BEST CONSTANTS FOR UNCENTERED MAXIMAL FUNCTIONS. Loukas Grafakos and Stephen Montgomery-Smith University of Missouri, Columbia
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1 BEST CONSTANTS FOR UNCENTERED MAXIMAL FUNCTIONS Loukas Gafakos and Stehen Montgomey-Smith Univesity of Missoui, Columbia Abstact. We ecisely evaluate the oeato nom of the uncenteed Hady-Littlewood maximal function on L (R ). Consequently, we comute the oeato nom of the stong maximal function on L (R n ), and we obseve that the oeato nom of the uncenteed Hady-Littlewood maximal function ove balls on L (R n ) gows exonentially as n. Fo a locally integable function f on R n, let (M n f)(x) = su f(y) dy, B x B B whee the suemum is taken ove all closedballs B that contain the oint x. M n f is calledthe uncenteedhady-littlewoodmaximal function of f on R n. In this ae we find the ecise value of the oeato nom of M on L (R ). It tuns out that this oeato nom is the solution of an equation. Ou main esult is the following: Theoem. Fo <<, the oeato nom of M : L (R ) L (R ) is the unique ositive solution of the equation () ( ) x x =. 99 Mathematics Classification Numbe 42B25 Reseach atially suoted by the NSF. Tyeset by AMS-TEX
2 In ode to ove ou Theoem, we fix a nonnegative f andwe intoduce the left and ight maximal functions: x (M L f)(x) = su f(t) dt a<x x a a b and(m R f)(x) = su f(t) dt. b>x b x x Fo the oof of the next esult, known oulaly as the sunise lemma, we efe the eade to Lemma (2.75) (i), Ch VI in [2]. Lemma. Let f be in L (R ). Fo each λ>, let C λ = {x :(M L f)(x) >λ} and D λ = {x :(M R f)(x) >λ}. Then (2) λ C λ = fdt and λ D λ = f dt. C λ D λ Now we ae eady to ove the main lemma that leads to ou Theoem. This next esult may be viewedas the coect weak tye estimate fo the maximal function M. Lemma 2. Let f be in L (R ). Fo each λ>, let A λ = {x :(M f)(x) >λ} and B λ = {x : f(x) >λ}. Then (3) λ( A λ + B λ ) fdt+ f dt. A λ B λ To ove (3), fist note that (4) su(m L,M R )=M. Fo, clealy su(m L,M R ) M. On the othe hand, it is easy to see that fo each eal numbe x, (M f)(x) is bounded by a convex combination of (M L f)(x) and(m R f)(x). Now we add the two equalities in (2). Then using the fact that A λ = C λ D λ which follows fom (4), we obtain (5) λ( A λ + C λ D λ )= fdt+ A λ f dt. C λ D λ 2
3 Clealy B λ (C λ D λ ) is a set of measue zeo, and f λ on (C λ D λ ) B λ. Theefoe (6) (C λ D λ ) B λ fdt λ (C λ D λ ) B λ. Equations (5) and(6) now imly equation (3), as equied. To ove the inequality in ou Theoem, we equie the following fact. Lemma 3. Let f and g be nonnegative functions on R. Then if >, we have λ 2 g(t)>λ f(t) dt dλ = fg dt, R and if >, we have λ {g >λ} dλ = g dt. R The fist equality is easily oved, since by Fubini s theoem, the left hand side is g(t) f(t) λ 2 dλ dt, which is eadily seen to equal the ight hand side. The second equality is the secial case of the fist when f =. We now continue the oof of ou Theoem. Multilying (3) by λ ( 2), integating λ fom to, andalying Lemma 3, we obtain M f + f f + f(x)[(m f)(x)] dx, R that is, ( ) M f f(x)[(m f)(x)] dx. R 3
