ON WEIGHTED ESTIMATES FO STEIN S MAXIMAL FUNCTION Hedra Guawa Abstract. Let φ deote the ormalized surface measure o the uit sphere S 1. We shall be iterested i the weighted L p estimate for Stei s maximal fuctio M φ f, amely M φ f L p (w) C p,w f L p (w), where w is a A p weight, especially for 1 < p 2. Usig the Melli trasformatio approach, we prove that the estimate holds for every weight w δ where w A p ad 0 δ < p( 1) (p 1), for 3 ad 1 < p 2. Itroductio Let φ be the ormalized surface measure o the uit sphere S 1. Cosider Stei s maximal fuctio M φ f, which is defied by M φ f(x) = sup φ r f(x), x, r>0 for ay ice fuctio f o. The we have the L p iequality M φ f p C p f p, f L p, for 2 ad 1 < p, which is kow to be best possible [1, 4]. I this paper, we are iterested i the weighted L p estimate for Stei s maximal fuctio, M φ f Lp (w) C p,w f Lp (w), where w A p, especially for 1 < p 2 (cosult [3] about A p weights). For 3, a positive result ca be foud i [3]; here we shall reprove ad exted it. AMS Classificatio Number: 42B25; 42B20 1
Usig the Melli trasformatio approach of Cowlig ad Mauceri [2], let K u (x) = C(u) x +iu, where C(u) = π 2 +iu Γ( iu 2 )/Γ( iu 2 ). (K u is the distributio o whose Fourier trasform is K u (ξ) = ξ iu.) The, formally, we have φ(x) = P 1 (x) + D(u)K u (x) du, x, where P 1 deotes the Poisso kerel at 1 ad D(u) satisfies 2πC(u)D(u) = 0 (ω 1 1 δ 1 P 1 )(s)s 1 iu ds, u, with δ 1 beig the poit mass at 1. Oe may observe that C(u) = O((1 + u ) 2 ) ad D(u) = O((1 + u ) 2 ). Now, for every r > 0, φ r (x) = P r (x) + D(u)K u (x)r iu du, x, ad accordigly, for every smooth fuctio f o, φ r f(x) = P r f(x) + D(u)K u f(x)r iu du, x. Hece M φ f(x) M P1 f(x) + D(u) K u f(x) du, x. Sice we kow that M P1 f is majorized by the Hardy-Littlewood maximal fuctio M HL f, we obtai M φ f L p (w) M HL f L p (w) + D(u) K u f L p (w) du. Thus, to verify the estimate, we eed to get a good weighted L p estimate for K u f, that is oe that makes for 1 < p 2. D(u) K u f L p (w) du < C p,w f L p (w), This work was iitiated whe the author visited the School of Mathematics, Uiversity of New South Wales, Sydey, Australia, i Jue 1994. May thaks are due to Professor M. Cowlig who suggested the problem ad helped the author with a lot of ideas. 2
Mai esults We obtai the followig results. The first lemma below is stadard. Lemma 1. For x 2 y ad for all γ (0, 1), K u (x y) K u (x) C(1 + u ) 2 +γ y γ x γ. Proof. For x 2 y, we have, as i [2], two estimates K u (x y) K u (x) C(1 + u ) 2 x ad K u (x y) K u (x) C(1 + u ) 2 +1 y x 1. Iterpolatig these estimates, we get K u (x y) K u (x) C(1 + u ) 2 +γ y γ x γ, for all γ (0, 1). Followig the work of Watso [6], we have Lemma 2. For 1 < p 2 ad for ay γ (0, 1), K u f L p (w) C p,w,γ (1 + u ) 2 +γ f L p (w), wheever w A p. Proof. First ote that K u (ξ) = 1 for all ξ. Next, we eed to show that the L r -Hörmader coditio : for > 2 y > 0, j=1(2 j ) r ( 2 j < x <2 j+1 ) 1 K u (x y) K u (x) r r dx C γ (1 + u ) 2 +γ, is satisfied for all r (1, ). (Here r deotes the dual expoet to r.) Havig doe this, we ca the choose r (1, ) sufficietly large such that w r A p. Thus, followig [6], we obtai K u f Lp (w) C p,w,γ (1 + u ) 2 +γ f L p (w), 3
