ON APPROXIMATE DIFFERENTIABILITY OF THE MAXIMAL FUNCTION
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1 ON APPROXIMATE DIFFERENTIABILITY OF THE MAXIMAL FUNCTION PIOTR HAJ LASZ, JAN MALÝ Dedicated to Professor Bogdan Bojarski Abstract. We prove that if f L 1 R n ) is approximately differentiable a.e., then the Hardy-Littlewood maximal function Mf is also approximately differentiable a.e. Moreover, if we only assume that f L 1 R n ), then any open set of R n contains a subset of positive measure such that Mf is approximately differentiable on that set. On the other hand we present an example of f L 1 R) such that Mf is not approximately differentiable a.e. 1. Introduction Juha Kinnunen [12] proved that the Hardy-Littlewood maximal function Mfx) = sup Bx, r) 1 fy) dy r>0 is a bounded operator in the Sobolev space W 1,p R n ), 1 < p <. Since the maximal function is not bounded in L 1, there is no apparent reason to expect any kind of boundedness of the maximal function in W 1,1 R n ). However, Tanaka [26] proved that in the one dimensional case the noncentered maximal function of f W 1,1 R) belongs locally to W 1,1 R). Since that time it has been an open problem to extend Tanaka s result to the case of the Hardy-Littlewood maximal function and to find analogous results in the higher dimensional case, cf. [10, Question 1]. To the best of our knowledge there are no known higher dimensional results in the case p = 1 and even in the one dimensional case it is still not known whether the Hardy-Littlewood maximal function i.e. the centered one) of f W 1,1 R) belongs locally to W 1,1 R), see, however, [2], [3]. The results proved in the paper are clearly motivated by this challenging problem. Theorem 1.1. If f L 1 R n ) is approximately differentiable a.e., then the maximal function Mf is approximately differentiable a.e Mathematics Subject Classification. Primary 42B25; Secondary 46E35, 31B05. P.H. was supported by NSF grant DMS J.M. was supported by the research project MSM and by grants GA ČR 201/06/0198, 201/09/
2 2 PIOTR HAJ LASZ, JAN MALÝ Since every function f W 1,1 R n ) is approximately differentiable a.e., the result implies a.e. approximate differentiability of Mf. This, in particular, implies see Lemma 2.1) that Mf coincides with a C 1 function off an open set of arbitrarily small measure. However, a.e. approximate differentiability of Mf is much less that weak differentiability of Mf which is still an open problem. On the other hand, the assumption about f in the theorem is much weaker than f W 1,1. In addition to this result, Theorem 1.2 provides a formula for the approximate derivative of Mf when f W 1,1. Let f L p R n ), 1 p <. It is easy to see cf. [22]) that for a.e. x R n either 1.1) Mfx) = fy) dy for some r x > 0 or Bx,r x) 1.2) Mfx) = fx). Denote by E and P the sets of points in R n for which 1.1) and respectively 1.2) is satisfied. The following result is due to Luiro [22] when p > 1 and is new when p = 1. Our proof is new and simpler even in the case p > 1. Theorem 1.2. Let f W 1,p R n ), 1 p <. Then the weak derivative, when p > 1, and approximate derivative, when p = 1, of the maximal function Mf satisfies 1.3) Mfx) = fy) dy for a.e. x E Bx,r x) Mfx) = fx) for a.e. x P. Remark 1.3. If x E, then r x > 0 is not necessarily uniquely defined and 1.3) holds for all such r x. In the next result we deal with differentiability properties of Mf for any f L 1 R n ). Theorem 1.4. If f L 1 R n ), then any open set Ω R n contains a subset E Ω of positive Lebesgue measure such that Mf is approximately differentiable a.e. in E. Again, Lemma 2.1 implies that for any open set Ω R n there is a function g C 1 R n ) such that the set {x Ω : fx) = gx)} has positive measure. In view of Theorem 1.4 it is natural to inquire whether for every f L 1 R n ) the maximal function Mf is approximately differentiable a.e. Unfortunately the answer is in the negative as an example presented at the end of the paper shows. While the proofs of Theorems 1.1 and 1.2 are completely elementary, the proof of Theorem 1.4 requires some advanced potential theory.
