Continuity of the gradient of the fractional maximal operator on W^{1,1}(\mathbb{R}^d)
David Beltran, Cristian González-Riquelme, José Madrid, Julian Weigt
CCONTINUITY OF THE GRADIENT OF THE FRACTIONAL MAXIMALOPERATOR ON W , ( R d ) DAVID BELTRAN, CRISTIAN GONZ ´ALEZ-RIQUELME, JOS´E MADRID AND JULIAN WEIGT
Abstract.
We establish that the map f (cid:55)→ |∇M α f | is continuous from W , ( R d ) to L q ( R d ), where α ∈ (0 , d ), q = dd − α and M α denotes either the centered or non-centeredfractional Hardy–Littlewood maximal operator. In particular, we cover the cases d > α ∈ (0 ,
1) in full generality, for which results were only known for radial functions. Introduction
Given f ∈ L ( R d ) and 0 ≤ α < d , the centered fractional Hardy–Littlewood maximaloperator is defined by M α f ( x ) := sup r> r α | B ( x, r ) | ˆ B ( x,r ) | f ( y ) | d y for every x ∈ R d . The non-centered version of M α , denoted by (cid:102) M α , is defined by taking thesupremum over all balls B ( z, r ) such that x is contained in the closure of B ( z, r ). In whatfollows, we use M α to denote either the centered or non-centered version, in the sense that ifwe formulate a result or a proof for M α , we mean that it holds for both M α and (cid:102) M α . Thenon-fractional case α = 0 corresponds to the classical maximal function, which we denote by M = M , (cid:102) M = (cid:102) M and M = M .The study of regularity properties for M and M α started with the influential works ofKinnunen [13] and Kinnunen and Saksman [14], where it was established that |∇M α f ( x ) | ≤ M α |∇ f | ( x ) (1.1)a.e. in R d . The mapping properties of M α then imply that the map f (cid:55)→ M α f is boundedfrom W ,p ( R d ) to W ,q ( R d ) when 1 < p ≤ d/α and q = p − αd . At the endpoint p = 1 thisboundedness fails since M α f / ∈ L q ( R d ) unless f = 0 a.e.. However, one can still consider thefollowing question:Is the map f (cid:55)→ |∇M α f | bounded from W , ( R d ) to L dd − α ( R d ) ? (1.2)By a dilation argument, this is equivalent to proving that there exists a constant C > (cid:107)∇M α f (cid:107) L d/ ( d − α ) ( R d ) ≤ C (cid:107)∇ f (cid:107) L ( R d ) . (1.3)This question was first explored in the classical case α = 0 and d = 1 [23, 1, 15] and, morerecently, for d >
1, radial functions and non-centered (cid:102) M [18]. For α >
0, this boundedness wasfirst considered in [6], where the case d = 1 was settled for (cid:102) M α . Moreover, they observed thatthe case d >
1, 1 ≤ α < d follows via Sobolev embedding and the smoothing property of M α Date : February 23, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Fractional maximal function, Sobolev spaces, Continuity. a r X i v : . [ m a t h . C A ] F e b D. BELTRAN, C. GONZ´ALEZ-RIQUELME, J. MADRID AND J. WEIGT obtained by Kinnunen and Saksman [14], which ensures, that if 1 ≤ α < d and f ∈ L p ( R d )with 1 ≤ p ≤ d/α , then |∇M α f ( x ) | ≤ ( d − α ) M α − f ( x ) (1.4)a.e. in R d .For 0 < α <
1, the first boundedness result in higher dimensions was established for (cid:102) M α in [19] for radial functions. Analogous results in both d = 1 and d > M α in [3], where a pointwise relation between ∇ M α and ∇ (cid:102) M α was observed for the firsttime for α >
0. That relation revealed that both operators behave quite similarly, unlike itwas previously thought; note that without taking the gradient the two maximal functions arecomparable. Very recently, the question (1.2) was established in full generality by the fourthauthor in [24] for α >
0, completing the remaining open cases in the fractional setting (thatis, d >
1, 0 < α < f ). He originally proved it for the uncentered operator (cid:102) M α ,but he observed shortly after that almost the same proof also works for the centered operator M α , see [24, Remark 1.3]. The proof in [24] is based on the corresponding bound for the dyadicmaximal operator in the non-fractional case α = 0 in [25]. Other interesting related results inthe context of fractional maximal functions have recently been proven in [4, 9, 12, 21, 22].In this manuscript we explore the continuity of the map f (cid:55)→ |∇M α f | for α >
0. Notethat this map is not sublinear, and thus its boundedness from W , ( R d ) to L d/ ( d − α ) ( R d ) doesnot immediately imply its continuity as a map between those function spaces. For p > M α . Once again, the endpoint case p = 1 is moreintricate. For d = 1 the continuity was established by the third author [20] for the non-centeredcase and by the first and third authors [2] for the centered case. For d >
1, similarly to theboundedness, we shall distinguish between the ranges 1 ≤ α < d and 0 < α <
1. For theformer range, the result can be obtained via the inequality (1.4) and dominated convergencetheorem arguments. This was proven in [2]. The range 0 < α < f were obtained by the first andthird authors in both the non-centered [2] and centered case [3]. We refer to [7, 5, 10, 17] forcomplementary results regarding the continuity of (cid:102) M .Here we establish the following complete result for α >
0, which in particular yields thecontinuity in the remaining open cases, that is, for d >
1, 0 < α < f ∈ W , ( R d ). Theorem 1.1.