4 Alying Hölde s inequality with exonents and /( ) to the secondtem, we obtain ( ) ( ) M f M f (7) ( ), fom which we conclude that M f c, whee c is the unique ositive solution of (). To show that c is in fact the oeato nom of M on L (R ), we give an examle. Note that equality in (3) is satisfiedwhen f is even symmetically deceasing and equality in (7) is satisfiedwhen M f is a multile of f. We ae theefoe ledto the following examle. Let f ε,n (t) = t χε,n ( t ), whee χ ε,n is the chaacteistic function of the inteval [ε, N]. It can be easily seen that (8) lim lim ε N M f = M (f )(), whee f (t) = t L loc. An easy calculation gives that (9) M (f )() = γ + γ +, whee γ is the unique ositive solution of the equation () γ + γ + = γ. Using (9) and(), it is a matte of simle aithmetic to now show that M (f )() is the unique ositive oot of equation (). This comletes the oof of ou Theoem. Befoe we conclude, we would like to make some emaks. Denote by x =(x,...,x n ) oints in R n. Fo a locally integable function f on R n, define (N n f)(x) = su su a <x b >x a n <x n b n >x n b bn f(y,...,y n ) dy n dy. (b a ) (b n a n ) a a n 4
5 N n is calledthe stong maximal function on R n. Clealy N = M. Obseve that N n M () M (n), whee M (j) denotes the maximal oeato M aliedto the x j coodinate. This shows that the oeato nom of N n on L (R n ) is less than o equal to c n. By consideing the function g(x) = n f ɛ,n (x j ), j= whee f ɛ,n is as above, we obtain the convese inequality. We have theefoe ovedthe following: Coollay. Fo <<, the oeato nom of N n : L (R n ) L (R n ) is c n, whee c is the unique ositive solution of equation (). One can show that <c < 2. This imlies that the oeato nom of N n on L (R n ) gows exonentially with n, asn. Next, we obseve that the same is tue fo the uncenteedmaximal function M n. Thee ae seveal ways to see this. One way is by consideing the sequence of functions h ɛ,n (x) = x n χɛ,n ( x ). Let U n be the oen unit ball in R n.fox R n, let B x = x 2 + x () ( M n (h ε,n ) ) (x) B x B x y n χε,n ( y ) dy = U n ( 2 x 2 U n. Then x B x and ) n y Bx n χε,n ( y ) dy. Theefoe fo << andfo all ε, N > wehave { M n (h ε,n ) L 2 n + [ ] h ε,n L h ε,n L U n n y n dφ χɛ,n ( y )dy n d (2) = { 2 n + = h ɛ,n L U n = [ n 5 B φ t t= S φ ( t ) } n χɛ,n (t)t n dt ] dφ t dθ n d },
6 whee S φ (t) ={θ : tθ φ 2 2 }. By a change of vaiables (2) is equal to (3) { 2 n h ε,n L U n { = 2n U n [ = + [ = t= S φ (t) (K φ χ ε,n )() d = χ d ε,n () χ ε,n (t) t n dt ] } t dθ d dφ ] dφ ω n }, whee K φ (t) =t n/ χ [,] (t) n S φ (t) θ n/ B dθ, ω n = (n )π 2 =, and denotes Γ( n+ 2 ) convolution on the multilicative gou G =(R +, dt t ). If K ong, the sequence of functions χ ɛ,n gives equality in the convolution inequality g K L (G) K L (G) g L (G) as ε and N. Theefoe, the exession inside backets in (3) conveges to K φ L (G) as ɛ and N, andwe obtain the estimate M n (h ε,n ) L lim ɛ h ε,n L N (4) { 2n U n =2 n ω n 2 ω n [ t n S φ (t) dθ dt ] } dφ t ω n = n2n ω n s n ( s 2 ) n 3 2 ds =2 n ω n 2 ω n B( n 2 t n θ t 2, n 3 2 ). dθ dt t Stiling s fomula gives that exession (4) is asymtotic to { 4( ) ( +) } n 2 as n, andsince the numbe inside the baces above is bigge than when <<, we get that the oeato nom of M n on L (R n ) gows exonentially as n. These emaks shouldbe comaedto the fact that fo <<, the oeato nom of the Hady-Littlewood maximal function on L (R n ) is bounded above by some constant A indeendent of the dimension n (see [3] and[4]). 6
7 Refeences. M. Chist and L. Gafakos, Best constants fo two non-convolution inequalities, Poc. Ame. Math. Soc. 23 (995), Hewitt and Stombeg, Real and Abstact Analysis, Singe-Velag, New Yok, NY, E.M. Stein, Some esults in hamonic analysis in R n, fo n, Bull. Ame. Math. Soc. 9 (983), E.M. Stein and J.O. Stombeg, Behavio of maximal functions in R n, fo lage n, Ak. Math. 2 (983), A. Zygmund, Tigonometic Seies, Cambidge Univ. Pess, Cambidge, UK, 959. Deatment of Mathematics, Univesity of Missoui, Columbia, MO 652 addess: loukas@math.missoui.edu, stehen@math.missoui.edu 7
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