u|²dx)^at−1 ≤ exp(cR^−4t/l₀ /2) (3.20) for all t ≥ t₀.

Let s1 be the smallest positive integer such that s1 ≥ t0. If

R^2s₀ ¹ ≤ R_j0 = (γ(j₀+ 1/2))^−l/2, (3.21) then we can find a j₁ ∈ N with j1 ≥ j₀ such that (3.11) holds, i.e.,

R_j1+1 < R^2s₀ ¹ ≤ R_j1. On the other hand, if

R^2s₀ ¹ > R_j0, (3.22) then we pick the smallest positive integer s2 > s1 such that R^2s₀ ² ≤ Rj0 and thus we can also find a j₁ ∈ N with j1 ≥ j₀ for which (3.11) holds. We now define

s =

(s1 if (3.21) holds, s2 if (3.22) holds.

It is important to note that with such s, (3.11) is satisfied for some j₁ and (3.19), (3.20) hold. Therefore, we set m₁ = n + 2(j₁+ 1/2) and m = (m₁− n)/2. Combining (3.18), (3.19) and (3.20) yields that

exp(−cR^−4s/l₀ ) Z

|x|≤R^2s₀

|u|²dx

≤ exp(−cR^−4s/l₀ )(C₀³(R₀)^−3β1)^{(−3 log R0)s−1} (

Z

|x|<R²₀

|u|²dx/

Z

|x|<R⁴₀

|u|²dx)^a(s−1) Z

|x|≤R₀^2s+2

|u|²dx.

≤ 1 2

Z

|x|≤R^2s+2₀

|u|²dx

which is (3.13). Using (3.13) in (3.12), we have that 1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx + 1

2m²(R²₀R_j1)^−2m−n Z

|x|≤R^2s+2₀

|u|²dx

≤ C˜⁰⁰δ^−2m−n Z

|x|≤δ

|u|²dx. (3.23)

From (3.23), we get that (m₁− n)²

8 ˜C⁰⁰ (R²₀Rj1)^−m1 Z

|x|≤R^2s+2₀

|u|²dx ≤ δ^−m1 Z

|x|≤δ

|u|²dx (3.24) and

1

2m²(2δ)^−2m−n Z

|x|≤2δ

|u|²dx ≤ ˜C⁰⁰δ^−2m−n Z

|x|≤δ

|u|²dx which implies

Z

|x|≤2δ

|u|²dx ≤ 8 ˜C⁰⁰ (m₁− n)²2^m1

Z

|x|≤δ

|u|²dx. (3.25) The estimates (3.24) and (3.25) are valid for all δ ≤ R^2s+2₀ /4. There- fore, (1.3) holds with R2 = R0. (1.4) holds with R3 = R^2s+2₀ /8 and C3 =

8 ˜C⁰⁰

(m1−n)²2^m1 and the proof is now complete.

2

Acknowledgements

The first and third authors are supported in part by the National Science Council of Taiwan.

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