Let M α ∈ { (cid:102) M α , M α } . If < α < d , the operator f (cid:55)→ |∇M α f | mapscontinuously W , ( R d ) into L d/ ( d − α ) ( R d ) . As observed in [2], it suffices to establish the continuity for any compact set K ⊂ R d . For anygiven δ >
0, we consider two types of points in K , depending on whether the ball with maximalaverage has large radius (larger than δ ) or small radius (smaller than δ ). The techniques from[2, 3] immediately apply to prove the continuity for the points whose maximal ball has large radius: the radiality assumption was not used in that situation.Thus, in order to establish continuity in Theorem 1.1, it suffices to bound contributionscoming from points whose maximal ball has small radius, i.e. radius smaller than δ , and showthat they go to zero for δ →
0. This is the main novelty of this paper. To obtain this boundfor points with small radius, we first note that on any compact set K , M α f is bounded awayfrom 0. Then we use the Poincar´e–Sobolev inequality, which becomes stronger the smaller theradius is and the larger the average of the function is. Then we apply a refined version of (1.4)which allows us to invoke a local version of the boundedness (1.3) in [24] on the subset of pointswith small radius. This yields the desired result. In the passage, we also use a refined versionof (1.1). ONTINUITY GRADIENT FRACTIONAL MAXIMAL FUNCTION 3
The proof of Theorem 1.1 is presented in Section 4. Auxiliary results which feature promi-nently in the proof are presented in Sections 2 and 3.
Notation.
Given a measurable set E ⊆ R d , we denote by E c := R d \ E the complementary setof E in R d . For c ∈ R , we denote by cE the concentric set to E dilated by c . The integralaverage of f ∈ L ( R d ) over E is denoted by f E ≡ ´ E f := | E | − ´ E f . Given a ball B ⊆ R d ,we denote its radius by r ( B ). The volume of the d -dimensional unit ball is denoted by ω d . Theweak derivative of f is denoted by ∇ f . Acknowledgments.
The authors would like to thank Juha Kinnunen for his encouragement.D.B. was partially supported by NSF grant DMS-1954479. J.W. has been supported by theVilho, Yrj¨o and Kalle V¨ais¨al¨a Foundation of the Finnish Academy of Science and Letters.2.
Families of good balls
In this section we develop some estimates and identities regarding the weak derivative of themaximal functions of interest. We shall only be concerned with 0 < α < d , although many ofthe arguments can also be extended to α = 0.2.1. The truncated fractional maximal function.
An important object for our purposesare the truncated fractional maximal operators which, for a given δ >
0, are defined as M δα f ( x ) := sup r>δ r α ˆ B ( x,r ) | f ( y ) | d y and (cid:102) M δα f ( x ) := sup ¯ B ( z,r ) (cid:51) xr>δ r α ˆ B ( z,r ) | f ( y ) | d y. We use M δα to denote either M δα or (cid:102) M δα . Note that if δ = 0, we recover the original operators M α = M α . The following is a well-known and elementary result; see for instance [3, Lemma2.4] and [11, Lemma 8]. Proposition 2.1.
Let < α < d and δ > . If f ∈ L ( R d ) , then M δα f is Lipschitz continuous(in particular, a.e. differentiable). Weak derivative and approximate derivative.