as desired. Ideed, usig Lemma 1, we observe that for all r (1, ), K u (x y) K u (x) r dx 2 j < x <2 j+1 C r (1 + u ) r 2 +γr y γr C r (1 + u ) r 2 +γr γr 2 j < x <2 j+1 2 j <t<2 j+1 C r (1 + u ) r 2 +γr γr (2 j ) (r 1) γr [ = C(1 + u ) 2 +γ (2 j ) r 2 γj] r. x r γr dx (r 1) γr dt t t Therefore the coditio is satisfied ad the lemma is proved. (We have actually proved that the estimate holds wheever w A p, for 1 < p <.) We are aware that the estimate i Lemma 2 is ot good eough. We have, however, the followig result of Cowlig ad Mauceri [2] for the uweighted case. Lemma 3 (Cowlig ad Mauceri). For 1 < p 2 ad for ay γ (0, 1), K u f p C p,γ (1 + u ) p 2 +γ f p, f L p. Now we have a better estimate for K u f, amely Theorem 4. For 1 < p 2 ad for ay γ (0, 1), K u f L p (w δ ) C p,w,γ,δ (1 + u ) p 2 +δ δ p +γ f L p (w δ ), f L p (w δ ), wheever w A p ad 0 δ 1. Proof. The proof follows directly from Lemma 2 ad Lemma 3 by the Stei-Weiss iterpolatio theorem [5]. Theorem 4 leads us to the weighted L p estimate for Stei s maximal fuctio. Theorem 5. For 3 ad 1 < p 2, the weighted Lp estimate M φ f L p (w δ ) C p,w,δ f L p (w δ ), f L p (w δ ), holds wheever w A p ad 0 δ < p( 1) (p 1). 4
Proof. Choose γ (0, 1) sufficietly small such that 0 δ < p( 1 γ) (p 1). The, by Theorem 4, we have D(u) K u f L p (w δ )du < C p,w,δ f L p (w δ ) < C p,w,δ f L p (w δ ), (1 + u ) p +δ δ p +γ du ad so the theorem follows immediately. For power weights w(x) = x a, we kow that w A p for some p > 1 if ad oly if < a < (p 1). So, Theorem 5 implies that the estimate holds for w(x) = x a with p( 1) p 1 < a < p( 1). Statig it i aother way, the estimate with respect to w(x) = x a holds for +a 1 < p 2 whe a 0, or for +a +a 1 < p 2 whe a < 0. Thus, for p 2, our result agrees with the oe stated i [3, p. 571] for the special case where w(x) = x a with a 0. Cocludig emarks We suspect that the same estimate also holds for p > 2, but we ecouter difficulties i verifyig it. Duality argumets will ot work sice the edpoits of the rage of allowable p s are ot symmetric. The Stei-Weiss iterpolatio theorem oly gives the estimate for 2 p provided that w A 2 ad 0 δ < 2. Also, sice the estimate holds oly for some but ot all w A p whe 1 < p 2, we caot use the existig extrapolatio theorem of ubio de Fracia ad Garcia-Cuerva. Some ovel techique seems to be eeded here ad we are still workig o it. efereces [1] J. Bourgai, Averages i the plae over covex curves ad maximal operators, J. Aalyse Math. 47 (1986), 69-85. [2] M. Cowlig ad G. Mauceri, O maximal fuctios, ed. Sem. Mat. Fis. Mil. 49 (1979), 79-87. [3] J. Garcia-Cuerva ad J.L. ubio de Fracia, Weighted Norm Iequalities ad elated Topics, North-Hollad, Amsterdam, 1985. 5
[4] E.M. Stei, Maximal fuctios: spherical meas, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2174-2175. [5] E.M. Stei ad G. Weiss, Iterpolatio of operators with chage of measures, Tras. Amer. Math. Soc. 87 (1958), 159-172. [6] D.K. Watso, Weighted estimates for sigular itegrals via Fourier trasform estimates, Duke Math. J. 60 (1990), 389-399. Departmet of Mathematics Istitut Tekologi Badug Gaesha 10 Badug 40132 INDONESIA 6