3 ON APPROXIMATE DIFFERENTIABILITY OF THE MAXIMAL FUNCTION 3 Let us also mention that the result of Kinnunen [12] has been applied and generalized by many authors [2], [3], [6], [7], [10], [13], [14], [16], [17], [18], [19], [20], [22], [23], [26]. Notation used in the paper is pretty standard. The volume of the unit ball in R n is denoted by ω n and we use barred integral to denote the integral average 1 fy) dy = fy) dy. Bx, r) By C we will denote a generic positive constant whose actual value may change even in a single string of estimates. 2. Approximate differentiability Let f be a real-valued function defined on a set E R n. We say that f is approximately differentiable at x 0 E if there is a vector L = L 1,..., L n ) such that for any ε > 0 the set A ε = { x : fx) fx 0) Lx x 0 ) x x 0 } < ε has x 0 as a density point. If this is the case x 0 is a density point of E and L is uniquely determined. The vector L is called the approximate differential of f at x 0 and is denoted by fx 0 ). In what follows we will need the following theorem of Whitney [27] which provides several characterizations of a.e. approximate differentiability of a function. We state it as a lemma. Lemma 2.1. Let f : E R be measurable, E R n. Then the following conditions are equivalent. a) f is approximately differentiable a.e. b) For any ε > 0 there is a closed set F E and a locally Lipschitz function f : R n R such that f F = g F and E \ F < ε. c) For any ε > 0 there is a closed set F E and a function g C 1 R n ) such that f F = g F and E \ F < ε. Investigating the positive an negative part of a function separately one can easily prove Lemma 2.2. A measurable function f : E R is a.e. approximately differentiable if and only if f is a.e. approximately differentiable. Recall that W 1,p R n ) is the space of all functions f L p R n ) such that weak distributional) partial derivatives f/ x i also belong to L p R n ), similarly for W 1,p loc Rn ). It is well known that f W 1,1 loc Rn ) is approximately
4 4 PIOTR HAJ LASZ, JAN MALÝ differentiable a.e. To prove this it suffices to assume that f W 1,1 R n ) and then this fact follows immediately from the inequality fx) fy) C x y M f x) + M f y)) a.e., see [1], [4], [5], which shows that f is Lipschitz continuous on the sets Thus we proved E t = {x : M f x) t}. Lemma 2.3. If f W 1,1 loc Rn ), then f is approximately differentiable a.e. 3. Proof of Theorem 1.1 We consider a restricted version of the maximal function M ε fx) = sup fy) dy. r ε Lemma 3.1. If f L 1 R n ), then M ε fx) M ε fy) n ε x y Mε fx) + M ε fy)) 2n ω n ε n+1 f 1 x y. Proof. The second inequality of the lemma is obvious because M ε fx) 1 ω n ε n fy) dy = 1 R n ω n ε n f 1. Thus we are left with the proof of the first inequality. For a, r > 0 the function ϕr) = r/r + a) is increasing and hence applying Bernoulli s inequality we have for r ε ) r n ) ε n x y /ε 1 n r + x y ε + x y 1 + x y /ε 1 n x y. ε Fix x, y R n. Then for any r ε we have By, r) Bx, r + x y ) and hence ) M ε r n fx) f 1 n ) r + x y By,r) ε x y f. By,r) Passing to supremum over r ε we obtain M ε fx) 1 n ε x y ) M ε fy). Since the inequality is also true if we replace x by y and y by x one easily conclude the first inequality from the lemma. Lemma 3.2. If f L 1 R n ), then {x : Mfx) > fx) } = Z k=1 E k