As mentioned in the introduction, thefourth author proved in [24], after partial contributions by many, the following result.
Theorem 2.2 ([24, Theorem 1.1 and Remark 1.3]) . Let < α < d and f ∈ W , ( R d ) . Then M α f is weakly differentiable and there exists a constant C d,α > such that (cid:107)∇M α f (cid:107) L d/ ( d − α ) ( R d ) ≤ C d,α (cid:107)∇ f (cid:107) L ( R d ) . It will be convenient in our arguments to also recall the concept of approximate derivative.A function f : R d → R is said to be approximately differentiable at a point x ∈ R if thereexists a vector Df ( x ) ∈ R d such that, for any ε >
0, the set A ε := (cid:26) x ∈ R : | f ( x ) − f ( x ) − (cid:104) Df ( x ) , x − x (cid:105)|| x − x | < ε (cid:27) (2.1)has x as a density point. In this case, Df ( x ) is called the approximate derivative of f at x and it is uniquely determined. It is well-known that if f is weakly differentiable, then f isapproximate differentiable a.e. and the weak and approximate derivatives coincide [8, Theorem6.4].The approximate derivative satisfies the following property, which will play a rˆole in Propo-sitions 2.4 and 2.6 below. D. BELTRAN, C. GONZ´ALEZ-RIQUELME, J. MADRID AND J. WEIGT ερ Df ( x ) ρ sin ε Γ ρε B (0 , ρ ) A ε Figure 1.
The sets Γ ε,ρ and A ε intersect. Lemma 2.3.
Let f be approximately differentiable at a point x ∈ R d . Then there exists asequence { h n } n ∈ N with | h n | → such that | Df ( x ) | = − lim n →∞ f ( x + h n ) − f ( x ) | h n | , where Df ( x ) denotes the approximate derivative of f at x .Proof. Let 0 < ε < π/
2. By the definition of the approximate derivative, there exists 0 < ρ < ε such that | A ε ∩ B (0 , ρ ) | ≥ (cid:16) − ω d − d ω d (sin ε ) d − (cos ε ) d (cid:17) | B (0 , ρ ) | (2.2)where A ε is as in (2.1).If Df ( x ) = 0, the result simply follows by the definition of A ε and taking ε = 1 /n .Assume next Df ( x ) (cid:54) = 0. For each h ∈ R d , let β h denote the angle formed by h and − Df ( x ),so that −(cid:104) Df ( x ) , h (cid:105) = | Df ( x ) || h | cos β h . The set Γ ε,ρ := { h ∈ B (0 , ρ ) : β h ≤ ε } has measure | Γ ε,ρ | > ˆ ρ cos ε ω d − ( r sin ε ) d − d r = ω d − d (sin ε ) d − (cos ε ) d ρ d . Thus, it follows from (2.2) that Γ ε,ρ ∩ A ε (cid:54) = ∅ , so by the definition of A ε there is an h ∈ R d such that | f ( x + h ) − f ( x ) − (cid:104) Df ( x ) , h (cid:105)|| h | < ε, β h ≤ ε and 0 < | h | < ρ < ε. (2.3)By the triangle inequality, for h satisfying (2.3), (cid:12)(cid:12)(cid:12)(cid:12) | Df ( x ) | + f ( x + h ) − f ( x ) | h | (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) | Df ( x ) | + (cid:104) Df ( x ) , h (cid:105)| h | (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) f ( x + h ) − f ( x ) | h | − (cid:104) Df ( x ) , h (cid:105)| h | (cid:12)(cid:12)(cid:12)(cid:12) < | Df ( x ) || − cos β h | + ε ≤ | Df ( x ) || − cos ε | + ε. ONTINUITY GRADIENT FRACTIONAL MAXIMAL FUNCTION 5 As | Df ( x ) | (cid:54) = 0, the result now follows taking ε = min { / n, / (cid:112) | Df ( x ) n } and the corre-sponding h n = h from the previous display. (cid:3) The approximate derivative of
M f for a.e. approximately differentiable functions f ∈ L ( R d )was studied by Haj(cid:32)lasz and Maly [11]. In particular, their arguments show that if f ∈ L isa.e. approximately differentiable, then M α f is a.e. approximately differentiable.2.3. The families of good balls.
