Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

Let 0<α<d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha <d$$\end{document} and 1≤p<d/α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<d/\alpha $$\end{document}. We present a proof that for all f∈W1,p(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in W^{1,p}({\mathbb {R}}^d)$$\end{document} both the centered and the uncentered Hardy–Littlewood fractional maximal operator Mαf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {M}}_\alpha f$$\end{document} are weakly differentiable and ‖∇Mαf‖p∗≤Cd,α,p‖∇f‖p,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Vert \nabla {\mathrm {M}}_\alpha f\Vert _{p^*} \le C_{d,\alpha ,p} \Vert \nabla f\Vert _p , $$\end{document} where p∗=(p-1-α/d)-1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p^* = (p^{-1}-\alpha /d)^{-1} . $$\end{document} In particular it covers the endpoint case p=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=1$$\end{document} for 0<α<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha <1$$\end{document} where the bound was previously unknown. For p=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=1$$\end{document} we can replace W1,1(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,1}({\mathbb {R}}^d)$$\end{document} by BV(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {BV}({\mathbb {R}}^d)$$\end{document}. The ingredients used are a pointwise estimate for the gradient of the fractional maximal function, the layer cake formula, a Vitali type argument, a reduction from balls to dyadic cubes, the coarea formula, a relative isoperimetric inequality and an earlier established result for α=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0$$\end{document} in the dyadic setting. We use that for α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} the fractional maximal function does not use certain small balls. For α=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0$$\end{document} the proof collapses.


