THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION. Juha Kinnunen. 1 f(y) dy, B(x, r) B(x,r)

Transcription

1 Appeared in Israel J. Math. 00 (997), 7 24 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION Juha Kinnunen Abstract. We prove that the Hardy Littlewood maximal operator is bounded in the Sobolev space W,p (R n ) for <p. As an application we study a weak type inequality for the Sobolev capacity. We also prove that the Hardy Littlewood maximal function of a Sobolev function is quasicontinuous.. Introduction. The Hardy Littlewood maximal function Mf : R n [0, ] of a locally integrable function f : R n [, ] is defined by (.) Mf(x) = sup f(y) dy, B(x, r) B(x,r) where the supremum is over all radii r>0. Here B(x, r) denotes the volume of the ball B(x, r). The maximal function is a classical tool in harmonic analysis but recently it has been successfully used in studying Sobolev functions and partial differential equations, see [] and [4]. The celebrated theorem of Hardy, Littlewood and Wiener asserts that the maximal operator is bounded in L p (R n ) for <p, (.2) Mf p A p f p, where the constant A p depends only on p and n, see [5, Theorem I.]. This theorem is one of the cornerstones of harmonic analysis but the applications to Sobolev functions and to partial differential equations indicate that it would also be useful to know how the maximal operator preserves the differentiability properties of functions. It is easy to show that maximal function of a Lipschitz function is again Lipschitz and hence, in that case, by Rademacher s theorem it is differentiable almost everywhere. Unfortunately, the maximal function of a differentiable function is not differentiable in general. The reason for this is twofold. First, the modulus of a differentiable function is not differentiable and, even though the function would not change signs, the supremum of differentiable functions may fail to be differentiable. The purpose of this note is to show that, however, certain weak differentiability properties are preserved under the maximal operator. Our main theorem is that the Hardy Littlewood maximal operator is bounded in the Sobolev space W,p (R n )for<p and hence, in that case, it has classical partial derivatives almost everywhere. The corresponding result for p = fails because then we don t have the Hardy Littlewood Wiener theorem available. 99 Mathematics Subject Classification. 42B25, 46E35, 3B5. Key words and phrases. Maximal function, Sobolev spaces, capacity. Typeset by AMS-TEX

2 2 JUHA KINNUNEN Recall, that the Sobolev space W,p (R n ), p consists of functions u L p (R n ), whose first weak partial derivatives D i u, i =, 2,...,n, belong to L p (R n ). We endow W,p (R n ) with the norm (.3) u,p = u p + Du p, where Du =(D u, D 2 u,...,d n u) is the weak gradient of u. For the basic properties of Sobolev functions we refer to [3, Chapter 7]. Now we are ready to formulate our main result..4. Theorem. Let <p<. Ifu W,p (R n ), then Mu W,p (R n ) and (.5) Di Mu MDi u, i =, 2,...,n, almost everywhere in R n. 2. The proof of Theorem.4. If χ B(0,r) is the characteristic function of B(0,r) and χ r = χ B(0,r) B(0,r), then u(y) dy = u χ r (x), B(x, r) B(x,r) where denotes the convolution. Now u χ r W,p (R n ) and D i ( u χ r )=χ r D i u, i =, 2,...,n, almost everywhere in R n. Let r j, j =, 2,..., be an enumeration of positive rationals. Since u is locally integrable, we may restrict ourselves in definition (.) to the positive rational radii. Hence Mu(x) = sup( u χ rj )(x). j We define functions v k : R n R, k =, 2,...,by v k (x) = max j k ( u χ r j )(x). Now (v k ) is an increasing sequence of functions in W,p (R n ) [3, Lemma 7.6] which converges to Mu pointwise and D i v k max Di ( u χ rj ) j k (2.) = max χrj D i u MDi u = MD i u, j k i =, 2,...,n, almost everywhere in R n. Here we also used the fact that Di u = D i u, i =, 2,...,n, almost everywhere. Thus Dv k p n D i v k p i= n MD i u p i=