Let 0 < α < d and δ ≥
0. For the uncentered maximaloperator, given a function f ∈ W , ( R d ) and a point x ∈ R d , define the family of good balls for f at x as B δα,x ≡ B δα,x ( f ) := (cid:110) B ( z, r ) : r ≥ δ, x ∈ B ( z, r ) , M δα f ( x ) = r α ˆ B ( z,r ) | f ( y ) | d y (cid:111) . For the centered maximal operator we use the same definition, except that z = x . Notethat B δα,x (cid:54) = ∅ for all x ∈ R d if δ >
0. Moreover, by the Lebesgue differentiation theorem B α,x ≡ B α,x (cid:54) = ∅ for a.e. x ∈ R d , and if B ( z, r ) ∈ B α,x , then r >
0. This immediately impliesthat for a.e. x there exists δ x > ≤ δ < δ x , then M δα f ( x ) = M α f ( x ) . This type of observation will be used at the derivative level in the forthcoming Lemma 3.5.2.4.
Luiro’s Formula.
An important tool for our purposes is the so called Luiro’s formula,which relates the derivative of the maximal function with the derivative of the original function.It corresponds to a refinement of Kinnunen’s inequality (1.1) and has its roots in [16, Theorem3.1].
Proposition 2.4.
Let < α < d , δ ≥ and f ∈ W , ( R d ) . Then, for a.e. x ∈ R d and B = B ( z, r ) ∈ B δα,x , the weak derivative ∇M δα f satisfies ∇M δα f ( x ) = r α ˆ B ∇| f | ( y ) d y. (2.4) Proof.
This essentially follows from an argument of Haj(cid:32)lasz and Maly [11, Theorem 2], whichwe include for completeness. By § M δα f equals its approximate gradientalmost everywhere, so it suffices to show (2.4) at a point x at which M δα f is approximatelydifferentiable and for which there exists B = B ( z x , r x ) ∈ B δα,x . Define the function ϕ : R d → R by ϕ ( y ) := M δα f ( y ) − r α ˆ B ( z x + y − x,r x ) | f ( t ) | d t = M δα f ( y ) − r α ˆ B ( z x − x,r x ) | f ( y + t ) | d t, which satisfies ϕ ≥ ϕ ( x ) = 0. Thus, ϕ has a minimum at x . Furthermore, ϕ isapproximately differentiable at x (note that one can differentiate under the integral sign) andby Lemma 2.3 there exists a sequence { h n } n ∈ N with | h n | → | Dϕ ( x ) | = − lim n →∞ ϕ ( x + h n ) − ϕ ( x ) | h n | . As ϕ has a minimum at x , the right-hand side is non-positive and thus Dϕ ( x ) = 0, which yieldsthe desired result. (cid:3) Remark . Proposition 2.4 continues to hold for α = 0, replacing the weak derivative by theapproximate derivative in the cases where the weak differentiability of M is currently unknown. D. BELTRAN, C. GONZ´ALEZ-RIQUELME, J. MADRID AND J. WEIGT
Refined Kinnunen–Saksman Inequality.
The Kinnunen–Saksman inequality (1.4) ad-mits a refinement in terms of the good balls, in the same spirit as Luiro’s formula (2.4) improvesover Kinnunen’s pointwise inequality (1.1). It is noted that further refinements involving bound-ary terms (that is, averages along spheres) have been obtained in [19] and [3] for (cid:102) M α and M α respectively, although these are not required for the purposes of this paper. Proposition 2.6.