5 ON APPROXIMATE DIFFERENTIABILITY OF THE MAXIMAL FUNCTION 5 where Z = 0 and Mf Ek is Lipschitz continuous for k = 1, 2,... In particular Mf is a.e. approximately diferentiable in the set {x : Mfx) > fx) }. Proof. Let Z be the set of points that are not Lebesgue points of f. Clearly Z = 0. Assume that x R n \ Z and Mfx) > fx). Let r i > 0 be a sequence such that f Mfx). Bx i,r i ) The sequence r i is bounded because Mfx) > 0 and f L 1 ) and hence we can select a subsequence still denoted by r i ) such that r i r. Clearly r > 0 as otherwise we would have Mfx) = fx). Thus Mfx) = f for some r > 0. This easily implies that {x : Mfx) > fx) } Z {x : Mfx) = M 1/k fx)}. k=1 Since the function M 1/k f is Lipschitz continuous by Lemma 3.1, the first part of the result follows. The second part is a direct consequence of Lemma 2.1. Now we can complete the proof of Theorem 1.1. Let f L 1 R n ) be approximately differentiable a.e. Then also f is approximately differentiable a.e. Lemma 2.2). According to Lemma 3.2 R n = {x : Mfx) = fx) } Z where Z = 0 and Mf Ek is Lipschitz continuous. Since Mf Ek is approximately differentiable a.e. and Mf = f is approximately differentiable a.e. in the set {x : Mfx) = fx) } the theorem follows. k=1 E k 4. Proof of Theorem 1.2 Since Mfx) = fx) in P, clearly Mfx) = fx) a.e. in P. Thus let x E and r x > 0 be such that the equality 1.1) holds. Assume also that Mf is approximately differentiable at x. Note that the function ϕy) = Mfy) fz) dz = Mfy) fy + z) dz By,r x) B0,r x)
6 6 PIOTR HAJ LASZ, JAN MALÝ is approximately differentiable at x and ϕx) = Mfx) = Mfx) B0,r x) Bx,r x) f x + z) dz fz) dz. Indeed, Mf is approximately differentiable at x and since f W 1,p we can differentiate in the second term under the sign of the integral. Note also that ϕ 0 and ϕx) = 0, so ϕ attains minimum at x and hence its approximate derivative at x must be equal to 0 which is the claim we wanted to prove. 5. Proof of Theorem 1.4 This proof requires some results from the potential theory. We say that a locally integrable function u : Ω [0, ] defined on an open set Ω R n is superharmonic if it is lower semicontinuous and 5.1) ux) uy) dy whenever Bx, r) Ω. We will need the following well known regularity result, see [21] for an even more general result. For the sake of completeness we will provide a proof. Lemma 5.1. If a locally integrable function u : Ω [0, ], Ω R n, is superharmonic, then u W 1,p loc Ω) for all 1 p < n/n 1). In particular u is a.e. approximately differentiable. Proof. Let u : Ω [0, ] be superharmonic and let U Ω. According to the Riesz decomposition theorem [24], [11], u restricted to U can be represented as 5.2) ux) = hx) Φx y) dµy), for x U R n where h is harmonic, Φ is the fundamental solution to the Laplace equation and µ is a finite positive measure supported in U. It is easy to see that we can compute the weak first order partial derivatives of u in U by differentiating the right hand side of 5.2) under the sign of the integral u x) = h x) 1 x i x i nω n x i y i ) R n x y n dµy). By Young s convolution inequality cf. [25, II.1.1., p. 27]), the convolution is as integrable as the kernel. Since the measure µ has a bounded support and the function x is clearly in L p loc Rn ), we deduce that u L p U). x x n
