1. Introduction. The non-centered fractionalhardy-littlewoodmaximal operatorm β is defined by setting for f L 1 loc (Rn ) and 0 β < n that r β

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1 THE VARIATION OF THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION arxiv: v [math.ca] 9 Oct 207 HANNES LUIRO AND JOSÉ MADRID Abstract. In this paper we study the regularity of the noncentered fractional maximal operator M β. 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 DM β f L q (R n ) C(n,β) Df L (R n ). The corresponding result was previously known only if n = 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, (R n ).. Introduction The non-centered fractionalhardy-littlewoodmaximal operatorm β is defined by setting for f L loc (Rn ) and 0 β < n that r β M β f(x) := sup f(y) dy =: sup r β f(y) dy (z,r) x (z, r) (z,r) (z,r) x (z,r) (.) for every x R n. The centered version of M β, denoted by Mβ c, is defined by taking the supremum over all balls centered at x. In the non-fractional case β = 0, we also denote M 0 = 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,p (R n ) for p > was established by Kinnunen in [Ki]. The analogous result in the fractional context was established by Kinnunen and Saksman in [KiSa]: forevery 0 < β < nwe havethat M β isboundedfromw,p (R n ) to W,q (R n ) under the relation /q = /p β/n (if p > ). For other interesting results on this theory we refer to [CHP], [CFS], [CaHu], [CMP], [CaMo], [CaSv], [HM], [HO], [L], [Ma] and [R]. Date: October 20, Mathematics Subject Classification. 4225, 26A45, 46E35, 46E39. Key words and phrases. Fractional maximal operator, Sobolev spaces, Radial functions.

2 2 HANNES LUIRO AND JOSÉ MADRID The case p = is particularly complicated and interesting. In the case n = it is known (see [Ta] and [AlPe] for the Non-Centered, and [Ku] for the Centered) that Mf is weakly differentiable and DMf L (R) C Df L (R), (.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 (.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 DM β f bounded from W, (R n ) to L q (R n )? The problem can be rather easily reduced to the case 0 β <, as it was also observed by Carneiro and Madrid (see [CaMa]). Indeed, in the case β < d, the positive answer follows by combining the boundedness property of the fractional maximal operator from L p to L q (under the condition /q = /p β/n), the Sobolev embedding Theorem and the result in [KiSa] which says: If f L r (R n ) with < r < n and β < n/r, then M β f is weakly differentiable and DM β f(x) C(n,β)M β f(x) for a.e. x R n. 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. (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, (R n ) we have that M β f is weakly differentiable and DM β f L q (R n ) C Df L (R n ). 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 2.0. 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

3 THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 3 case n =, in other words our argument also gives a new proof for Theorem in [CaMa]. 2. Preliminaries Let us introduce some notation. The boundary of the n-dimensional unit ball is denoted by S n. The s-dimensional Hausdorff measure is denoted by H s. The volume of the n-dimensional unit ball is denoted by ω n and the H n -measure of S n by σ n. The integral average of f L loc (Rn ) over a measurable set A R n is sometimes denoted by f A. The weak derivative of f (if exists) is denoted by Df. If v S n, then D v f(x) := lim h 0 h (f(x+hv) f(x)), in the case the limit exists. For f W, (R n ), 0 β < n, let us define x β (f) } = x := {(z,r) : x (z,r), M β f(x) = r β f(y) dy. (z,r) We use to call x as the collection of the best balls at x. It is easy to see that x is non-empty set for every x R n (since f L (R n )) and also it is compact in the sense that if (z k,r k ) x and z k z R n and r k r (0, ) as k, then (z,r) x. Proposition 2.. Given f W, (R n ), a ball, a family of affine mappings L i (y) = a i y+b i, a i R, b i R n, i := L i () and a sequence {h i } i N R such that h i 0 as i, L i (y) y lim i h i = g(y) and lim i a β i h i = γ, where γ R,g : R n R, it holds that ( ) lim r β i f(y) dy r β f(y) dy i h i i =r β D f (y) g(y)dy+γr β f(y) dy, where r denotes the radius of and r i the radius of i for every i. With slight modifications in the proof in the case f is not symmetric with respect to the origin