Let < α < d , δ ≥ and f ∈ W , ( R d ) . Then, for a.e. x ∈ R d and B = B ( z, r ) ∈ B δx,α , the weak derivative ∇M δα f satisfies |∇M δα f ( x ) | ≤ ( d − α ) r α − ˆ B | f ( y ) | d y. (2.5) Proof. By § M δα f equals its approximate gradient almost everywhere,so it suffices to show (2.5) at a point x at which M δα f is approximately differentiable and forwhich there exists B = B ( z x , r x ) ∈ B δα,x . By Lemma 2.3 there is a sequence { h n } n ∈ N with | h n | → |∇M δα f ( x ) | = lim n →∞ M δα f ( x ) − M δα f ( x + h n ) | h n | . Now the proof follows from the classical Kinnunen–Saksman [14] reasoning, which we includefor completeness. Note that x + h n ∈ B ( z + h n , r + | h n | ), and that for the centered maximaloperator we have z = x . This implies M δα f ( x + h n ) ≥ ( r + | h n | ) α ˆ B ( z + h n ,r + | h n | ) | f ( y ) | d y. Therefore M δα f ( x ) − M δα f ( x + h n ) | h n |≤ ω d | h n | (cid:16) r α − d ˆ B ( z,r ) | f ( y ) | d y − ( r + h n ) α − d ˆ B ( z + h n ,r + | h n | ) | f ( y ) | d y (cid:17) ≤ ω d | h n | (cid:16) r α − d ˆ B ( z + h n ,r + | h n | ) | f ( y ) | d y − ( r + | h n | ) α − d ˆ B ( z + h n ,r + | h n | ) | f ( y ) | d y (cid:17) = r α − d − ( r + | h n | ) α − d ω d | h n | ˆ B ( z + h n ,r + | h n | ) | f ( y ) | d y → ( d − α ) r α − d − ω d ˆ B ( z,r ) | f ( y ) | d y for n → ∞ , which concludes the proof. (cid:3) Remark . Proposition 2.6 continues to hold for α = 0, replacing the weak derivative by theapproximate derivative in the cases where the weak differentiability of M is currently unknown.2.6. A refined fractional maximal function.
In view of the Kinnunen–Saksman type in-equality (2.5), it is instructive to define the operator M α, − f ( x ) = sup B ∈B α,x ( f ) r ( B ) α − ˆ B | f ( y ) | d y, so that for any 0 < α < d , |∇M α f ( x ) | ≤ ( d − α ) M α, − f ( x ) for a.e. x ∈ R d . (2.6)Furthermore, this extends to the case δ >
0, that is, |∇M δα f ( x ) | ≤ ( d − α ) M α, − f ( x ) for a.e. x ∈ R d . (2.7) ONTINUITY GRADIENT FRACTIONAL MAXIMAL FUNCTION 7
Indeed, let δ > B ∈ B δα,x . Then, there exists C ∈ B α,x such that r ( C ) ≤ r ( B ). Thisimmediately yields r ( B ) α − ˆ B | f | ≤ r ( C ) α − ˆ C | f | ≤ M α, − f ( x ) , which implies (2.7) via Proposition 2.6.The proof of Theorem 2.2 in [24] is obtained through the analogous bound on M α, − . Indeed,such a bound is of local nature. The following is a local version of [24, Theorem 1.2]; see [24,Remark 1.9]. Theorem 2.8.
Let < α < d and E ⊆ R d . There exist constants c > and C d,α > suchthat the inequality (cid:107)M α, − f (cid:107) L d/ ( d − α ) ( E ) ≤ C d,α (cid:107)∇ f (cid:107) L ( D ) holds for all f ∈ W , ( R d ) , where D = (cid:91) B ∈I E cB and I E := { B ∈ B α,x ; for some x ∈ E } . Remark . For 0 < α < d one has, combining (2.6) and Theorem 2.8, that (cid:107)∇M α f (cid:107) L d/ ( d − α ) ( E ) ≤ ( d − α ) C d,α (cid:107)∇ f (cid:107) L ( D ) , where C d,α is the constant in Theorem 2.8.2.7. Poincar´e–Sobolev Inequality.
Another important tool for our purposes is the following.
Lemma 2.10.
Let < α < d , f ∈ W , ( R d ) , x ∈ R d , B = B ( z, r ) ∈ B α,x ( f ) and c > . Thenthere is a constant C d,α,c such that ˆ cB | f ( y ) | d y ≤ C d,α,c r ˆ cB |∇ f ( y ) | d y. Proof.
By the triangle inequality and the Poincar´e-Sobolev inequality there is a C d such that ˆ cB (cid:12)(cid:12) | f ( y ) | − | f | cB (cid:12)(cid:12) d y ≤ ˆ cB | f ( y ) − f cB | d y ≤ C d r ˆ cB |∇ f ( y ) | d y. Since B ∈ B α,x we have c α | f | cB < | f | B . This and the triangle inequality yield c d ˆ cB (cid:12)(cid:12) | f ( y ) | − | f | cB (cid:12)(cid:12) d y ≥ ˆ B (cid:12)(cid:12) | f ( y ) | − | f | cB (cid:12)(cid:12) d y ≥ | f | B − | f | cB ≥ ( c α − ˆ cB | f ( y ) | d y. Then, combining the above, we obtain ˆ cB | f ( y ) | d y ≤ c d C d c α − r ˆ cB |∇ f ( y ) | d y, as desired. (cid:3) Convergences
In this section we review some auxiliary convergence results established in the series of papers[7, 2] which reduce the proof of Theorem 1.1 to the convergence of the difference M α f j − M δα f j on a compact set.3.1. A Sobolev space lemma.
