arXiv:1710.07233v1 [math.CA] 19 Oct 2017 THE VARIATION OF THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION HANNES LUIRO AND JOSÉ MADRID Abstract. In this paper we study the regularity of the noncentered fractional maximal operator Mβ . As the main result, we prove that there exists C(n, β) such that if q = n/(n − β) and f is radial function, then kDMβ f kLq (Rn ) ≤ C(n, β)kDf kL1 (Rn ) . The corresponding result was previously known only if n = 1 or β = 0. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all f ∈ W 1,1 (Rn ). 1. Introduction The non-centered fractional Hardy-Littlewood maximal operator Mβ is defined by setting for f ∈ L1loc (Rn ) and 0 ≤ β < n that Z Z rβ β |f (y)| dy Mβ f (x) := sup |f (y)| dy =: sup r B(z,r)∋x |B(z, r)| B(z,r) B(z,r)∋x B(z,r) (1.1) n for every x ∈ R . The centered version of Mβ , denoted by Mβc , is defined by taking the supremum over all balls centered at x. In the non-fractional case β = 0, we also denote M0 = M. The study of the regularity of maximal operators has strongly attracted the attention of many authors in recent years. The boundedness of the classical maximal operator on the Sobolev space W 1,p (Rn ) for p > 1 was established by Kinnunen in [Ki]. The analogous result in the fractional context was established by Kinnunen and Saksman in [KiSa]: for every 0 < β < n we have that Mβ is bounded from W 1,p (Rn ) to W 1,q (Rn ) under the relation 1/q = 1/p − β/n (if p > 1). For other interesting results on this theory we refer to [BCHP], [CFS], [CaHu], [CMP], [CaMo], [CaSv], [HM], [HO], [L], [Ma] and [R]. Date: October 7, 2018. 2010 Mathematics Subject Classification. 42B25, 26A45, 46E35, 46E39. Key words and phrases. Fractional maximal operator, Sobolev spaces, Radial functions. 1 2 HANNES LUIRO AND JOSÉ MADRID The case p = 1 is particularly complicated and interesting. In the case n = 1 it is known (see [Ta] and [AlPe] for the Non-Centered, and [Ku] for the Centered) that Mf is weakly differentiable and kDMf kL1 (R) ≤ CkDf kL1(R) , (1.2) but even in this case there are still some interesting open questions. The proofs of these theorems strongly exploit the simplicity of onedimensional topology. Indeed, the situation in higher dimension is quite unknown, only a few results have been obtained (see [L2], [S]). The analogous result to (1.2) for the fractional non-centered maximal operator was established by Carneiro and Madrid in [CaMa]. In full generality the next question was posed by them. Main Question. Let 0 ≤ β < n and q = n/(n − β). Is the operator f 7→ |DMβ f | bounded from W 1,1 (Rn ) to Lq (Rn )? The problem can be rather easily reduced to the case 0 ≤ β < 1, as it was also observed by Carneiro and Madrid (see [CaMa]). Indeed, in the case 1 ≤ β < d, the positive answer follows by combining the boundedness property of the fractional maximal operator from Lp to Lq (under the condition 1/q = 1/p − β/n), the Sobolev embedding Theorem and the result in [KiSa] which says: If f ∈ Lr (Rn ) with 1 < r < n and 1 ≤ β < n/r, then Mβ f is weakly differentiable and |DMβ f (x)| ≤ C(n, β) Mβ−1 f (x) for a.e. x ∈ Rn . In the case β = 0 (non-fractional