7 ON APPROXIMATE DIFFERENTIABILITY OF THE MAXIMAL FUNCTION 7 For an open set Ω R n and f L 1 loc Ω) we define a local maximal function M Ω fx) = sup fy) dy where the supremum is over all balls Bx, r) Ω. The following characterization of superharmonic functions will be very useful, see [15]. Lemma 5.2. A locally integrable function u : Ω [0, ], Ω R n, is superharmonic if and only if M Ω ux) = ux) for all x Ω. Proof. If u is superharmonic, then taking supremum over all balls in 5.1) gives ux) M Ω ux) for all x Ω. On the other hand lower semicontinuity of u yields M Ω ux) lim sup r 0 uy) dy lim inf y x for all x Ω. Hence ux) = M Ω ux) for all x Ω. uy) ux) Now suppose that M Ω ux) = ux) for all x Ω. Since the maximal function is lower semicontinuous we conclude lower semicontinuity of u and the superharmonicity of u follows from the inequality uy) dy M Ω ux) = ux) which is satisfied on every ball Bx, r) Ω. Corollary 5.3. If f L 1 R n ) and fx) = Mfx) a.e. in an open set Ω R n, then we can redefine f on a set of measure zero in such a way that f becomes superharmonic in Ω. Proof. It follows from the Lebesgue differentiation theorem that fx) M Ω f x) a.e. in Ω. Hence M Ω f x) M f x) = Mfx) = fx) M Ω f x) a.e. in Ω and thus fx) = M Ω f x) a.e. in Ω. Now it is clear that we can modify f on a set of measure zero in such a way that fx) = M Ω f x) which makes the function f superharmonic. everywhere in Ω Now we can complete the proof of the theorem. Let f L 1 R n ) and let Ω R n be open. According to Lemma 3.2, Mf is a.e. approximately differentiable in the set {x Ω : fx) < Mfx)}.
8 8 PIOTR HAJ LASZ, JAN MALÝ If this set has positive measure the theorem follows. If it has measure zero, then fx) = Mfx) a.e. in Ω and hence fx) coincides a.e. with a superharmonic function in Ω, see Corollary 5.3. Now Lemma 5.1 gives a.e. approximate differentiability of f in Ω and hence that of f, see Lemma 2.2. The proof is complete. 6. Example In this section we will construct a bounded integrable function f L 1 R) such that the maximal function Mf is not approximately differentiable a.e. In our construction Mf will coincide with f on a contact set P of positive length and f will not be approximately differentiable on P. This will imply the lack of approximate differentiability of Mf at the Lebesgue points of P. In the first step we will construct a bounded periodic function f with period 1 such that Mf is not approximately differentiable a.e. and then it will be clear that also for f = fχ [0,1] L 1 R) the maximal function is not approximately differentiable a.e Construction. We denote r k = 3 kk+1), α k = exp 9 k 2 ). For k = 1, 2,..., on the interval [ ) 0, r k 1 we define 1, x [ ) i 1)r k, ir k, i {2, 4,..., 9 k 1}, g k x) = α k, x [ ) i 1)r k, ir k, i {3, 5,..., 9 k 2}, 0, x [ ) i 1)r k, ir k, i {1, 9 k }. We extend g k to R periodically with the period r k 1. Finally we set n f 0 = 1, f n = g k, f = lim f n. n k=1 Observe that the function g k is constant on the intervals [i 1)r k, ir k ), i Z, and hence f n is constant on the intervals [i 1)r n, ir n ), i Z Maximal function. We will now estimate the maximal function of f. We denote P = {f k > 0}. k Let x R and ρ > 0. We consider smallest n N such that M n x, ρ) = x ρ, x + ρ) r n Z. In this situation x ρ, x + ρ) is contained in one of the intervals [i 1)r n 1, ir n 1 ) and hence f n 1 equals to a constant β on x ρ, x + ρ). Let us write M = M n x, ρ). Now we will distinguish two cases.