We start recalling an auxiliary result concerning the con-vergence of the modulus of a sequence in W , ( R d ). This is useful in view of the identity(2.4). Lemma 3.1 ([2, Lemma 2.3]) . Let f ∈ W , ( R d ) and { f j } j ∈ N ⊂ W , ( R d ) be such that (cid:107) f j − f (cid:107) W , ( R d ) → as j → ∞ . Then (cid:13)(cid:13) | f j | − | f | (cid:13)(cid:13) W , ( R d ) → as j → ∞ . D. BELTRAN, C. GONZ´ALEZ-RIQUELME, J. MADRID AND J. WEIGT
Convergence outside a compact set.
By Theorem 2.2 and the work of the first andthird author in [2] we have that it suffices to study the convergence in a compact a set.
Proposition 3.2 ([2, Proposition 4.10]) . Let < α < d , f ∈ W , ( R d ) and { f j } j ∈ N ⊂ W , ( R d ) such that (cid:107) f j − f (cid:107) W , ( R d ) → . Then, for any ε > there exists a compact set K and j ε > such that (cid:107)∇M α f j − ∇M α f (cid:107) L d/ ( d − α ) ((3 K ) c ) < ε for all j ≥ j ε . Continuity of M δα in W , ( R d ) , δ > . A key observation is the a.e. convergence of themaximal function M δα f j at the derivative level. Lemma 3.3.
Let < α < d , δ ≥ , f ∈ W , ( R d ) and { f j } j ∈ N ⊂ W , ( R d ) be such that (cid:107) f j − f (cid:107) W , ( R d ) → as j → ∞ . Then ∇M δα f j ( x ) → ∇M δα f ( x ) a.e. as j → ∞ . A version of this result for the full M α is given in [2, Lemma 2.4]. The proof for M δα is identi-cal (in fact, it slightly simplifies), and relies on Luiro’s formula for M δα , that is, Proposition 2.4.We omit further details. For δ >
0, we have the following norm convergence.
Proposition 3.4.
Let < α < d , δ > , f ∈ W , ( R d ) and { f j } j ∈ N ⊂ W , ( R d ) be such that (cid:107) f j − f (cid:107) W , ( R d ) → as j → ∞ . Let K ⊂ R d be a compact set. (cid:107)∇M δα f − ∇M δα f j (cid:107) L d/ ( d − α ) ( K ) → as j → ∞ . Proof.
By Proposition 2.4 and Lemma 3.1 there exists j ∈ N such that |∇M δα f j ( x ) | ≤ ω d δ d − α (cid:107)∇| f j |(cid:107) ≤ ω d δ d − α (cid:107)∇| f |(cid:107) + 1 for all j ≥ j , a.e. x ∈ K. Furthermore, by Lemma 3.3 ∇M δα f j ( x ) → ∇M δα f ( x ) a.e. as j → ∞ . The convergence on L d/ ( d − α ) ( K ) then follows from the dominated convergence theorem. (cid:3) δ -convergence of ∇M δα f . Here we establish that ∇M δα f provides a good approximationfor ∇M α f in L d/ ( d − α ) ( K ) when δ →
0. This relies on the Theorem 2.2.
Lemma 3.5.
Let < α < d and f ∈ W , ( R d ) . Then (cid:107)∇M α f − ∇M δα f (cid:107) L d/ ( d − α ) ( K ) → as δ → . Proof.