4 4 HANNES LUIRO AND JOSÉ MADRID Proof of Proposition 2.. ( r β i f(y) dy r β h i i = ( a β i h rβ f(y) dy r β i = r β ( L i () ) f(y) dy ) f(y) dy a β i f(y +(L i(y) y)) f(y) h i ( = r β a β i f(y +(L i(y) y)) a β i f(y) dy h i ) f(y) (a β i + ) dy h i r β D f (y) g(y)dy+γr β f(y) dy as i. ) Lemma 2.2. Let f W, (R n ), x R n, x, δ > 0, and let L h (y) = a h y +b h, h [ δ,δ], be affine mappings such that x L h () and Then L h (y) y lim h 0 h 0 = (rad()) β a β h = g(y) and lim = γ. h 0 h D f (y) g(y)dy+γm β f(x). (2.3) Proof of Lemma 2.2. y the previous Proposition 2. the right hand side of (2.3) equals to ) lim (rad(l h ()) β f(y) dy r β f(y) dy h 0 h L h () =: lim h 0 h s h. (2.4) Since x and x L h ( ) for all h, it follows that s h 0 for all h. Since h can take positive and negative values, the existing limit must equal to zero.

5 THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 5 Corollary 2.3. In particular, if L h (y) = y +h(y x) we obtain that g(y) = y x, γ = β, and x = L h (x) L h ( ). Therefore (rad()) β D f (y) xdy = (rad()) β D f (y) ydy +βm β f(x). (2.5) The following lemma is a counterpart of Lemma 2.2 in [L2]. It was proved by Carneiro and Madrid in [CaMa, Theorem ] that if g W, (R) thus M β g is absolutely continuous, therefore if f W, (R n ) is a radial functions we can apply the next lemma. Lemma 2.4. Suppose that f W, (R n ) and M β f is differentiable at x. Then () For all v S n and x, it holds that D v M β f(x) = (rad()) β D v f (y)dy. (2) If x for some x, then DM β f(x) = 0. (3) If x, = (z,r) x and DM β f(x) 0, then (4) If x, then f (y)dy = β DM β f(x) DM β f(x) = z x z x. D f (y) (y x)dy. (2.6) Proof. () Let = (z,r) x and h := (z +hv,r). Then it holds for every v S n that M β f(x+hv) M β f(x) r β lim lim h f(y) dy r β f(y) dy h 0 h h 0 h = r β D v f (y)dy r β = lim h 0 f(y) dy rβ h f(y) dy h M β f(x) M β f(x hv) lim h 0 h (2) If x and x, then M β f(x) M β f(y) for every y. (3) Let = (z,r) x, v S n such that v (z x) = 0, and let usdenoteforallh (0, )thatx h := x+hv, r h := z x h, and h := (z,r h ). These definitions guarantee that x h h \.

6 6 HANNES LUIRO AND JOSÉ MADRID for all h, and h. Moreover, since v (z x) = 0, it is elementary fact that r h = z x hv z x + h2 2r. Therefore, r/r h ( h r )2, and M β f(x h ) r β h f(z) dz r β h h r β h rβ f(z) dz ( ) n β ) n β r = r β f(z) dz ( h2 M r h r 2 β f(x). This implies that D v M β f(x) 0 for all v S n such that v (z x) = 0. Since we assumed that M β f is differentiable at x, it follows that D v M β f(x) = 0 if v S n,v (z x) = 0. In particular, it follows that DM β f(x) is parallel to z x or x z. The final claim follows easily by the fact that M β f(x+ h(z x)) M β f(x) if 0 h 2. (4) This is an immediate consequence of Corollary 2.3. Proposition 2.5. If f W, loc (Rn ), z R n, r > 0, then [ D f (y) (z y)dy = n f (z,r) (z,r) f (z,r) ]. (2.7) Proof. In the case of radial functions the previous proposition follows from the one dimensional case (that is enough in order to get Theorem.). 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 R n a bounded open set with C boundary and ν denotes the outward unit normal to Ω if u W,p (Ω) and v W,q (Ω) for exponents p,q with thus the following identity holds u v = y i Ω p + q + n Ω uvν i where ν i is the i component of the vector ν. Ω v u y i