9 ON APPROXIMATE DIFFERENTIABILITY OF THE MAXIMAL FUNCTION 9 Case 1. Let ρ r n+1. Then there is only one point z M. By a symmetry reason we may assume that x z. Then x [z, z + r n+1 ). Since g n+1 = 0 on [z, z + r n+1 ), f n+1 = 0 on that interval and hence x / P. Case 2. Let ρ > r n+1. We split x ρ, x + ρ) into intervals x ρ, x + ρ) [i 1)r n, ir n ). For each interval I of the partition, with an endpoint z M, either [z, z + r n+1 ) I, [z r n+1, z) I, or I z r n+1, z + r n+1 ). Since f n+1 = 0 on z r n+1, z + r n+1 ) we have z r n+1, z + r n+1 ) P =. In each case I P 1 r ) n+1 I. r n Summing over I we obtain x ρ, x + ρ) P 2ρ 1 r ) n+1. r n It follows x+ρ x ρ On the other hand, if x P, then Since f β 2ρ x ρ, x + ρ) P β 1 r ) n+1. r n fx) βα n α n r n+1 = 1 9 n 1 e 9 n 1 < e 9 n 2 9 n 3 9 n 4... = α n α n+1..., r n we obtain Mfx) fx) on P, and hence Mfx) = fx) a.e. in P The contact set. On the set P we have fx) β := α 1 α 2 α 3 = exp ) > 0. We will estimate the size of the set P [0, 1]. We see that {f1 > 0} [0, 1] = 1 2r1, {f 2 > 0} [0, 1] = {f1 > 0} [0, 1] 1 2 r ) 2 = 1 2r 1 ) r 1..., so that 6.1) P [0, 1] = 1 2r 1 ) 1 2 r ) r ) 3 > 0 r 1 r 2 as r k+1 = 9 k 1 < +. r k k k 1 2 r ) 2, r 1
10 10 PIOTR HAJ LASZ, JAN MALÝ 6.4. Differentiability. Let us consider x P, k IN and an interval [z, z + r k ) such that z r k Z and x [z, z + r k ) {f k > 0}. This interval is contained in an interval [i 1)r k 1, ir k 1 ) where the function f k 1 has constant value β β, 1]. Since z i 1)r k 1 + r k otherwise f k z) = 0) we have that [z r k, z + r k ) [i 1)r k 1, ir k 1 ) and hence f k 1 = β on [z r k, z + r k ). There are three possibilities { 0 on [z r k, z) = I, f k = β on [z, z + r k ) = J, In each case whereas f k = f k = { β on [z r k, z) = J, α k β on [z, z + r k ) = I, { α k β on [z r k, z) = I, β on [z, z + r k ) = J. f βα k = β exp 9 k 2 ) on I, f βα k+1 α k+2 α k+3... = β exp 9 k 2 ) 8 on J P. Hence 6.2) f fx) 1 2 β exp 9 k 2 ) ) exp 9 k 2 ) 8 on at least one of the sets I or J P. Since the infinite product at 6.1) converges, for sufficiently large k we have J P = J 1 2 r ) k r ) k+2 > 1 r k r k+1 2 J = r k 2, and hence inequality 6.2) is satisfied on a set E k [z r k, z + r k ) of length E k > r k /2. To estimate the right hand side of 6.2), observe that e x e y e y e y x 1) 1 y)y x), 0 < x < y, and thus exp 9 k 2 ) exp 9 k 2 ) k 2) 9 k 2 > 4 9 k Accordingly 6.3) fy) fx) y x 4β 9 k 3 4r k = β 3 k2 k 6 for y E k.