operator) the main question for radial functions was recently proven by Luiro [L2]. Our main Theorem is a counterpart of this result in the case β > 0. Theorem 1.1 (Main Theorem). Given 0 < β < n and q = n/(n − β), there is a constant C = C(n, β) > 0 such that for every radial function f ∈ W 1,1 (Rn ) we have that Mβ f is weakly differentiable and kDMβ f kLq (Rn ) ≤ CkDf kL1 (Rn ) . The proof adapts some basic ideas from [L2], like in Lemma 2.4. However, as we will see (and as one can see in [CaMa] as well), some new difficulties arise with respect to the case of the classical maximal operator. The key element to overcome these problems is Lemma 2.10. We believe that the modification of this result may play a crucial role in the solution of the problem in its full generality. In addition, we point out that the presented argument also gives a new proof for the THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 3 case n = 11, in other words our argument also gives a new proof for Theorem 1 in [CaMa]. 2. Preliminaries Let us introduce some notation. The boundary of the n-dimensional unit ball is denoted by S n−1 . The s-dimensional Hausdorff measure is denoted by Hs . The volume of the n-dimensional unit ball is denoted by ωn and the Hn−1 -measure of S n−1 by σn . The integral average of f ∈ L1loc (Rn ) over a measurable set A ⊂ Rn is sometimes denoted by fA . The weak derivative of f (if exists) is denoted by Df . If v ∈ S n−1 , then 1 Dv f (x) := lim (f (x + hv) − f (x)) , h→0 h in the case the limit exists. For f ∈ W 1,1 (Rn ), 0 ≤ β < n, let us define Bxβ (f ) = Bx :=  B(z, r) : x ∈ B̄(z, r) , Mβ f (x) = r β Z  |f (y)| dy . B(z,r) We use to call Bx as the collection of the best balls at x. It is easy to see that Bx is non-empty set for every x ∈ Rn (since f ∈ L1 (Rn )) and also it is compact in the sense that if B(zk , rk ) ∈ Bx and zk → z ∈ Rn and rk → r ∈ (0, ∞) as k → ∞, then B(z, r) ∈ Bx . Proposition 2.1. Given f ∈ W 1,1 (Rn ), B a ball, a family of affine mappings Li (y) = ai y +bi , ai ∈ R, bi ∈ Rn , Bi := Li (B) and a sequence {hi }i∈N ⊂ R such that hi → 0 as i → ∞, aβ − 1 Li (y) − y = g(y) and lim i = γ, i→∞ i→∞ hi hi lim where γ ∈ R, g : Rn → R , it holds that  Z  Z 1 β β ri |f (y)|dy − r |f (y)|dy lim i→∞ hi Bi B Z Z β β D|f |(y) · g(y)dy + γr |f (y)| dy , =r B B where r denotes the radius of B and ri the radius of Bi for every i. 1With slight modifications in the proof in the case f is not symmetric with respect to the origin 4 HANNES LUIRO AND JOSÉ MADRID Proof of Proposition 2.1.  Z  Z 1 β β r |f (y)|dy − r |f (y)|dy hi i Bi B   Z Z 1 β β β = ai r |f (y)|dy − r |f (y)|dy hi Li (B) B ! Z β a |f (y + (L (y) − y))| − |f (y)| i i = rβ hi B Z β ai |f (y + (Li (y) − y))| − aβi |f (y)| β =r dy hi B ! Z β |f (y)|(ai − 1) + dy hi B Z Z β β D|f |(y) · g(y)dy + γr |f (y)|dy →r B B as i → ∞.  Lemma 2.2. Let f ∈ W 1,1 (Rn ), x ∈ Rn , B ∈ Bx , δ > 0, and let Lh (y) = ah y + bh , h ∈ [−δ, δ], be affine mappings such that x ∈ Lh (B) and Lh (y) − y aβh − 1 lim = g(y) and lim = γ. h→0 h→0 h h Then 0 = (rad(B)) β Z D|f |(y) · g(y)dy + γMβ f (x). (2.3) B Proof of Lemma 2.2. By the previous Proposition 2.1 the right hand side of (2.3) equals to   Z Z 1 1 β β rad(Lh (B)) lim |f (y)|dy − r |f (y)|dy =: lim sh . h→0 h h→0 h Lh (B) B (2.4) Since B ∈ Bx and x ∈ Lh (B̄) for all h, it follows that sh ≤ 0 for all h. Since h can take positive and negative values, the existing limit must equal to zero.  THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 5 Corollary 2.3. In particular, if Lh (y) = y + h(y − x) we obtain that g(y) = y − x, γ = β, and x = Lh (x) ∈ Lh (B̄). Therefore Z Z β β D|f |(y) · xdy = (rad(B)) D|f |(y) · y dy + βMβ f (x). (rad(B)) B B (2.5) The following lemma is a counterpart of Lemma 2.2 in [L2]. It was proved by Carneiro and Madrid in [CaMa, Theorem 1] that if g ∈ W 1,1 (R) thus Mβ g is absolutely continuous, therefore if f ∈ W 1,1 (Rn ) is a radial functions we can apply the next lemma. Lemma 2.4. Suppose that f ∈ W 1,1 (Rn ) and Mβ f is differentiable at x. Then (1) For all v ∈ S n−1 and B ∈ Bx , it holds that Z β Dv Mβ f (x) = (rad(B)) Dv |f |(y) dy . B (2) If x ∈ B for some B ∈ Bx , then DMβ f (x) = 0 . (3) If x ∈ ∂B, B = B(z, r) ∈ Bx and DMβ f (x) 6= 0, then z−x DMβ f (x) = . |DMβ f (x)| |z − x| (4) If B ∈ Bx , then Z Z 1 D|f |(y) · (y − x) dy . |f |(y)dy = − β B B (2.6) (1) Let B = B(z, r) ∈ Bx and Bh := B(z + hv, r). Then it holds for every v ∈ S n−1 that R R r β Bh |f (y)|dy − r β B |f (y)|dy Mβ f (x + hv) − Mβ f (x) lim ≥ lim h→0 h→0 h h Z = rβ Dv |f |(y)dy B R R r β B |f (y)|dy − r β B−h |f (y)|dy = lim h→0 h Mβ f (x) − Mβ f (x − hv) . ≥ lim h→0 h (2) If B ∈ Bx and x ∈ B, then Mβ f (x) ≤ Mβ f (y) for every y ∈ B. (3) Let B = B(z, r) ∈ Bx , v ∈ S n−1 such that v ·(z −x) = 0, and let us denote for all h ∈ (0, ∞) that xh := x+hv, rh := |z−xh |, and Bh := B(z, rh ). These definitions guarantee that xh ∈ B̄h \ B Proof. 6 HANNES LUIRO AND JOSÉ MADRID for all h, and B ⊂ Bh . Moreover, since v · (z − x) = 0, it is elementary fact that rh = |z − x − hv| ≤ |z − x| + h2 . 2r Therefore, r/rh ≥ 1 − ( hr )2 , and Z Z rhβ |B| β β Mβ f (xh ) ≥ rh r |f (z)| dz |f (z)| dz ≥ β r |Bh | Bh B   n−β Z n−β h2 r β |f (z)| dz ≥ 1 − 2 = r Mβ f (x) . rh r B This implies that Dv Mβ f (x) ≥ 0 for all v ∈ S n−1 such that v · (z − x) = 0 . Since we assumed that Mβ f is differentiable at x, it follows that Dv Mβ f (x) = 0 if v ∈ S n−1 , v · (z − x) = 0 . In particular, it follows that DMβ f (x) is parallel to z − x or x − z. The final claim follows easily by the fact that Mβ f (x + h(z − x)) ≥ Mβ f (x) if 0 ≤ h ≤ 2. (4) This is an immediate consequence of Corollary 2.3.  1,1 Proposition 2.5. If f ∈ Wloc (Rn ), z ∈ Rn , r > 0, then Z  Z Z D|f |(y) · (z − y) dy = n |f | − |f | . B(z,r) B(z,r) (2.7) ∂B(z,r) Proof. In the case of radial functions the previous proposition follows from the one dimensional case (that is enough in order to get Theorem 1.1). In general, the proof of this proposition is based in the following fact, which is a consequence of Gauss Divergence Theorem. Remark 2.6 (Integration by parts). Given Ω ⊂ Rn a bounded open set with C 1 boundary and ν denotes the outward unit normal to ∂Ω if u ∈ W 1,p (Ω) and v ∈ W 1,q (Ω) for exponents p,q with 1 1 1 + ≤1+ p q n thus the following identity