7 THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 7 Using this we get = = (z,r) n i= n ( i= = = n D f (y) (z y)dy (z,r) (z,r) (z,r) f (y)(z i y i )dy y i f(y) (z i y i ) (y ) i z i ) y z dy + f(y) dy (z,r) n y i z i 2 n f(y) dy + f(y) dy y z i= i= (z,r) f(y) dy f(y) y z dy (z,r) (z,r) [ = n f(y) dy r (z,r) n (z,r) [ = n f(y) dy rn w n (z,r) r n σ n [ = n f(y) dy (z,r) (z,r) ] f(y) dy (z,r) (z,r) ] f(y) dy ] f(y) dy. y dividing both sides of the last equality by (z,r) we arrived in the desired identity. y using Proposition 2.5 we yet state one more formula related to the derivative of the fractional maximal operator. Lemma 2.7. Suppose that f W, loc (Rn ), 0 < β < n, β x for some x R n, and r := rad(). Then D f (y)dy = n [ ( β/n) r f(y) dy ] f(y) dy. (2.8)

8 8 HANNES LUIRO AND JOSÉ MADRID Proof. Suppose that = (z,r). y Lemma 2.4 and Proposition 2.5 it follows that ( ) D f (y)dy z x = D f (y) dy r = [ ] D f (y) (z y)dy + D f (y) (y x)dy r = [ ] D f (y) (z y)dy β f (y)dy r = [ [ ] ] n f f β f (y)dy r = n [ ] ( β/n) f(y) dy f(y). r We will use the following elementary property for radial functions. The proof is left for an interested reader. Proposition 2.8. Suppose that f L loc (Rn ) satisfies f(x) = F( x ), F : (0, ) [0, ), := (z,r) (0,2 z ) \ (0, z ), and 2 a := z r, b := z +r. Then it holds that F(t)dt C(n) f(y)dy. (2.9) [a,b] (z,2r) The following two lemmas contain the key estimates for the proof of the main theorem. Lemma 2.9. Suppose that f W, loc (Rn ) is radial and x β for some x R n \{0} such that (0, x ). Then D f (y)dy Df(y) y dy. (2.0) x Proof. If DM β f(x) = 0, the claim is trivial. If DM β f(x) 0, Lemma 2.4 yields that D f (y)dy x = D f (y)dy x, (2.)

9 THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 9 and = Df(y)dy = This proves the claim. Df(y) y x dy β x D f (y) x x dy f(y) dy Df(y) y x. Given a ball = (z,r) we define 2 to be equal to (z,2r). Lemma 2.0. Suppose that f W, loc (Rn ) is radial, 0 < β < n, x β for some x R n, r := rad() x, and 4 E := {z 2 : 2 f f(z) 2 f }. (2.2) Then D f (y)dy C(n,β) Df(z) χ E (z)dz. (2.3) Proof. First observe that by Lemma 2.7 it holds that D f (y)dy n [ ] f f. (2.4) r Let then f(x) = F( x ), where F : R\{0} [0, ), let z denote the center point of, a := z r, b := z +r, and A := {t 2[a,b] : 2 f F(t) 2 f }. (2.5) Then we show that f 2 f 2 F (t) χ A (t)dt. (2.6) [a,b] The above inequality is more or less trivial: To prove it, choose t 0 [a,b] such that F(t 0 ) = f and choose t [a,b] such that f = F(t ). y (2.4) we havethat F(t 0 ) F(t ). InthecaseF(t ) f 2 in[a,b]theclaimfollowsbyusing thecontinuity, because inthiscaseby thecontinuity we canassume without loss of generality that [t 0,t ] A (or [t,t 0 ] A). Otherwise, if F(t ) < f 2, there exists t 2 [a,b] between t 0 and t such that F(t 2 ) = f 2, by the continuity of F it clearly follows that f f f 2 F (t) χ A (t)dt 2 F (t) χ A (t)dt. [t 0,t 2 ] [a,b] (2.7)