11 ON APPROXIMATE DIFFERENTIABILITY OF THE MAXIMAL FUNCTION 11 Set E n = k=n E k. Since lim sup h 0+ E n x h, x + h) 2h lim sup k E k 4r k 1 8, the approximate limes-superior of fy) fx) y x as y x is at least β 3 n2 n 6 for each n, and thus it is. Hence f cannot be approximately differentiable at x. If x is a density point of P, then also Mf cannot be approximately differentiable at x An integrable function. Finally let f = fχ 0,1). Then M fx) Mfx) fx) = fx) in P 0, 1) and hence M fx) = fx) = fx) a.e. in P 0, 1). Since f is not approximately differentiable on P 0, 1), M f cannot be approximately differentiable a.e. References [1] Acerbi, E., Fusco, N.: An approximation lemma for W 1,p functions. In: Material instabilities in continuum mechanics Edinburgh, ), pp. 1 5, Oxford Sci. Publ., Oxford Univ. Press, New York, [2] Aldaz, J. M., Pérez Lázaro, J.: Boundedness and unboundedness results for some maximal operators on functions of bounded variation. J. Math. Anal. Appl ), [3] Aldaz, J. M., Pérez Lázaro, J.: Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities. Trans. Amer. Math. Soc ), [4] Bojarski, B.: Remarks on some geometric properties of Sobolev mappings. In: Functional analysis & related topics Sapporo, 1990), pp , World Sci. Publ., River Edge, NJ, [5] Bojarski, B., Haj lasz, P.: Pointwise inequalities for Sobolev functions and some applications. Studia Math ), [6] Buckley, S. M.: Is the maximal function of a Lipschitz function continuous? Ann. Acad. Sci. Fenn. Math ), [7] Carneiro, E., Moreira, D.: On the regularity of maximal operators. Proc. Amer. Math. Soc ), [8] Haj lasz, P.: Change of variables formula under minimal assumptions. Colloq. Math ), [9] Haj lasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc ), no [10] Haj lasz, P., Onninen, J.: On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn. Math ), [11] Hayman, W. K., Kennedy, P. B.: Subharmonic functions. Vol. I. London Mathematical Society Monographs, No. 9. Academic Press, London-New York, 1976 [12] Kinnunen, J.: The Hardy-Littlewood maximal function of a Sobolev function. Israel J. Math ), [13] Kinnunen, J., Latvala, V.: Lebesgue points for Sobolev functions on metric spaces. Rev. Mat. Iberoamericana ), [14] Kinnunen, J., Lindqvist, P.: The derivative of the maximal function. J. Reine Angew. Math ),
12 12 PIOTR HAJ LASZ, JAN MALÝ [15] Kinnunen, J., Martio, O.: Maximal operator and superharmonicity. In: Function spaces, differential operators and nonlinear analysis Pudasjärvi, 1999), pp , Acad. Sci. Czech Repub., Prague, [16] Kinnunen, J., Saksman, E.: Regularity of the fractional maximal function. Bull. London Math. Soc ), [17] Kinnunen, J., Tuominen, H.: Pointwise behaviour of M 1,1 Sobolev functions. Math. Z ), [18] Korry, S.: A class of bounded operators on Sobolev spaces. Arch. Math. Basel) ), [19] Korry, S.: Extensions of Meyers-Ziemer results. Israel J. Math ), [20] Korry, S.: Boundedness of Hardy-Littlewood maximal operator in the framework of Lizorkin-Triebel spaces. Rev. Mat. Complut ), [21] Lindqvist, P.: On the definition and properties of p-superharmonic functions. J. Reine Angew. Math ), [22] Luiro, H.: Continuity of the maximal operator in Sobolev spaces. Proc. Amer. Math. Soc ), [23] MacManus, P.: Poincaré inequalities and Sobolev spaces. In: Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations El Escorial, 2000). Publ. Mat. 2002, Vol. Extra, pp [24] Ransford, T.: Potential theory in the complex plane. London Mathematical Society Student Texts, 28. Cambridge University Press, Cambridge, [25] Stein, Elias M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J [26] Tanaka, H.: A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function. Bull. Austral. Math. Soc ), [27] Whitney, H.: On totally differentiable and smooth functions. Pacific J. Math. 1, 1951), [28] Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc ), Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA, hajlasz@pitt.edu Department KMA of the Faculty of Mathematics and Physics, Charles University, Sokolovská 83, CZ Praha 8, Czech Republic, and Department of Mathematics of the Faculty of Science, J. E. Purkyně University, České mládeže 8, Ústí nad Labem, Czech Republic, maly@karlin.mff.cuni.cz
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