Introduction
For f ∈ L 1 loc (R d ) and a ball or cube B, we denote The centered Hardy-Littlewood maximal function is defined by where the supremum is taken over all balls that contain x. The regularity of a maximal operator was first studied by Kinnunen in 1997. He proved in [18] that for each p > 1 and f ∈ W 1, p (R d ) the bound holds for M = M c . Formula (1.1) also holds for M = M. This implies that both Hardy-Littlewood maximal operators are bounded on Sobolev spaces with p > 1. His proof does not apply for p = 1. Note that unless f = 0 also M f 1 ≤ C d,1 f 1 fails since M f is not in L 1 (R d ). In [16] Hajłasz and Onninen asked whether formula (1.1) also holds for p = 1 for the centered Hardy-Littlewood maximal operator. This question has become a well known problem for various maximal operators and there has been lots of research on this topic. So far it has mostly remained unanswered, but there has been some progress. For the uncentered maximal function and d = 1 it has been proved in [28] by Tanaka and later in [22] by Kurka for the centered Hardy-Littlewood maximal function. The proof for the centered maximal function turned out to be much more complicated. Aldaz and Pérez Lázaro obtained in [3] the sharp improvement ∇ M f L 1 (R) ≤ ∇ f L 1 (R) of Tanaka's result.
For the uncentered Hardy-Littlewood maximal function Hajłasz's and Onninen's question already also has a positive answer for all dimensions d in several special cases. For radial functions Luiro proved it in [24], for block decreasing functions Aldaz and Pérez Lázaro proved it in [2] and for characteristic functions the author proved it in [30]. As a first step towards weak differentiability, Hajłasz and Malý proved in [15] that for f ∈ L 1 (R d ) the centered Hardy-Littlewood maximal function is approximately differentiable. In [1] Aldaz et al. proved bounds on the modulus of continuity for all dimensions. A related question is whether the maximal operator is a continuous operator. Luiro proved in [23] that for p > 1 the uncentered maximal operator is continuous on W 1, p (R d ). There is ongoing research for the endpoint case p = 1. For example Carneiro et al. proved in [11] that f → ∇ M f is continuous W 1,1 (R) → L 1 (R) and in [14] González-Riquelme and Kosz recently improved this to continuity on BV. Carneiro et al. proved in [8] that for radial functions f , the operator f → ∇ M f is continuous as a map W 1, The regularity of maximal operators has also been studied for other maximal operators and on other spaces. We focus on the endpoint p = 1. In [12] Carneiro and Svaiter and in [7] Carneiro and González-Riquelme investigated maximal convolution operators M associated to certain partial differential equations. Analogous to the Hardy-Littlewood maximal operator they proved In [9] Carneiro and Hughes proved ∇M f l 1 (Z d ) ≤ C d f l 1 (Z d ) for centered and uncentered discrete maximal operators. This bound does not hold on R d , but because in the discrete setting we have In [21] Kinnunen and Tuominen proved the boundedness of a discrete maximal operator in the metric Hajłasz Sobolev space M 1,1 . In [27] Pérez et al. proved the boundedness of certain convolution maximal operators on Hardy-Sobolev spacesḢ 1, p for a sharp range of exponents, including p = 1. In [29] the author proved var M d f ≤ C d var f for the dyadic maximal operator for all dimensions d.
For 0 ≤ α ≤ d the centered fractional Hardy-Littlewood maximal function is defined by For a ball B we denote the radius of B by r (B). The uncentered fractional Hardy-Littlewood maximal function is defined by where the supremum is taken over all balls that contain x. Note that M α does not make much sense for α > d. For α = 0 it is the Hardy-Littlewood maximal function. The following is the fractional version of formula (1.1).
where the constant C d,α, p depends only on d, α and p. In the endpoint p = 1 we can replace for almost every x ∈ R d , and then concluded formula (1.2) from the L ( p −1 −α/d) −1boundedness of M α , which fails for p = 1. Another result by Kinnunen and Saksman in [20] is that for all In [10] Carneiro and Madrid used this, the L d/(d−α) -boundedness of M α−1 , and Sobolev embedding to concluded formula (1.2). All of this also works for the uncentered fractional maximal function M α . The strategy fails for α < 1.
Our main result is the extension of formula (1.2) to the endpoint p = 1 for 0 < α < 1 which has been an open problem. Our proof of Theorem 1.1 also works for 1 ≤ α ≤ d, and further extends to 1 ≤ p < ∞, 0 < α ≤ d/ p. We present the proof for this range of parameters here, since it also smoothens out the blowup of the constants for p → 1 which occurs in the previous proof for p > 1. Note that interpolation is not immediately available for results on the gradient level. Our approach fails for α = 0. The corner point α = 0, p = 1 is the earlier mentioned question by Hajłasz and Onninen and remains open. Similarly to Carneiro and Madrid, we begin the proof with a pointwise estimate |∇M α f (x)| ≤ (d − α)M α,−1 f (x) which holds for all 0 < α < d for bounded functions. We estimate M α,−1 f in Theorem 1.2 and from that conclude Theorem 1.1.
For the centered fractional maximal function define where r is the largest radius such that M c α f (x) = r α f B(x,r ) and for the uncentered fractional maximal function define Then for almost every x ∈ R d the sets B c α (x) and B α (x) are nonempty, i.e. the supremum in the definition of the maximal function is attained in a largest ball B with x ∈ B, see Lemma 2.2.
For β ∈ R with −1 ≤ α + β < d this allows us to define the following maximal functions for almost every x ∈ R d . Note that also for the centered version the supremum is all balls B ∈ B c α whose closure contains x, not only over those centered in x. Theorem 1.2 Let 1 ≤ p < ∞ and 0 < α < d and β ∈ R with 0 ≤ α + β + 1 < d/ p and We prove Theorem 1.2 in Sect. 4. There had also been progress on 0 < α ≤ 1 similarly as for the Hardy-Littlewood maximal operator. For the uncentered fractional maximal function Carneiro and Madrid proved Theorem 1.1 for d = 1 in [10], and Luiro proved Theorem 1.1 for radial functions in [25]. Beltran and Madrid transferred Luiros result to the centered fractional maximal function in [5]. In [6] Beltran et al. proved Theorem 1.1 for d ≥ 2 and a centered maximal operator that only uses balls with lacunary radius and for maximal operators with respect to smooth kernels. The next step after boundedness is continuity of the gradient of the fractional maximal operator, as it implies boundedness, but doesn't follow from it. In [4,26] Beltran and Madrid already proved it for the uncentered fractional maximal operator in the cases where the boundedness is known.
For a dyadic cube Q we denote by l(Q) the sidelength of Q. The fractional dyadic maximal function is defined by where the supremum is taken over all dyadic cubes that contain x. The dyadic maximal operator has enjoyed a bit less attention than its continuous counterparts, such as the centered and the uncentered Hardy-Littlewood maximal operator. The dyadic maximal operator is different in the sense that formula (1.2) only holds for α = 0, p = 1 and only in the variation sense, for which formula (1.2) has been proved in [29]. But for any other α and p formula (1.2) fails because ∇M d α f is not a Sobolev function. We can however prove Theorem 1.4, the dyadic analog of Theorem 1.2. For α ≥ 0 and a function f ∈ L 1 (R d ) define Q α to be the set of all cubes Q such that for all dyadic cubes P Q we have l(P) α f P < l(Q) α f Q .