holds Z Z Z ∂v ∂u u = uvνi − v Ω ∂yi ∂Ω Ω ∂yi where νi is the i−component of the vector ν. THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 7 Using this we get Z D|f |(y) · (z − y)dy B(z,r) n Z X = i=1 B(z,r) n Z X ∂|f | (y)(zi − yi )dy ∂yi  Z (yi − zi ) = |f (y)|(zi − yi ) · dy + |f (y)|dy |y − z| ∂B(z,r) B(z,r) i=1 Z n Z n X X |yi − zi |2 =− |f (y)| dy + |f (y)|dy |y − z| B(z,r) ∂B(z,r) i=1 i=1 Z Z =n |f (y)|dy − |f (y)||y − z|dy B(z,r) ∂B(z,r)  Z Z r |f (y)|dy |f (y)|dy − =n n ∂B(z,r) B(z,r)  Z Z r n wn |f (y)|dy =n |f (y)|dy − n−1 r σn ∂B(z,r) B(z,r) Z  Z =n |f (y)|dy − |B(z, r)| |f (y)|dy . B(z,r) ∂B(z,r) By dividing both sides of the last equality by |B(z, r)| we arrived in the desired identity.  By using Proposition 2.5 we yet state one more formula related to the derivative of the fractional maximal operator. 1,1 Lemma 2.7. Suppose that f ∈ Wloc (Rn ), 0 < β < n, B ∈ Bxβ for n some x ∈ R , and r := rad(B) . Then Z n D|f |(y) dy = r B  1 − β/n) Z |f (y)| dy − B Z  |f (y)|dy . (2.8) ∂B 8 HANNES LUIRO AND JOSÉ MADRID Proof. Suppose that B = B(z, r). By Lemma 2.4 and Proposition 2.5 it follows that   Z Z z−x dy D|f |(y) dy = D|f |(y) · r B B Z  Z 1 = D|f |(y) · (z − y) dy + D|f |(y) · (y − x) dy r B B Z  Z 1 = D|f |(y) · (z − y) dy − β |f |(y) dy r B B  Z   Z Z 1 |f | − |f | − β |f |(y) dy = n r B ∂B B   Z Z n = 1 − β/n) |f (y)| dy − |f (y)| . r B ∂B  We will use the following elementary property for radial functions. The proof is left for an interested reader. Proposition 2.8. Suppose that f ∈ L1loc (Rn ) satisfies f (x) = F (|x|), F : (0, ∞) → [0, ∞), B := B(z, r) ⊂ B(0, 2|z|) \ B(0, 12 |z|) , and a := |z| − r, b := |z| + r. Then it holds that Z Z F (t) dt ≤ C(n) f (y) dy . (2.9) [a,b] B(z,2r) The following two lemmas contain the key estimates for the proof of the main theorem. 1,1 Lemma 2.9. Suppose that f ∈ Wloc (Rn ) is radial and B ∈ Bxβ for n some x ∈ R \ {0} such that B ⊂ B(0, |x|). Then Z Z |y| (2.10) D|f |(y) dy ≤ |Df (y)| dy . |x| B B Proof. If |DMβ f (x)| = 0, the claim is trivial. If |DMβ f (x)| = 6 0, Lemma 2.4 yields that R D|f |(y) dy −x B , (2.11) = R |x| D|f |(y) dy B THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 9 and −x dy D|f |(y) · |x| B B Z Z Z |y| β −y |f (y)| dy ≤ |Df (y)| . dy − = Df (y) · |x| |x| B |x| B B This proves the claim. Z Df (y) dy = Z  Given a ball B = B(z, r) we define 2B to be equal to B(z, 2r). 1,1 Lemma 2.10. Suppose that f ∈ Wloc (Rn ) is radial, 0 < β < n, B ∈ Bxβ for some x ∈ Rn , r := rad(B) ≤ |x| , and 4 1 E := {z ∈ 2B : |f |B ≤ |f (z)| ≤ 2|f |B } . (2.12) 2 Then Z Z D|f |(y) dy ≤ C(n, β) |Df (z)|χE (z) dz . (2.13) 2B B Proof. First observe that by Lemma 2.7 it holds that   Z Z n |f |B − D|f |(y) dy ≤ |f | . r B ∂B (2.14) Let then |f (x)| = F (|x|), where F : R \ {0} → [0, ∞), let z denote the center point of B, a := |z| − r, b := |z| + r, and 1 A := {t ∈ 2[a, b] : |f |B ≤ F (t) ≤ 2|f |B }. (2.15) 2 Then we show that Z Z |f |B − |f | ≤ 2 |F ′(t)|χA (t) dt . (2.16) [a,b] ∂B The above inequality is more or less trivial: To prove it, choose t0 ∈ [a, b] such that F (t0 ) = |f |B and choose t1 ∈ [a, b] such that |f |∂B = F (t1 ). By (2.14) we have that F (t0 ) ≥ F (t1 ). In the case F (t1 ) ≥ 21 |f |B in [a, b] the claim follows by using the continuity, because in this case by the continuity we can assume without loss of generality that [t0 , t1 ] ⊂ A (or [t1 , t0 ] ⊂ A). Otherwise, if F (t1 ) < 21 |f |B , there exists t2 ∈ [a, b] between t0 and t1 such that F (t2 ) = 21 |f |B , by the continuity of F it clearly follows that Z Z Z ′ |f | ≤ |f |B ≤ 2 |F (t)|χA (t) dt ≤ 2 |F ′(t)|χA (t) dt . |f |B − ∂B [t0 ,t2 ] [a,b] (2.17) 10 HANNES LUIRO AND JOSÉ MADRID Since |Df (y)|χE (y) = |F ′(|y|)|χA(|y|), Proposition 2.8 yields that Z Z ′ |F (t)|χA (t) dt ≤ C(n) |Df (y)|χE (y) dy . (2.18) [a,b] B(z,2r) Combining this with( 2.14) and (2.16) implies the desired result.  Proposition 2.11. Suppose that β ≥ 0, f ∈ L1loc (Rn ), and B1 := B(z1 , r1 ) and B2 := B(z2 , r2 ) are best balls for Mβ f such that B2 ⊂ B(z1 , 2r1 ). Then it holds that  β 1 r1 |f |B2 ≥ n |f |B1 . (2.19) 2 r2 Proof. Let B := B(z1 , 2r1). Since B2 is best ball and B2 , B1 ⊂ B, it holds that r2β |f |B2 ≥ (2r1 )β |f |B ≥ (r1 )β 1 |f |B1 . 2n This implies the claim.  3. Proof of the main Theorem Let us fix Bx := B(zx , rx ) ∈ Bx for (almost) every x ∈ E , such that rx is the smallest possible (then by Lemma 2.4 item (3) we have DM f (x) zx = x + rx |DMββ f (x)| ), where E := { x : DMβ f (x) 6= 0 } . (3.20) By the choise of radius we can see that x → rx is an upper semicontinuous function then it is measurable function, thus x → zx is also a measurable function. By Lemma 2.4, it holds for almost all x ∈ E that Bx is of type Bx = B(cx x, |cx − 1||x|) , where cx ∈ R . (3.21) In the other words, this means that the center point of Bx lies on the line containing x and the origin, and x lies on the boundary of Bx . For simplicity, let us yet denote the radius of Bx by rx , thus rx = |cx −1||x| . Observe first that for all x ∈ E it holds that cx ≥ 0. To see this, observe that otherwise (since Mβ f (x) = Mβ f (−x)) it follows that Bx ∈ B−x and −x ∈ Bx , implying that 0 = |DMβ f (−x)| = |DMβ f (x)|, which is a contradiction. We are going to use different type of estimates for THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 11 |DMβ f (x)| depending on how Bx is located with respect to the origin. Indeed, let 3 5 E1 : = {x ∈ E : cx > } , E2 := {x ∈ E : 0 ≤ cx < } , and 4 4 3 5 E3 : = {x ∈ E : ≤ cx ≤ } . 4 4 Then we can estimate Z Z Z Z q β q q dx Df (y) dy rx |DMβ f (x)| dx = |DMβ f (x)| dx = Rn = Z E rxqβ E n(q−1) (ωn )q−1 rx ≤ C(n, β) ||Df ||1q−1 = C(n, β) ||Df ||1q−1 Z E q−1 Df (y) dy Bx Z Z E Bx Z 3 X i=1 Ei Z Bx Df (y) dy dx Bx Df (y) dy dx Z Df (y) dy dx , Bx where we used the fact qβ = n(q − 1) . Especially, the claim follows, if we can show that Z Z Df (y) dy dx ≤ C(n, β) ||Df ||1 , for i = 1, 2, 3 . (3.22) Ei Bx The case of E1 . In this case the easiest type of estimate turns out to be sufficient. Indeed, Z Z Z Z Df (y) dy dx ≤ |Df (y)| dy dx E1 E1 Bx Bx Z Z χBx (y)χE1 (x) dx dy . = |Df (y)| |Bx | Rn Rn and y ∈ Bx , then rx ≥ |y|/4. For every y ∈ Rn it holds that if |x| ≤ |y| 2 |y| Moreover, if 2 ≤ |x| ≤ |y|, then x ∈ E1 implies that rx ≥ 14 |x| ≥ |y| . 