Recall from § x ∈ R d one has that if B ( z, r ) ∈ B δα,x , then r >
0. Thisand Luiro’s formula (2.4) imply that for a.e. x ∈ R d there exists δ x > ∇M δα f ( x ) = ∇M α f ( x ) for all 0 ≤ δ < δ x , and thus ∇M δα f ( x ) → ∇M α f ( x ) for a.e. x ∈ R d as δ →
0. Furthermore, as proven in (2.7),for a.e. x ∈ R d we have that |∇M δα f ( x ) | ≤ M α, − f ( x ) for all δ ≥ f ∈ W , ( R d ), Theorem 2.8 ensures that M α, − f ∈ L d/ ( d − α ) ( R d ) and we can thenconclude the result by the dominated convergence theorem. (cid:3) ONTINUITY GRADIENT FRACTIONAL MAXIMAL FUNCTION 9 Proof of Theorem 1.1
Let f ∈ W , ( R d ) and { f j } j ∈ N ⊂ W , ( R d ) be a sequence of functions such that (cid:107) f j − f (cid:107) W , ( R d ) → j → ∞ . If f = 0 then the result follows directly from the boundedness, thatis Theorem 2.2. From now on we assume that f (cid:54) = 0. Let ε >
0. Then by Proposition 3.2 it issufficient to prove that there exists j ∗ ∈ N such that (cid:107)∇M α f − ∇M α f j (cid:107) L d/ ( d − α ) ( K ) < ε (4.1)for all j ≥ j ∗ . To this end, for any δ >
0, use the triangle inequality to bound (cid:107)∇M α f − ∇M α f j (cid:107) L d/ ( d − α ) ( K ) ≤ (cid:107)∇M α f − ∇M δα f (cid:107) L d/ ( d − α ) ( K ) (4.2)+ (cid:107)∇M δα f − ∇M δα f j (cid:107) L d/ ( d − α ) ( K ) + (cid:107)∇M δα f j − ∇M α f j (cid:107) L d/ ( d − α ) ( K ) . To finish the proof, it suffices to show that for ε > δ ∗ and a j ∗ such thatfor δ = δ ∗ and all j ≥ j ∗ , each of the summands on the right hand side of (4.2) is bounded by ε . We choose δ ∗ depending on ε, K and f , and j ∗ depending on δ ∗ , ε , K , f and the sequence { f j } j ∈ N .For the first term, we know by Lemma 3.5 that there exists a δ (cid:48) > (cid:107)∇M α f − ∇M δα f (cid:107) L d/ ( d − α ) ( K ) < ε for all 0 ≤ δ ≤ δ (cid:48) . For the second term, we have by Proposition 3.4 that for every δ > j ( δ ) ∈ N such that (cid:107)∇M δα f − ∇M δα f j (cid:107) L d/ ( d − α ) ( K ) < ε for all j ≥ j ( δ ). The rest of the section is devoted to proving a favourable bound for the thirdterm. More precisely, we will show that there are ˜ δ > j ∈ N such that for all 0 ≤ δ ≤ ˜ δ and j ≥ ˜ j , (cid:107)∇M δα f j − ∇M α f j (cid:107) L d/ ( d − α ) ( K ) < ε. (4.3)Temporarily assuming this, we can then conclude that for δ = δ ∗ := min { δ (cid:48) , ˜ δ } and j ≥ j ∗ :=max { j ( δ ∗ ) , ˜ j } , the right-hand side of (4.2) is bounded by at most 3 ε , as desired for (4.1).We now turn to the proof of (4.3). We start by noting that there exists a λ > j ∈ N such that for all j ≥ j and x ∈ K we have M α f j ( x ) > λ . Indeed, as f ∈ L ( R d ), there existsa ball B that contains K with ´ B | f | > ´ R d | f | . As (cid:107) f j − f (cid:107) → j →
0, by the triangleinequality, there exists j > j ≥ j we have ´ B | f j | > ´ B | f | > ´ R d | f | . Then, for every j ≥ j and x ∈ K we have M α f j ( x ) ≥ α r ( B ) α ˆ B ( x, r ( B )) | f j | > (2 r ( B )) α − d ω d ˆ R d | f | , where in the last inequality we have used that B ( x, r ( B )) ⊃ B for all x ∈ K . Thus, we cantake λ to be the right-hand side of the inequality above. Furthermore by Proposition 2.4, ifthere exists a B ∈ B α,x ( f j ) such that r ( B ) ≥ δ then ∇M α f j ( x ) = ∇M δα f j ( x ). Define E λ ,δ,j := (cid:110) x ∈ K : if B ∈ B α,x ( f j ) , then r ( B ) < δ and r ( B ) α ˆ B | f