SHARP INEQUALITIES FOR MAXIMAL FUNCTIONS ASSOCIATED WITH GENERAL MEASURES L. Grafakos Department of Mathematics, University of Missouri, Columbia, MO 65203, U.S.A. (e-mail: loukas@math.missouri.edu) and J. Kinnunen Department of Mathematics, P.O. Box 4, FIN-0004 University of Helsinki, Finland (e-mail: Juha.Kinnunen@Helsinki.Fi) Abstract. Sharp weak type (, ) and L p estimates in dimension one are obtained for uncentered maximal functions associated with Borel measures which do not necessarily satisfy a doubling condition. In higher dimensions uncentered maximal functions fail to satisfy such estimates. Analogous results for centered maximal functions are given in all dimensions.. Introduction Let µ be a nonnegative Borel measure on R n and let f : R n [0, ] beaµlocally integrable function. The uncentered maximal function of f with respect to µ is defined by Mf(x) = sup B f dµ, (.) µ(b) B where the supremum is taken over all closed balls B containing x. Let B(x, r) denote the closed ball with center x and radius r > 0. The centered maximal function of f with respect to µ is defined by Mf(x) = sup f dµ, (.2) r>0 µ(b(x, r)) B(x,r) 99 Mathematics Subject Classification. 42B25. Research partially supported by the National Science Foundation and by the University of Missouri Research Board. Typeset by AMS-TEX
with the interpretation that the integral averages in (.) and (.2) are equal to f(x) ifµ(b) =0orµ(B(x, r)) = 0. If µ is Lebesgue measure, these definitions give the usual uncentered and centered Hardy Littlewood maximal operators. It is a classical result, see [9, p. 3], that if µ satisfies a doubling condition, µ(b(x, 2r)) Cµ(B(x, r)) for all x R n and r>0, (.3) both of these operators are of weak type (,) and they map L p (R n,µ), p>, into itself. Omitting the doubling requirement, it is still true that M maps L p (R n,µ), p>, into itself, but the corresponding result for M is false if n 2. An example indicating this statement is given in section 3. Examples showing that M is not of weak type (,) if n 2 can be found in [8]. It is a geometrical phenomenon, however, that such counterexamples do not exist in dimension one. In fact in dimension one, M maps L p (R,µ), p>, into itself without the doubling assumption about µ, see [2] and [8]. This is a consequence of a special covering argument available only on the real line. In this article we give sharp L p and weak type (,) estimates for M with constants independent of µ. In higher dimensions we obtain an improvement of the known estimate µ({mf >λ}) c n λ {Mf>λ} f dµ, λ > 0, (.4) where c n is the Besicovitch constant. See section 3 for details. 2. The one-dimensional case On R, fix a nonnegative Borel measure µ. The inequality below was first proved 2
in [7] when µ is the usual Lebesgue measure. The proof given there is different and doesn t generalize to this context. Theorem 2.. For any λ>0 and any µ-locally integrable function f : R [0, ] we have µ({ Mf >λ})+µ({f >λ}) λ { f Mf>λ} fdµ+ λ {f>λ} f dµ. (2.) Proof. Fix λ>0 and denote E λ = { Mf > λ}. If µ({f > λ}) =, then by Chebyshev s inequality the right side of (2.) is infinity and there is nothing to prove. Hence we may assume that µ({f >λ}) <. For every x E λ there is an interval I x containing x such that fdµ>λ. (2.2) µ(i x ) I x By Lindelöf s theorem there is a countable subcollection I j, j =, 2,..., such that I j = j= Let I = {I j : j =, 2,..., N} and write x E λ I x. F N = I I I. By Lemma 4.4 in [6] we obtain two subcollections I and I 2 of I so that the intervals in each of these are pairwise disjoint and that 2 F N = I. i= I I i We denote F i = I I i I, i =, 2. Since the intervals in I i, i =, 2, are pairwise disjoint and (2.2) holds we obtain µ(f i )= I I i µ(i) < λ I I I i fdµ= fdµ for i =, 2. (2.3) λ F i 3