3 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION 3 and the Hardy Littlewood Wiener inequality (.2) implies v k,p Mu p + n MD i u p i= A p u p + A p n i= D i u p c< for every k =, 2,... Hence (v k ) is a bounded sequence in W,p (R n ) which converges to Mu pointwise. The weak compactness of Sobolev spaces implies Mu W,p (R n ), v k converges to Mu weakly in L p (R n ) and D i v k converges to D i Mu weakly in L p (R n ). Since D i v k MD i u almost everywhere by (2.), the weak convergence implies almost everywhere in R n. D i Mu MD i u, i =, 2,...,n, 2.2. Remarks. (i) If we only want to prove that that Mu W,p (R n ), <p<, whenever u belongs to W,p (R n ), there is a simple proof based on the characterization of W,p (R n ) by integrated difference quotients, see [3, 7.]. If f : R n [, ] and h R n, h 0, we denote (2.3) f h : R n [, ], f h (x) =f(x + h). The sublinearity of the maximal operator implies M(u h ) Mu M(u h u) and hence (Mu) h Mu p = M(u h ) Mu p M(u h u) p A p u h u p A p Du p h, from which the claim follows using [3, Lemma 7.24]. Unfortunately, this argument does not seem to give the pointwise inequality (.5) for the partial derivatives. (ii) Inequality (.5) implies (2.4) DMu(x) M Du (x) almost every x R n. To see this, let x R n. If DMu(x) = 0, then the claim is obvious. Hence we may assume that DMu(x) 0. Now D h u = Du h for every h R n with h =, where D h denotes the derivative to the direction h. We choose h = DMu(x)/ DMu(x) and rotate the coordinates so that h coincides with some of the coordinate directions, we get DMu(x) = D h Mu(x) MD h u(x) M Du (x). Now we may use the Hardy Littlewood Wiener theorem together with (2.4) and obtain Mu,p = Mu p + DMu p A p u p + M Du p A p u,p,

4 4 JUHA KINNUNEN where A p is the constant in (.2). (iii) If u W, (R n ), then a slight modification of our proof shows that Mu belongs to W, (R n ). Moreover, Mu, = Mu + DMu u + M Du u,. Recall, that after a redefinition on a set of measure zero u W, (R n ) is bounded and Lipschitz continuous. If we are not interested in pointwise estimates, there is a simple proof of the fact that the maximal function maps bounded Lipschitz continuous functions into themselves. Indeed, suppose that u is Lipschitz continuous with constant L, that is u h (x) u(x) L h for every x, h R n where u h is defined by (2.3). The same argument as in (i) shows that (Mu) h (x) Mu(x) = M(u h )(x) Mu(x) M(u h u)(x) = sup u h (y) u(y) dy L h, r>0 B(x, r) B(x,r) which means that the maximal function is Lipschitz continuous with constant L. Observe, that this proof applies to Hölder continuous functions as well. (iv) Finally we remark that our method applies to other maximal and maximal singular integral operators as well. 3. A capacity inequality. We show that a weak type inequality for the Sobolev capacity follows immediately from our Theorem.4. The standard proof depends on some extension properties of Sobolev functions, see [2]. Let <p<. The Sobolev p-capacity of the set E R n is defined by ( C p (E) = inf u p + Du p) dx, u A(E) R n where A(E) = { u W,p (R n ):u on a neighbourhood of E }. If A(E) =, wesetc p (E) =. The Sobolev p-capacity is a monotone and a countably subadditive set function [2]. Let u W,p (R n ), suppose that λ>0 and denote E λ = {x R n : Mu(x) >λ}. Then E λ is open and Mu/λ is admissible for E λ. Using (2.4) we get ( ) ( C p Eλ Mu p λ p + DMu p) dx R n (3.) Ap p ( u p + Du p) dx λ p R n Ap p λ p u p,p. This inequality can be used in studying the pointwise behaviour of Sobolev functions by the standard methods, see [2], but we shall use it to prove that the Hardy Littlewood maximal function of a Sobolev function is quasicontinuous.