Remark 1.3
In the uncentered setting one could also define B α in a similar way as Q α .
Then Our main result in the dyadic setting is the following.
where the constant C d,α, p depends only on d, α and p. In the endpoint p = 1 we can replace The endpoint result for p = ∞ holds true as well.
In Sect. 2.2 we conclude Theorem 1.4 from Theorem 1.5, and in Sect. 3 we prove Theorem 1.5.
Theorem 1.4 holds with the dyadic version of E and Theorem 1.5 where the sum on the left hand side is over any subset Q ⊂ Q α and the integral on the right is over {cQ : Q ∈ Q}. These localized results directly follow from the same proof as the global results, if one keeps track of the balls and cubes which are being dealt with. The respective localized version of Theorem 1.1 can be proven if one has Lemma 2.4 without the differentiability assumption. Then in the reduction of Theorem 1.1 to Theorem 1.2 one could apply Theorem 1.2 to the same function f and Q α for which one is showing Theorem 1.1, bypassing the approximation step and therefore preserving the locality of Theorem 1.2. This is in contrast to the actual local fractional maximal operator, for whom Theorem 1.1 fails by [17,Example 4.2], which works for α > 0. However if α = 0 and p > 1 then the local fractional maximal operator is again bounded due to [19], and by [30] for α = 0 and p = 1 and characteristic functions.
Dyadic cubes are much easier to deal with than balls, but the dyadic version still serves as a model case for the continuous versions since both versions share many properties. This can be observed in [30], where we proved var M 0 1 E ≤ C d var 1 E for the dyadic maximal operator and the uncentered Hardy-Littlewood maximal operator. The proof for the dyadic maximal operator is much shorter, but the same proof idea also works for the uncentered maximal operator. Also in this paper a part of the proof of Theorem 1.4 for the dyadic maximal operator is used also in the proof of Theorem 1.2 for the Hardy-Littlewood maximal operator.
The plan for the proof of Theorem 1.1 is the following. For simplicity we write it down for p = 1.
where σ d is the volume of the d-dimensional unit ball. In the second step we apply the layer cake formula, in the forth step we pass from a union of arbitrary balls to very disjoint balls B α with a Vitali covering argument, in the eighth step we pass from those balls to comparable dyadic cubes and as the last step use a result from the dyadic setting.
We use α > 0 as follows. Let A be a ball and B(x, r ) be a smaller ball that inter- We use this in the forth step of the proof strategy above. We use a dyadic version of these alternatives in last step. Note that for α = 0 optimal balls B of arbitrarily different sizes with similar values f B can intersect.

Remark 1.9
There is a proof of Theorem 1.1 which has a structure parallel to the one presented above, but three steps are replaced. The estimate Otherwise it is identical to the proof presented in this paper.
Note that formally Remark 1. 10 In the proof of Theorems 1.1, 1.2, 1.5 and 1.4 we do not a priori need f ∈ However from ∇ f p < ∞ we can then anyways conclude f ∈ L p (R d ) by Sobolev embedding.

Reformulation
In order to avoid writing absolute values, we consider only nonnegative functions f for the rest of the paper. We can still conclude Theorems 1.1, 1.2, 1.4 and 1.5 for signed functions because Such a function comes with a measure μ and a function ν : → R d that has |ν| = 1 μ-a.e. such that for all ϕ ∈ C 1 c ( ; R d ) we have u div ϕ = ϕν dμ.
If ∇u is a locally integrable function we call u weakly differentiable. Proof By the weak compactness of L p (R d ) there is a subsequence, for simplicity also denoted by (u n ) n , and a Then

