NORM INEQUALITIES FOR OFF-CENTERED MAXIMAL OPERATORS

NORM INEQUALITIES FOR OFF-CENTERED MAXIMAL OPERATORS


Introduction
In [CWW], weighted norm inequalities are derived for some integral and maximal operators associated with starlike sets in Euclidean space 1[8''x .Our aim now is to extend the results which deal with analogues of the Hardy-Littlewood and fractional maximal functions .The situation we will consider is closely adapted to both the geometry of the sets used in the definitions of these functions and to the relationship between these sets and the point at which the maximal functions are formed .The study of averages of functions over sets other than balls or cubes has a long history, and for some other weighted results we refer to [J] and [P3], and the referentes cited there.
To be more precise, given 0 < p, < n and two (possibly unrelated) sets S and E in Rn, we consider the maximal function MS,E,wf(x) = sup t,`-n l .f(y) ¡ dy, t>0, ZERn f_ ,+tS xEz+tE where z+tE denotes the set {z+t~: ~E E}, and similarly for z+tS .We are most interested in the case when S is starlike about the origin and E is bounded .If M = 0, (1.1) is a special case of an operator considered 1Supported in part by NSF grant DMS91-04195.
If we pick co so that Q1 C S.y , then the requirement in (1.1) that x E z+tE simply amounts to requiring that x belong to the cube-like central portion of z+tS.y as opposed to lying farther out in the "skirt" of z+tS,y .
The weak-type behavior of (1 .1)was described in [CWW], and we now want to study its strong-type behavior.For purposes of comparison, we define the centered maximal function (1 .3)Ms,/,f (x) = sup t"-n 1 f (y) ¡ dy, 0 < p < n, t>o f .+tswhich corresponds to choosing E to be the empty set in (1.1) .Both weak and strong-type results for (1.3) are derived in [CWW], and to put our results in perspective, we recall these for the model case S = S.y given in (1.2) .For a >_ 1 and fixed -y > 0, we associate the linear operators SQ x = (ax1, . . . .axn_1, a-7xn) and the rectangles Ra = SaQl .These rectangles are naturally adapted to S.y since UR,,,CS,C UKR2; a>1 for some geometric constant r .Given a rectangle R, we denote by B(R) the collection of all translates and dilates of R, Le., Let 1 < p < q < co, p' = p/(p -1), and w(x), v(x) be nonnegative locally integrable weight functions.Denote o, = v-1 y(P -1) .It is proved in [CWW] that if the centered maximal operator MS,,, satisfies the weak-type estimate w{x : MS' ,mf (x) > A} <-( C A II fIIP,v ) q with c independent of f and A, A > 0, then for all R E B(Ra) and all a > 1 .Here we have used the standard notations w(A) = fA w dx, IIfllp,v = (f If(x)Ipv(x)dx l p , and c for a constant which may be different at different occurrences .Conversely, the weak-type estimate holds if there exists a monotone function C(a), a > 1, such that and (1 .5)C(a) aa < oo.
1 Moreover, we have the strong-type estimate II MS,,jil q,w :5 CII f p,v, 1 < p < q < oo, if there exists r > 1 so that for all R E B(Ra), a _> 1, and C(a) is a monotone function which satisfies (1.5) .Of course, (1 .6) is stronger than (1 .4)due to H51der's inequality.
To each R E B(Ra), associate a rectangle R* as follows : if R = z + tRa, then R* = z + tR* .Thus the pair (R, R*) is a joint translation and dilation of (Ra , Ra*) by the same z, t.It is proved in [CWW] that if 1<p<q<ooand for all pairs (R, R*), R E 13(R), and all a > 1. Conversely, suppose 1 < p <q < oo and there is a monotone function C(a) such that {RI ñ -1 w(R*)9Q(R)P' < C(a)IRalñ-1 for all R E B(Ra) and all a >_ 1 .If C(a) also satisfies (1.5) then the weak-type estimate (1 .7)holds .