j | > λ (cid:111) . By the previous two observations, Proposition 2.6 and a crude application of the triangleinequality, one has (cid:107)∇M δα f j − ∇M α f j (cid:107) L d/ ( d − α ) ( K ) = (cid:107)∇M δα f j − ∇M α f j (cid:107) L d/ ( d − α ) ( E λ ,δ,j ) ≤ d − α ) (cid:107)M α, − f j (cid:107) L d/ ( d − α ) ( E λ ,δ,j ) . for all j ≥ j . Define the indexing set I λ ,δ,j := (cid:110) B ∈ B α,x ( f j ) : x ∈ K, r ( B ) < δ and r ( B ) α ˆ B | f j | > λ (cid:111) and consider the set D λ ,δ,j := (cid:91) B ∈I λ ,δ,j cB, where c is the constant from Theorem 2.8. Then, by Theorem 2.8, we have (cid:107)M α, − f j (cid:107) L d/ ( d − α ) ( E λ ,δ,j ) ≤ C d,α (cid:107)∇ f j (cid:107) L ( D λ ,δ,j ) for any δ >
0. Thus, the proof of (4.3) is reduced to showing that there exist a ˜ δ > j ∈ N such that for all j ≥ j and 0 ≤ δ ≤ ˜ δ we have (cid:107)∇ f j (cid:107) L ( D λ ,δ,j ) < ε d − α ) C d,α , (4.4)as one can then take ˜ j := max { j , j } .In order to prove (4.4), we first use the triangle inequality and that (cid:107)∇ f j − ∇ f (cid:107) L ( R d ) → j → ∞ to find a j ∈ N such that (cid:107)∇ f j (cid:107) L ( D λ ,δ,j ) ≤ (cid:107)∇ f (cid:107) L ( D λ ,δ,j ) + ε d − α ) C d,α . (4.5)for any δ > j ≥ j .Next, let x ∈ D λ ,δ,j . Then there is a B ∈ I λ ,δ,j with x ∈ cB . So, by Lemma 2.10, we have λ ≤ c d r ( B ) α ˆ cB | f j | ≤ C d,α,c c d +1 r ( B ) α +1 ˆ cB |∇ f j | ≤ C d,α,c c d − α +1 δ (cid:102) M α |∇ f j | ( x ) , where (cid:102) M α in the above inequality denotes the uncentered fractional maximal operator. Hence,by the weak (1 , d/ ( d − α )) inequality for (cid:102) M α , | D λ ,δ,j | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x : (cid:102) M α |∇ f j | ( x ) ≥ λ C d,α,c c d − α +1 δ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C d,α,c,λ δ d/ ( d − α ) (cid:107)∇ f j (cid:107) d/ ( d − α )1 ≤ C d,α,c,λ δ d/ ( d − α ) (cid:16) (cid:107)∇ f (cid:107) d/ ( d − α )1 (cid:17) (4.6)if j ≥ j for some j ∈ N , using that (cid:107)∇ f j − ∇ f (cid:107) L ( R d ) → j → ∞ .Finally, note that as ∇ f ∈ L ( R d ), there exists ρ > A ⊆ R d satisfying | A | < ρ , one has (cid:107)∇ f (cid:107) L ( A ) < ε d − α ) C d,α . (4.7)As the right-hand side of (4.6) goes to zero for δ → j , there exists ˜ δ > | D λ ,δ,j | < ρ for all j ≥ j and δ < ˜ δ . Thus, taking j := max { j , j } , (4.4) follows fromcombining (4.5) and (4.7) with A = D λ ,δ,j . This implies the claimed inequality (4.3) andtherefore finishes the proof of Theorem 1.1. (cid:3) Remark.
Note that in the above proof, instead of using Lemma 3.5 to bound the first term in(4.2), we could have also bounded it running the same scheme as for the third term.
ONTINUITY GRADIENT FRACTIONAL MAXIMAL FUNCTION 11
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Email address : [email protected] Cristian Gonz´alez-Riquelme: IMPA - Instituto de Matematica Pura e Aplicada, Estrada DonaCastorina, 110, Jardim Botanico, Rio de Janeiro - RJ, Brazil, 22460-320.
Email address : [email protected] Jos´e Madrid: Department of Mathematics, University of California, Los Angeles (UCLA), Por-tola Plaza 520, Los Angeles, California, 90095, USA
Email address : [email protected] Julian Weigt: Aalto University, Department of Mathematics and Systems Analysis, Otakaari 1,Espoo, Finland
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