Norm inequalities for off-centered maximal operators

Publicacions Matemátiques, Vol 37 (1993), 429-441 . Abstract NORM INEQUALITIES FOR OFF-CENTERED MAXIMAL OPERATORS RiCHARD L . WHEEDEN I Sufficient conditions are derived in order that there exist strongtype weighted norm inequalities for some off-centered maximal functions . The maximal functions are of Hardy-Littlewood and fractional types taken over starlike sets in Rn . The sufficient conditions are close to necessary and extend some previously known weak-type results . 1 . 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, ZER n 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 1 Supported in part by NSF grant DMS91-04195 . 43 0  R . L . WHEEDEN in [Cor], although the normalizations are different, and for 0 < t, <n it was studied in [CWW] . A simple example occurs by writing x = (x', xn) with x' = (x1, . . . , x n _1) and choosing E to be the unit cube Q1 centered at 0 and S to be the unbounded set (1 .2) S,={x :Ixnl<min{1, Ix l7 }}, -y>0 . 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 .+ts which corresponds to choosing E to be the empty set in (1 .1) . Bothweak 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 S Q x = (ax1, . . . . axn _1, a-7xn) and the rectangles R a = Sa Ql . These rectangles are naturally adapted to S . y since UR,,,CS,C UKR 2 ; 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 ., B(R) = {z + tR : z E R n , t > 0} . 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 f IIP,v ) q with c independent of f and A, A > 0, then IRIñ-1w(R)'u(R)PL < CIRa1ñ_1 INEQUALITIES FOR OFF-CENTERED OPERATORS  431 for all R E B(R a ) and all a > 1 . Here we have used the standard notations w(A) = f A 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 (1 .4)  IRA ñ -l w(R) 4 o,(R) -' < C(a)¡Ra l Ty -1 ,  R E ,13(R a ), 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 (1 .6)  IR¡ 1  p w ( R ) 1 (  <' IR¡ IR for all R E B(R a ), 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 . For the uncentered operator (1 .1), only a weak-type estimate is proved in [CWW] . To describe it, let 6* be defined by S*(x) _ (ax1, . . . , axn_1, xn) for a _> 1, and let R* = 6*Q1 . Note for future reference that R* is the smallest rectangle which contains both R a and Q1 . To each R E B(R a ), 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 (R a , Ra*) by the same z, t .  It is proved in [CWW] that if 1<p<q<ooand (1 .7)  w{x : MS 7 ,Q,mf(x) > A} <_ C Cli .fllp,v  q then IR1 ñ -1 w(R*)9Q(R)P' < CIRaiñ-1 43 2  R . L . WHEEDEN 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(R a ) 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 >n1 if 1 < p < n/p , , but that even weak-type for the uncentered maximal function requires p >  y  (> y - (n - 1) 1 1 - ñ) a positive result being guaranteed when strict inequality holds . For the model case S = S y , we will prove the following strong-type result for (1 .1) . Theorem 1 . Let y > 0 and S y be defined by (1 .2) . Let 0 < tt < n, 1 < p <_ q < oo and assume there exists r > 1 so that ( .8)  IR¡!-lw(R*)1  1  u'dx p, <-C(a)IR .I!-1 1  p  9  IR¡ IR for all R E 13(R a ) and all a >_ 1, where C(a) is a monotone which satisfies (1 .5) . Then ~IMS,,Q1,wfliq,w < CIIfIIp,v . 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 aresult 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(R a ) 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 R l , R Z E 13 and R 1 C R Z then Ri C R2 . For example, the pairs (R, R*) of joint translates and dilates of (R o , R*) defined earlier have this property . More generally, if R* is defined to be any rectangle containing R 13 , 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 (1 .10)  M a f (x) = sup IR¡' - '  I f (y)¡ dy . REC3 IR R *Bx 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 (1 .11)  IRI a  P w ( R* )° C IR¡  o,' dx/_  p  < C for all R E ,t3, then INEQUALITIES FOR OFF-CENTERED OPERATORS  433 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 IIMaf1I9,w < CIIfIIp,v I RI a-l w(R*) 9o,(R) p~ < c,  R E I3, 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 . 43 4  R . L . WHEEDEN 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(R L3 ) 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 B dy be the corresponding grid of dyadic rectangles of the form  1 and for z ER', define 2 . Proof of Theorem 2 [ m,el (ml + 1)el  [m r¿ e 7¿ (m .,+ 1)e n , l 2i '  2i  X . . . X  2i '  2i for j, ml, . . . , m,, = 0, ±1, ±2, ...  .  Each rectangle in B d y is also in 1i . Define M . d y f (x) =  sup  IR1  1  I f (y)¡ dy, REI3dy IR R'gx Máy,zf (x) =  Sup  I R+ zI a -1  I f (Y)¡ d . RE 13dy  f R+z Of course, IR + zi = IR¡ . (R+z)`Dx Lemma (2 .1) . If 1 < q < oo and w is a weight, then JIM«flIq, . < C Sup IIMa y ' z fIIq,w ZERn with c depending on a and n but not on B or f . 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 .13 dy whose 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, 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 R l + z, R* C (R l + z)*}, INEQUALITIES FOR OFF-CENTERED OPERATORS  435 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), then (2 .2)  IRIa -1 f R I f (y)I dy <_ cI R1 + zia -1 f Rl+z I f (y) I dy <cMá y, 'f (x) since IR, +zi = IR,¡ zIRI, R CR1+z and x E (R1+z)* . The constant c depends only on n, a . The key points to observe are that if we denote SZ = UE(R 1 ), then R 1 the inequality between the first and third terms in (2 .2) holds if x E R* and zE 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 f o , , 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) . To prove (2 .3), fix z and f > 0, and for k = 0, fl, f2, . . . . let S2 k = {x E R' : M d ,, y,z (for)(x) > 2k,} . Then x E SZ k if and only if theie exists R E B d y 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 13 d y 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 {R j 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 (2 .5)  2kn < IR ; la-1 f  fvdy < 2n(1-n)2kn . Rk+z 436  R . L . WHEEDEN We claim that (2 .6) SZk = U(R jk + z)* . j We have already observed that each (R i + 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 Rh o for some jo (since R is maximal or not), and consequently (R + z)* C (R~ + z)* by (1 .9) . Hence, xE (R 3 ~ o + z)* and the claim follows . By (2 .6), where E j' = (R j + z)*\S2k + 1 . Thus II M a d ` (fo , )Ilq,w where for a rectangle R, II May,z(fo,)Ilq,w < 2'l Qk\Qk+l = U[(Rj + z) * \ 9 k+l1 = U E j f [M .,,,(fa)(x)14w(x) dx E f9 k k\Qk+1 E2 (k+1)ngw(Ek) kj q < 2nq 1 : (IR j k j` l fQdy~ w(E j ) k j  R~ +z =2nq1 :w(Ejk)[IRkI«-lA(Rjk +z)]q . k,j 1  q dy l , A(R~ +z) .  R 3 +z fu A(R) = IR¡ T (f Qr dy) T . R We estimate the last sum by using hypothesis (1 .11) for the rectangles R~k + z and the fact that E j k C (R jk + z)*, obtaining that (2 .7) \q A(R~ k + z) P  k  I  f u dy l  , A(R j + z)  R~+z INEQUALITIES FOR OFF-CENTERED OPERATORS  437 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 B dy + 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, II  1 II  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 ¡ < (~ ai)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) 2 mn < IR¿ l a-1  f Q dy < 2kn Rm+z A(R 7 ~ + z) . =m j :R ; CR-