Even in the unweighted case w = v = 1, it follows that the results for the centered and uncentered maximal operators associated with S.y are different .In fact, it is easy to check that the conditions then require 1/q = 1/p -p,/n, that the centered maximal function is strong-type for y > n -1 if 1 < p < n/p,, but that even weak-type for the uncentered maximal function requires a positive result being guaranteed when strict inequality holds .
function A result for general starlike S is given in Section 3. Condition (1 .8) is analogous to (1.6) for the centered maximal function .
To prove Theorem 1, we use a covering technique given by C .P. Calderón in [Ca] together with a result we now describe.Let 13 be the family of all translates and dilates of a fixed rectangle Rz3 (Le ., 13 = 13(R,3) in our previous notation) .Of course R13 is not uniquely determined by 13 but its eccentricities (ratios of edgelengths) are, and we may assume without loss of generality that its first edgelength is 1 .Thus, for example, we may view the basic rectangle in 13(Ra) as having edgelengths 1, . . ., 1, a--í -1 rather than a, . . ., a, a--í .To each R E C3, associate a set (not necessarily a rectangle) R* so that the following holds : (1.9) If Rl , RZ E 13 and R1 C RZ then Ri C R2 .
For example, the pairs (R, R*) of joint translates and dilates of (Ro , R*) defined earlier have this property.More generally, if R* is defined to be any rectangle containing R13, and given R E 13, R = z + tR,3, we define R* = z + tR* then the pairs (R, R*) satisfy (1 .9) .
For such a collection of pairs and 0 < a < 1, define Of course this depends on 13 and on the choice of the sets R*, although for simplicity our notation does not reflect this dependence.We will need the following result.
Theorem 2 .Let 1 < p < q < co and 0 <_ a < 1, and let M, be defined by (1 .10),assuming that (1 .9)holds.If there exists r > 1 such that with a constant C which is a multiple depending on a, n, p, and not on 13 or f, of the constant in (1.11).
We note that the condition q, but is necessary for (1 .12)(even for the corresponding weak-type result), as can be seen by choosing f = XRU in (1.12) and using a standard argument.
In case R* = R and R is a cube, Theorem 2 is due to C .Pérez [P1], [P2] .Our proof will be modeled on ideas in [SW] and is given in Section 2 .
The proof of Theorem 2 uses some ideas from [SW].The details which are either the same or nearly the same as ones there will be omitted .
Let X3 = 13(RL3) be a family a rectangles R as in the introduction, with associated sets R* which satisfy (1.9) .Let e l , . . ., en, (el = 1, say) be the edgelengths of R13, and let Bdy be the corresponding grid of dyadic rectangles of the form Proof. .We argue as in [W] and [SW], and earlier [FS].The important part of the argument is as follows .Fix R E .t3 and consider the collection of those Rl E .13dywhose edgelengths are about twice those of R, respectively, and think of Rn as partioned into the union of such Rl .Of course, IR,¡ Pz:~IR¡ for each Rl with constants of equivalente depending only on n.A simple geometric argument using translations shows that for each Rl, The key points to observe are that if we denote SZ = UE(R 1 ), then R1 the inequality between the first and third terms in (2.2) holds if x E R* and z E 9, that IE(R1)I >_ cIR1I for each R1, and that the E(R1) are essentially disjoint for different Rl .The rest of the proof then proceeds as in [SW] or [W], and is omitted .
To prove Theorem 2, it is enough by Lemma (2.1) to prove the analogue of (1 .12)for each Máy,z f, with a constant independent of z .If we replace f by fo, , this amounts to showing that (2.3) IIM«y'z (fU)Ilq,w <ClIfIIp,a with c equal to a multiple depending only on a, n, p and q of the constant in (1 .11) .
Then x E SZ k if and only if theie exists R E B dy such that x E (R + z)* and (2.4) IRI" f fv dy > 2kn .