5 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION 5 4. Quasicontinuity. First we recall some terminology. A property holds p- quasieverywhere if it holds outside a set of the Sobolev p-capacity zero. A function u is p -quasicontinuous in R n if for every ε>0 there is a set F such that C p (F ) <ε and the restriction of u to R n \F is continuous and finite. It is well known that each Sobolev function has a quasicontinuous representative, see [2]. To be more precise, for each u W,p (R n ) there is a p-quasicontinuous function v W,p (R n ) such that v = u a.e. in R n. Moreover, this representative is unique in the following sense: If v and w are p-quasicontinuous and v = w a.e., then w = up-quasieverywhere in R n. 4.. Theorem. If u W,p (R n ), <p<, then Mu is p-quasicontinuous. Proof. We begin with showing that if u C(R n ) L p (R n ), then Mu C(R n ) L p (R n ). Indeed, if x, h R n and ε>0, then there is r ε < such that ( u h (y) u(y) dy B(x, r) B(x,r) B(x, r) B(x,r) u h u p B(x, r) p 2 u p B(x, r) p <ε ) /p u h (y) u(y) p dy whenever r>r ε. On the other hand, if 0 <r r ε, then there is δ>0 such that u h (y) u(y) dy sup u h u <ε B(x, r) B(x,r ε ) B(x,r) whenever h <δ. Thus M(u h u)(x) ε for h <δand (Mu) h (x) Mu(x) M(u h u)(x) ε whenever h <δ. This shows that Mu is continuous at x. Suppose then that u W,p (R n ) and let (ϕ i ) be a sequence of functions ϕ i C 0 (Rn ), i =, 2,..., so that ϕ i u in W,p (R n ). By the weak type inequality (3.) there is a set F with C p (F ) = 0 so that Mu is finite in R n \ F. We choose a subsequence, which is denoted by (ϕ i ), such that ϕ i u p,p < (4i A p ) p. Set E i = {x R n \ F : M(ϕ i u)(x) > 2 i }, i =, 2,... Then using inequality (3.) we get C p (E i ) 2 ip A p p ϕ i u p,p 2 ip. If F j = i=j E i, then by subadditivity C p (F j ) 2 ip < i=j and hence lim j C p (F j ) = 0. Moreover, for x R n \ F j we have Mϕ i (x) Mu(x) M(ϕ i u)(x) 2 i

6 6 JUHA KINNUNEN whenever i j, which shows that the convergence is uniform in R n \ F j. As a uniform limit of continuous functions Mu is continuous in R n \ F j. This implies that Mu is p-quasicontinuous Remark. If p>n, then every non-empty set has a positive p capacity and hence the maximal function of a function u W,p (R n ) is continuous. In fact, by the Sobolev imbedding theorem [3, Theorem 7.7] it is Hölder continuous, (Mu) h (x) Mu(x) c h n/p for every x, h R n.ifp =, the maximal function is Lipschitz continuous. Acknowledgements. The author would like to thank Professor Peter Lindqvist and Professor Olli Martio for inspiring discussions. References. Bojarski, B. and Haj lasz, P., Pointwise inequalities for Sobolev functions and some applications, Studia Math. 06 () (993), Federer, H. and Ziemer, W., The Lebesgue set of a function whose distribution derivatives are p-th power summable, Indiana Univ. Math. J. 22 (972), Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order (2nd edn), Springer Verlag, Berlin, Lewis, J.L., On very weak solutions of certain elliptic systems, Comm. Partial Differential Equations 8(9&0) (993), Stein, E.M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 970. Department of Mathematics, P.O.Box 4, FIN-0004 University of Helsinki, Finland address: Juha.Kinnunen@ Helsinki.Fi