10 0 HANNES LUIRO AND JOSÉ MADRID Since Df(y) χ E (y) = F ( y ) χ A ( y ), Proposition 2.8 yields that F (t) χ A (t)dt C(n) Df(y) χ E (y)dy. (2.8) [a,b] (z,2r) Combining this with( 2.4) and (2.6) implies the desired result. Proposition 2.. Suppose that β 0, f L loc (Rn ), and := (z,r ) and 2 := (z 2,r 2 ) are best balls for M β f such that 2 (z,2r ). Then it holds that f 2 ( ) β r f. (2.9) 2 n r 2 Proof. Let := (z,2r ). Since 2 is best ball and 2,, it holds that r β 2 f 2 (2r ) β f (r ) β 2 n f. This implies the claim. 3. Proof of the main Theorem Let us fix x := (z x,r x ) x for (almost) every x E, such that r x is the smallest possible (then by Lemma 2.4 item (3) we have DM z x = x+r β f(x) x DM β ), where f(x) E := {x : DM β f(x) 0}. (3.20) y the choise of radius we can see that x r x is an upper semicontinuous function then it is measurable function, thus x z x is also a measurable function. y Lemma 2.4, it holds for almost all x E that x is of type x = (c x x, c x x ), where c x R. (3.2) In the other words, this means that the center point of x lies on the line containing x and the origin, and x lies on the boundary of x. For simplicity, let us yet denote theradiusof x byr x, thusr x = c x x. Observefirst thatforallx E itholdsthatc x 0. Toseethis, observe that otherwise (since M β f(x) = M β f( x)) it follows that x x and x x, implying that 0 = DM β f( x) = DM β f(x), which is a contradiction. We are going to use different type of estimates for

11 THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION DM β f(x) depending on how x is located with respect to the origin. Indeed, let E : = {x E : c x > 5 4 }, E 2 := {x E : 0 c x < 3 }, and 4 3 E 3 : = {x E : 4 c x 5 4 }. Then we can estimate q DM β f(x) q dx = DM β f(x) q dx = rβ x Df(y)dy dx R n E E x r qβ q x = E (ω n ) q rx n(q ) Df(y)dy Df(y)dy dx x x C(n,β) Df q Df(y)dy dx E x 3 =C(n,β) Df q Df(y)dy dx, x i= E i where we used the fact qβ = n(q ). Especially, the claim follows, if we can show that Df(y)dy dx C(n,β) Df, for i =,2,3. (3.22) x E i The case of E. In this case the easiest type of estimate turns out to be sufficient. Indeed, Df(y)dy dx Df(y) dydx E x E x χ x(y)χ E(x) = Df(y) dxdy. R n R n x For every y R n it holds that if x y and y 2 x, then r x y /4. Moreover, if y x y, then x E 2 implies that r x y x. 4 8 Finally, if x > y and x E, then x R n \(0, y ), thus y x. y combining these, we conclude that for every y R n χ x(y)χ E(x) dx dx C(n) = C(n). (3.23) R n x (0, y ) (0, y ) The case of E 2. In this case we recall the estimate from Lemma 2.9, which yields that Df(y)dy Df(y) y x x x 4n Df(y) y (0, x ) x. (3.24)

12 2 HANNES LUIRO AND JOSÉ MADRID Then the claim follows by Df(y)dy dx 4n E 2 x E 2 ( =4 R n χ (0, x )(y)χ E2(x) Df(y) y n R ω n n x n+ ( ) 4n dx Df(y) y dy ω n R n R n \(0, y ) x n+ =C(n) Df(y) dy. R n Df(y) y (0, x ) ) dx dy x dydx The case of E 3. Inthiscase we will exploit theestimate fromlemma 2.0. For this, let us denote for every x E 3 that A x := {y 2 x : 2 f x f(y) 2 f x }. (3.25) Since x E 3 implies that r x x, Lemma 2.0 yields that for every 4 x E 3 it holds that Df(y)dy C(n,β) Df(y) χ Ax (y)dy. (3.26) x 2 x Therefore, Df(y)dy dx C Df(y) χ Ax (y)dydx E 3 x E 3 2 ( x ) χ 2x(y)χ Ax(y)χ E3(x) =C Df(y) dx dy. R n R 2 n x Consider above the inner integral for fixed y R n. Firstly, suppose that χ 2x0 (y)χ Ax0 (y) 0 and χ 2x (y)χ Ax (y) 0, for some x 0,x R n. (3.27) Observe that if this kind of points does not exist, the desired estimates are trivially true. y the definition, the above means that 2 f x0 f(y) 2 f x0, and 2 f x f(y) 2 f x. Especially, it follows that 4 f x0 f x 4 f x0. (3.28)