Hardy-Littlewood maximal operator
In this section we reduce Theorem 1. Proof We formulate one proof that works both for the centered and uncentered fractional maximal operator. Let (B n ) n a sequence of balls with x ∈ B n and Assume there is a subsequence (n k ) k with r (B n k ) → 0. Then f B n k → f (x) and thus lim sup Hence there is a subsequence (n k ) k such that r (B n k ) converges to some value r ∈ (0, ∞). We can conclude that there is a ball B with x ∈ B and r (B) = r and B n k f → B f . So we have A similar argument shows that there exist a largest ball B for which sup B x r (B) α f B is attained.
Then M α f is locally Lipschitz.
Proof If f = 0 then the statement is obvious, so consider f = 0. Let B be a ball. Then there is a ball A ⊃ B with f A > 0. Define 2r (A)) .
That means that on B the maximal function M α f is the supremum over all functions σ −1 d r α−d f * 1 B(z,r ) with r ≥ r 0 and z such that 0 ∈ B(z, r ) for the uncentered operator and z = 0 for the centered. Those convolutions are weakly differentiable with The following has essentially already been observed in [17,20,23,25].
Proof Let B(z, r ) ∈ B α (x) and let e be a unit vector. Note that for the centered maximal operator we have z = x. Then for all h > 0 we have x + he ∈ B(z, r + h). Thus Now we reduce Theorem 1.1 to Theorem 1.2. We prove Theorem 1.2 in Sect. 4.
Proof of Theorem 1.1 For each n ∈ N define a cutoff function ϕ n by Then |∇ϕ n (x)| = 2 −n 1 2 n ≤|x|≤2 n+1 and thus Then since M α f n → M α f pointwise from below, M α f n converges to M α f in L 1 loc (R d ). So from Lemma 2.1 it follows that By Lemma 2.3 we have that M α f n is weakly differentiable and differentiable almost everywhere, so that by Lemmas 2.2, 2.4 and Theorem 1.2 we have which by formula (2.2) converges to ∇ f p . for n → ∞. For the endpoint p = d/α the proof works the same.

Dyadic maximal operator
In this section we reduce Theorem 1.4 to Theorem 1.5. Let 1 ≤ p < d/α and f ∈ L p (R d ).
Recall that we denote by Q α the set of all dyadic cubes Q such that for every dyadic cube ball P Q we have l(P) α f P < l(Q) α f Q . For x ∈ R d , we denote by Q α (x) the set of dyadic cubes Q with x ∈ Q and

Lemma 2.5 Let 1 ≤ p < d/α and f ∈ L p (R d ) and x ∈ R d be a Lebesgue point of f . Then Q α (x) contains a dyadic cube Q x with
and that cube also belongs to Q α .
Proof Let (Q n ) n be a sequence of cubes with l(Q n ) → ∞. Then Let (Q n ) n be a sequence of cubes with l(Q n ) → 0. Then since f Q n → f (x) and l(Q n ) α → 0, we have l(Q n ) α f Q → 0. Thus since for each k there are at most 2 d many cubes Q with l(Q) = 2 k and whose closure contains x, the supremum has to be attained for a finite set of cubes from which we can select the largest. Now we reduce Theorem 1.4 to Theorem 1.5. We prove Theorem 1.5 in Sect. 3.
Proof of Theorem 1.4 By Lemma 2.5, M d α,β f is defined almost everywhere. We have where the last step follows from Theorem 1.5. In the endpoint case we have by Theorem 1.5

Dyadic maximal operator
In this section we prove Theorem 1.5. For a measurable set E ⊂ R d we define the measure theoretic boundary by We denote the topological boundary by ∂ E. As in [29,30], our approach to the variation is the coarea formula rather then the definition of the variation, see for example [13,Theorem 5.9]. Then Lemma 3.2 Let f ∈ L 1 loc (R d ) be weakly differentiable and U ⊂ R d and λ 0 < λ 1 . Then Recall also the relative isoperimetric inequality for cubes.

Lemma 3.3 Let Q be a cube and E be a measurable set. Then
We will use a result from the case α = 0. For a subset Q ⊂ Q 0 and Q ∈ Q 0 , we denote Proposition 3.4 Let 1 ≤ p < ∞ and f ∈ L 1 loc (R d ) and |∇ f | ∈ L p (R d ). Then for every set For p = 1 it also holds with ∇ f 1 replaced by var f .