Therefore µ(f N )+µ(f F 2 )=µ(f )+µ(f 2 ) < fdµ+ fdµ λ F λ F 2 = fdµ+ f dµ. λ F λ N F F 2 For any µ-measurable set E such that µ(e) < we have (2.4) To see this, we observe that (f λ) dµ = E fdµ+ µ({f >λ}) fdµ+ µ(e). (2.5) λ E λ {f>λ} {f λ} E {f>λ} Using (2.4) and (2.5) we deduce that µ(f N )+µ({f >λ}) λ (f λ) dµ + (f λ) dµ. F N fdµ+ λ {f>λ} E {f>λ} (f λ) dµ f dµ. Since F N is an increasing sequence of µ measurable sets whose union is E λ, inequality (2.) follows by letting N. Remarks 2.2. () Inequality (2.) is stronger than the standard weak type (,) estimate obtained, for example, in [2]. In particular, estimate (2.) implies that M is of weak type (,) with constant 2. (2) Equality can actually occur in (2.). For instance this is the case when f even, symmetrically decreasing about the origin and µ is Lebesgue measure, see [7]. Now we show that the sharp weak type estimate (2.) implies a sharp version of the Hardy Littlewood Theorem. Corollary 2.3. Let <p< and let A p be the unique positive solution of the equation (p ) x p px p =0. (2.6) 4
Then Mf p,µ A p f p,µ. (2.7) Proof. We may suppose that f is not zero µ-almost everywhere and that f L p (R,µ) since otherwise there is nothing to prove. Fubini s theorem and (2.) imply that ( Mf) p dµ + f p dµ = p R R and hence R ( Mf) p dµ Hölder s inequality gives and hence or equivalently p 0 0 = p p λ p µ({ Mf >λ}) dλ + p λ p 2 { f Mf>λ} fdµdλ+ p R ( Mf) p fdµ+ p p 0 0 λ p µ({f >λ}) dλ λ p 2 fdµdλ R f p dµ p ( p Mf) p fdµ+ f p dµ. R p R ( Mf) ( p fdµ ( Mf) ) (p )/p ( p dµ f p dµ R R R (p ) Mf p p,µ p Mf p p,µ f p,µ + f p p,µ ) /p {f>λ} ( Mf p,µ (p ) f p,µ The claim follows from this inequality. ) p p ( Mf p,µ f p,µ ) p 0. Remarks 2.4. () When µ is Lebesgue measure, then the L p -bound above is the best possible, see [7]. (2) The bound A p in (2.7) is independent of the measure µ. We close this section by studying the reverse inequality to (2.). 5
Proposition 2.5. Suppose that f : R [0, ] is a locally µ-integrable function. Then for every λ ess inf R Mf. { f Mf>λ} fdµ λµ({ Mf >λ}) (2.8) Proof. Let λ>ess inf R Mf and denote Eλ = { Mf >λ}. Then R \ E λ has a positive measure. On the other hand, E λ is an open set on the real line and hence it is a union of countably many pairwise disjoint open intervals E λ = j= I j.for an interval I and σ>0, let σi be the interval with the same center whose length is multiplied by σ. Since every σi j intersects R \ E λ when σ>, we see that fdµ λ, for j =, 2,... µ(σi j ) σi j By letting σ we obtain fdµ λ, for j =, 2,..., µ(i j ) I j and hence by summing up we deduce that fdµ E λ j= I j fdµ λ µ(i j )=λµ(e λ ). j= This implies that (2.8) is true for every λ ess inf R Mf and the proof is now complete. Remark 2.6. Suppose that f L (R,µ). If λ<ess inf R Mf, then µ(eλ )= µ(r ) and (2.8) holds for every λ µ(r f dµ. ) R In particular, if µ(r )=, then (2.8) holds for every λ>0. 6
3. The higher dimensional case If n 2, the uncentered maximal function associated to a general measure is not bounded on L p (R n ) for <p<. To see this, select closed balls B,B 2,... so that the origin is on the boundary of each ball and such that for every B i, i =, 2,..., there is a point x i B i \ j i B j.set µ = δ xi, i=0 where x 0 = 0 and δ xi denotes Dirac mass at x i. Let f be the characteristic function of B. Clearly f p,µ 2 /p, but Mf(x i ) fdµ µ(b i ) B i 2 for all i =, 2,..., and hence Mf p,µ 2 µ(rn ) /p =. A similar counterexample for the strong maximal operator was given in [4]. Next we discuss an improvement of (.4). Here we need the following Besicovitch s covering theorem. Theorem 3.. Suppose that E is a bounded subset of R n and that B is a collection of of closed balls such that each point of E is a center of some ball in B. Then there exists an integer c n 2 (depending only on the dimension) and subcollections B,...,B cn Bof at most countably many balls such that the balls in each family B i are pairwise disjoint and such that E c n B. i= B B i For the proof of Besicovitch s covering theorem we refer to [3, Theorem.]. Some estimates for the constant c n are obtained in [5]. 7