13 THE FRACTIONAL MAXIMAL FUNCTION OF A RADIAL FUNCTION 3 Let r 0 := rad( x0 ) and r := rad( x ) and assume that r r 0. Since y 2 x0 2 x, it follows that x0 8 x. y Proposition 2., it follows that f x0 ( ) β r f 8 n r x 0 8 n ( r r 0 ) β 4 f x0, (3.29) implying that r 8 n+ β r0. If r r 0, symmetric argument gives that r 0 8 n+ β r. Summing up, it follows that 8 n+ β rad( x 0 ) rad( x ) 8n+ β. (3.30) Indeed, this means that if χ 2x (y)χ Ax (y) 0, then x y C(n,β)rad( x0 ) and x C(n,β) x0. (3.3) Naturally, (3.3) holds also if x 0 is replaced by x. Finally, this implies that χ 2x(y)χ Ax(y)χ E3(x) dx dx C(n,β) R 2 n x (y,c(n,β)rad( x0 )) x0 C(n,β). Since this holds for all y R n, 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. The authors thank Emanuel Carneiro for suggesting to think about this problem. The authors also acknowledge the referee for the valuable comments and suggestions. [AlPe] [CHP] [CFS] References J.M. Aldaz and J. Pérez Lázaro. Functions of bounded variation, the derivative of the one-dimensional maximal function, and applications to inequalities Trans. Amer. Math. Soc. 359 (2007), no. 5, J. ober, E. Carneiro, K. Hughes and L.. Pierce, On a discrete version of Tanaka s theorem for maximal functions, Proc. Amer. Math. Soc. 40 (202), E. Carneiro, R. Finder and M. Sousa, On the variation of maximal operators of convolution type II, preprint at To appear in Revista Matematica Iberoamericana.

14 4 HANNES LUIRO AND JOSÉ MADRID [CaHu] [CaMa] [CMP] [CaMo] [CaSv] [HM] [HO] [Ki] [KiSa] [Ku] [L] [L2] [Ma] [R] [S] [Ta] E. Carneiro and K. Hughes. On the endpoint regularity of discrete maximal operators Math. Res. Lett. 9 (202), no. 6, E. Carneiro and J. Madrid. Derivative bounds for fractional maximal operators Trans. Amer. Math. Soc. 369 (207), E. Carneiro, J. Madrid and L.. Pierce, Endpoint Sobolev and V Continuity for Maximal Operators, J. Funct. Anal. 273 (207), no. 0, E. Carneiro and D. Moreira, On the regularity of maximal operators, Proc. Amer. Math. Soc. 36 (2008), no. 2, E. Carneiro and. F. Svaiter, On the variation of maximal operators of convolution type, J. Funct. Anal. 265 (203), P. Haj lasz and J. Maly. On approximative differentiability of the maximal function, Proc. Amer. Math. Soc. 38 (200), no., P. Haj lasz and J. Onninen. On oundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fen. Math. 29 (2004), J. Kinnunen. The Hardy-Littlewood maximal function of a Sobolevfunction. Israel J.Math. 00 (997), J. Kinnunen and E. Saksman. Regularity of the fractional maximal function ull. London Math. Soc. 35 (2003), no. 4, O. Kurka. On the variation of the Hardy-Littlewood maximal function. Ann. Acad. Sci. Fenn. Math. 40 (205), H. Luiro. Continuity of the maximal operator in Sobolev spaces. Proc. Amer. Math. Soc. 35 (2007), no., H. Luiro The variation of the maximal function of a radial function. To appear in Arkiv för Matematik. J. Madrid, Sharp inequalities for the variation of the discrete maximal function, ull. Aust. Math. Soc. 95 (207), no., J. P. G. Ramos, Sharp total variation results for maximal functions, preprint at O. Saari, Poincaré inequalities for the maximal function, preprint at H. Tanaka. A remark on the derivative of the one-dimensional Hardy- Littlewood maximal function. ull. Aust. Math. Soc. 65 (2002), no. 2, Department of Mathematics and Statistics, University of Jyvaskyla,P.O.ox 35 (MaD), 4004 University of Jyvaskyla, Finland address: hannes.s.luiro@jyu.fi Department of Mathematics, Aalto University, P.O. ox 00, FI Aalto University, Finland address: jose.madridpadilla@aalto.fi The Abdus Salam International Centre for Theoretical Physics, Str. Costiera, 345 Trieste, Italy address: jmadrid@ictp.it