R+z
In particular, if R E 13d" and (2.4) holds then (R+z)* C 1? k .Let {R jk } j be the maximal (with respect to inclusion) rectangles in 13dy which satisfy (2 .4) ; their existence is assured if f has compact support, which we may assume to be the case without loss of generality .By maximality, the {Rj k + z}j are nonoverlapping for each k .Moreover, if R~is the next largest dyadic rectangle containing R.~, then IR~Ia-1 f fvdy < 2kn R~-I-z by maximality, so that since I R~I = 2n I R~I , we have We have already observed that each (Ri + z)* must lie in Qk .On the other hand, if x E Qk there is a dyadic R with x E (R + z)* such that (2.4) holds .Thus R C Rho for some jo (since R is maximal or not), and consequently (R + z)* C (R~+ z)* by (1 .9) .Hence, x E (R 3 ~o + z)* and the claim follows .

R
We estimate the last sum by using hypothesis (1 .11)for the rectangles R~k + z and the fact that Ejk C (Rjk + z)*, obtaining that where c is the constant in (1.11) .
The remainder of the proof is based on using the next lemma to estimate the sum in (2 .7) .Lemma 2.8.Let {Ri}jEI be a collection of rectangles from a fixed dyadic grid (e .g., from Bdy + z for fixed z), let ~3 _> 1, and let {ai}iEI be positive numbers which satisfy (i) a(Ri) <_ coa¡ (ii) E ¿ < coaQ j:RjCR; for each i, with co independent of i.Then if 1 < p < oc and q = pp, I I 1 I I aá (-f Iflo,dy)q

LiEl
al Ri 9 <_ ellfllp,a, with c depending on co, p and q, but not on f or the particular grid.
The proof is virtually the same as that of Lemma (2 .10) of [SW], which deals with the case of dyadic cubes, and is therefore omitted .
If we apply Lemma (2.8) to the sum in (2.7) and note that o,(R) < A(R) by lldlder's inequality, we immediately obtain (2 .3)from (2.7) if we verify (2.9) A(R y ~+ z)g1p < cA(R-+ z)q/p k,j:R~CR-1 for each R' and 0 < p < q < oc, with c independent of l, m and z.We argue as in the proof of (2 .11) in [SW] .Using the simple inequality a¡ < (~a i)qlp, q > p, al > 0, we may prove just the case q = p.If Rk is a proper subset of Rm then where the second inequality follows from the maximality of R~.Therefore, we must have k >_ m in (2.9), and we may rewrite the left side of (2 .9)(with q/p = 1) as (2 .10)2mn < IR¿ l a-1 fQdy < 2kn Rm+z A(R 7 ~+ z) .=m j:R; CR-By Hblder's inequality and the definition of A(R), the inner sum in (2 .10) is at most Here, since {Rj k +z}j are nonoverlapping for fixed z, the second factor is at most (fR,-+z Q' dy) .Also, for the first factor, by the first inequality as desired .This preves (2.9), and also completes the proof of Theorem

Proof of Theorem 1
We now show how Theorem 1 implies Theorem 2. In fact, we will prove a more general result based en the following fact from [CWW] .Let S be a set in Rn which is starlike with respect to the origin, Le., except at most for a set of measure 0 en the boundary, S = {rO : 0 E Sn-1 , 0 < r < p(B)}, where p is the boundary function of S. Lemma 3.1 .If S is starlike with respect to the origin, there exist rectangles {Rj} (with varying orientations) such that each Rj contains the origin on its major axis, S C URj, and r_ I R; I < CISI for some 7 C>0 .
For more facts concerning such covers, we refer to [CWW] .Given a starlike S with respect to the origin, fix a cover {Rj } as above.Given a bounded set E, define Rj * to be a rectangle containing both E and Rj .Also, for each j, let Bj denote the collection of all dilates and translates of Rj , and if R E 13, say R = z + tRj , define R* = z + tRj* .The pairs (R, R*), R E Bj , then satisfy (1 .9) .Theorem 3. Let 0 <_ M, < n, 1 < p < q < oo, S be a starlike set with respect to the origin, and E be a bounded set.With the notation abone, assume there exists r > 1 so that (3.1) IR¡ Ty -Pw(R*)9 1 -P-1 (RI IfR v for all R E ,Cid, and that 1: CR~< oo.Then lI MS,E, N,f l iq, .< CIIf llp,v- Before giving the proof, we note that Theorem 3 includes Theorem 1 by picking S = S,,, E = Q1 and Rj = SaQl for a = 2j, j = 1, 2, . . . .Le ., R2; in the notation of the introduction .Recall that in the introduction R* is the smallest rectangle containing both Ra and Q1 .The requirement in Theorem 3 that E CRj < oo then amounts to (1.5) .
I{zER n :RCR1+z}I>cIR11 with c > 0 depending only on n.Also, with R still fixed, the sets {z E Rn : R C Rl + z} are essentially disjoint for different (nonoverlapping) Rl .Let E(R 1 ) = {z E Rn : R C Rl + z, R* C (Rl + z)*}, and note by (1.9) that E(R1) is the same as the set {z E R' : R C R, +z} above .Also if x E R* and z E E(R1)y) I dy <cMá y, 'f (x) since IR, +zi = IR,¡ z-IRI, R C R1+z and x E (R1+z)* .The constant c depends only on n, a .