Remark 3.5
We have that α < β implies Q β ⊂ Q α . This is because for l(Q) < l(P), l(Q) α f Q > l(P) α f P becomes a stronger estimate the larger α becomes.
By Remark 3.5 we can apply Proposition 3.4 to Q = Q α . For p = 1 Proposition 3.4 is Proposition 2.5 in [29]. For the proof for all p ≥ 1 we follow the strategy in [29]. In particular we use the following result. For Q ∈ Q 0 we denotē Lemma 3.6 (Corollary 3.3 in [29]) Let f ∈ L 1 loc (R d ). Then for every Q ∈ Q 0 we have Note that f P >λ P implies P ∈ Q 0 .
Proof of Proposition 3.4 By Lemmas 3.3, 3.2 we have for each P ∈ Q 0 and P Q that We note that for any Q ∈ Q we have λ Q Q ≥ λ ∅ Q and use Lemma 3.6. Then we apply the above calculation, Hölder's inequality and use that (λ P , f P ) and (λ Q , f Q ) are disjoint for P Q, For p = 1 with var f instead of ∇ f 1 we do not use Lemma 3.2 or Hölder's inequality, but interchange the order of summation first and then apply Lemma 3.1.
For a dyadic cube Q denote by prt(Q) the dyadic parent cube of Q. Lemma 3.7 Let 1 ≤ p < d/α and f ∈ L p (R d ) and let ε > 0. Then there is a subsetQ α of Q α such that for each Q ∈ Q α with l(Q) α f Q > ε there is a P ∈Q α with Q ⊂ prt(P) and f Q ≤ 2 d f P . Furthermore for any two Q, P ∈Q α one of the following holds. sup{ f P : P ∈ Q, P Q} ≤ 2 −ε f Q For the endpoint p = ∞ let Q ∈ Q α . Then we use f prt(Q) ≤ 2 −α f Q and copy the proof of the endpoint in Lemma 3.8 with p(Q) = prt(Q) and ε = 1/2.

Hardy-Littlewood maximal operator
In this section we prove Theorem 1.2.

Making the balls disjoint
and let ε > 0. Then for any c 1 ≥ 2, c 2 ≥ 1 there is a set of balls B ⊂ B α such that for two balls which means that r (B) is bounded by .
Then for all B ∈ B 0 we have that r (B) α f B is uniformly bounded. Inductively define a sequence of balls as follows. For B 0 , . . . , B k−1 already defined choose a ball B k ∈ B 0 such that c 1 B k is disjoint from c 1 B 0 , . . . , c 1 B k−1 and which attains at least half of Then B 0 ⊂ B 0 . We proceed by induction. For each n ∈ N define as above greedily select a sequence B n of balls B ∈ B n with almost maximal f B such that for every already selected A ∈ B n we have c 1 B ∩ c 1 A = ∅, and define Note that we have B n ⊂ B n . Finally set B = B 0 ∪ B 1 ∪ . . .. For A ∈ B, we denote Let λ > ε and B ∈ B α with r (B) α+β f B > λ. Then there is an n with B ∈ B n , and hence a A ∈ B n with B ∈ U A,λ . Let A ∈ B and B(x, r ) ∈ U A,λ . Then A ⊂ B(x, 5c 1 r (A)). Since r ∈ R α f (x) we have r α f B(x,r ) ≥ (5c 1 r (A)) α f B(x,5c 1 r (A)) ≥ (5c 1 r (A)) α (5c 1 ) −d f A which implies r ≥ (5c 1 ) 1−d/α r (A)( f A / f B(x,r ) ) 1/α ≥ (5c 1 ) 1−d/α c 1/α 2 r (A). Since r ≤ 5c 1 r (A) it follows that Acknowledgements I would like to thank my supervisor, Juha Kinnunen, for all of his support. I would like to thank Olli Saari for introducing me to this problem. I am also thankful for the discussions with Juha Kinnunen, Panu Lahti and Olli Saari who made me aware of a version of the coarea formula [13,Theorem 3.11], which was used in the first draft of the proof, and for discussions with David Beltran, Cristian González-Riquelme and Jose Madrid, in particular about the centered fractional maximal operator. The author has been supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.
Funding Open Access funding provided by Aalto University.
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