Theorem 3.2. For any λ>0and any µ-locally integrable function f : R n [0, ] we have µ({mf >λ})+(c n )µ({f >λ}) fdµ+(c n ) λ {Mf>λ} λ Here c n is the Besicovitch constant. {f>λ} f dµ. (3.) Proof. We fix λ>0 and denote E λ = {Mf > λ}. We may assume that µ(e λ ) <, since otherwise by (.4) the right side of (3.) is infinity. For every x E λ there is a ball B(x, r x ) so that We have that and that B(x,r x ) fdµ>λ. (3.2) µ(b(x, r x )) B(x,r x ) fdµ= fdµ+ fdµ B(x,r x ) (R n \E λ ) B(x,r x ) E λ λµ(b(x, r x ) (R n \ E λ )) + fdµ (3.3) B(x,r x ) E λ µ(b(x, r x )) = µ(b(x, r x ) (R n \ E λ )) + µ(b(x, r x ) E λ ). (3.4) Combining (3.2), (3.3), and (3.4) we obtain fdµ>λµ(b(x, r x ) E λ ). (3.5) B(x,r x ) E λ Let B R = B(0,R) be a fixed ball and denote B = {B(x, r x ): x B R E λ }. By Besicovitch s covering theorem there are subfamilies B,...,B cn,ofb such that each of these subfamilies consists of at most countably many pairwise disjoint balls and that B R E λ c n B. i= B B i 8
We denote F i = B B i B, i =, 2,...,c n, and F = c n i= F i. Since the balls in each B i are pairwise disjoint, it follows from (3.5) that µ(f i E λ ) < λ F i E λ f dµ, for i =, 2,...,c n. (3.6) Then we use the elementary fact that for any measure ν we have c n ν(f i E λ )=ν(f E λ )+ c n i= j=2 ν(g j E λ ), (3.7) where G j = {k,...,k j } {,...,c n }(F k F kj ), j =2, 3,...,c n. Using (3.6) and (3.7) we deduce that c n µ(f E λ )+ µ(g j E λ ) < fdµ+ c n f dµ. λ j=2 F E λ λ j=2 G j E λ Inequality (2.5) then implies that µ(b R E λ )+(c n )µ({f >λ}) fdµ+(c n ) f dµ, λ E λ λ {f>λ} and by letting R we prove the desired conclusion. As in Corollary 2.3 we obtain an estimate for the constant in the Hardy Littlewood Theorem. Corollary 3.3. Let A p,n be the unique positive solution of the equation (p ) x p px p (c n ) = 0, (3.8) where c n is the Besicovitch constant. Then the estimate Mf p,µ A p,n f p,µ, (3.9) 9
holds. The constant A p,n given by (3.8) tends to one as p goes to infinity. This shows that it is asymptotically sharp near. However, A p,n grows as n. It is still unknown to us whether the constant A p,n in (3.9) can be replaced with a constant both independent of the measure µ and of the dimension n. References. K. F. Andersen, Weighted inequalities for maximal functions associated with general measures, Trans. Amer. Math. Soc. 326 (99), 907 920. 2. A. Bernal, A note on the one-dimensional maximal function, Proc. Roy. Soc. Edinburgh A (989), 325 328. 3. M. deguzmán, Differentiation of integrals in R n, Lecture Notes in Mathematics 48, Springer Verlag, 975. 4. R. Fefferman, Strong differentiation with respect to measures, Amer. J. Math. 03 (98), 33 40. 5. Z. Füredi and P.A. Loeb, On the best constant for the Besicovitch covering theorem, Proc. Amer. Math. Soc. 2 (994), 063 073. 6. J. Garnett, Bounded analytic functions, Pure and Applied Mathematics, Academic Press, 98. 7. L. Grafakos and S. Montgomery Smith, Best constants for uncentered maximal functions, Bull. London Math. Soc. 29 (996), 60 64. 8. P. Sjögren, A remark on the maximal function for measures in R n, Amer. J. Math. 05 (983), 23 233. 9. E. M. Stein, Harmonic Analysis: Real-Variable Theory, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 993. 0