8 n Finally, if |x| > |y| and x ∈ E1 , then Bx ⊂ R \ B(0, |y|), thus y 6∈ Bx . By combining these, we conclude that for every y ∈ Rn Z Z χBx (y)χE1 (x) dx dx ≤ C(n) = C(n) . (3.23) |Bx | Rn B(0,|y|) |B(0, |y|)| The case of E2 . In this case we recall the estimate from Lemma 2.9, which yields that Z Z Z |y| |y| n ≤4 . (3.24) Df (y) dy ≤ |Df (y)| |Df (y)| |x| |x| Bx Bx B(0,|x|) 12 HANNES LUIRO AND JOSÉ MADRID Then the claim follows by Z Z Z Z |y| n Df (y) dy dx ≤ 4 |Df (y)| dy dx |x| E2 E2 B(0,|x|) Bx Z  Z χB(0,|x|) (y)χE2 (x) |Df (y)||y| dx dy =4n ωn |x|n+1 Rn Rn Z  Z 4n dx ≤ |Df (y)||y| dy n+1 ω n Rn Rn \B(0,|y|) |x| Z =C(n) |Df (y)| dy . Rn The case of E3 . In this case we will exploit the estimate from Lemma 2.10. For this, let us denote for every x ∈ E3 that 1 Ax := {y ∈ 2Bx : |f |Bx ≤ |f (y)| ≤ 2|f |Bx } . (3.25) 2 , Lemma 2.10 yields that for every Since x ∈ E3 implies that rx ≤ |x| 4 x ∈ E3 it holds that Z Z (3.26) Df (y) dy ≤ C(n, β) |Df (y)|χAx (y) dy . 2Bx Bx Therefore, Z E3 =C Z Z Rn Z Z Df (y) dy dx ≤ C |Df (y)|χAx (y) dy dx E3 2Bx Z  χ2Bx (y)χAx (y)χE3 (x) |Df (y)| dx dy . |2Bx | Rn Bx Consider above the inner integral for fixed y ∈ Rn . Firstly, suppose that χ2Bx0 (y)χAx0 (y) 6= 0 and χ2Bx1 (y)χAx1 (y) 6= 0 , for some x0 , x1 ∈ Rn . (3.27) Observe that if this kind of points does not exist, the desired estimates are trivially true. By the definition, the above means that 1 |f |Bx0 ≤ |f (y)| ≤ 2|f |Bx0 , and 2 1 |f |Bx1 ≤ |f (y)| ≤ 2|f |Bx1 . 2 Especially, it follows that 1 (3.28) |f |Bx0 ≤ |f |Bx1 ≤ 4|f |Bx0 . 4 THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 13 Let r0 := rad(Bx0 ) and r1 := rad(Bx1 ) and assume that r1 ≥ r0 . Since y ∈ 2Bx0 ∩ 2Bx1 , it follows that Bx0 ⊂ 8Bx1 . By Proposition 2.11, it follows that  β  β 1 r1 1 1 r1 (3.29) |f |Bx0 , |f |Bx1 ≥ n |f |Bx0 ≥ n 8 r0 8 r0 4 n+1 implying that r1 ≤ 8 β r0 . If r1 ≤ r0 , symmetric argument gives that n+1 r0 ≤ 8 β r1 . Summing up, it follows that n+1 n+1 rad(Bx0 ) 8− β ≤ ≤8 β . (3.30) rad(Bx1 ) Indeed, this means that if χ2Bx (y)χAx (y) 6= 0 , then |x − y| ≤ C(n, β)rad(Bx0 ) and |Bx | ≥ C(n, β)|Bx0 | . (3.31) Naturally, (3.31) holds also if x0 is replaced by x1 . Finally, this implies that Z Z χ2Bx (y)χAx (y)χE3 (x) dx dx ≤ C(n, β) |2Bx | Rn B(y,C(n,β)rad(Bx0 )) |Bx0 | ≤ C̃(n, β) . n Since this holds for all y ∈ R , the proof is complete. 4. Acknowledgments H.L acknowledges M. Parviainen, J. Kinnunen and the Academy of Finland for the financial support. J.M. acknowledges J. Kinnunen, Aalto University and Academy of Finland for the support. The authors are thankful to Juha Kinnunen for helpful discussions and guidance during the preparation of this manuscript. 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Department of Mathematics and Statistics, University of Jyvaskyla,P.O.Box 35 (MaD), 40014 University of Jyvaskyla, Finland E-mail address: hannes.s.luiro@jyu.fi Department of Mathematics, Aalto University, P.O. Box 11100, FI– 00076 Aalto University, Finland E-mail address: jose.madridpadilla@aalto.fi The Abdus Salam International Centre for Theoretical Physics, Str. Costiera 11, 34151 Trieste, Italy E-mail address: jmadrid@ictp.it