Norm inequalities for off-centered maximal operators

Publicacions Ma emá iques, Vol 37 (1993), 429-441 . Abs ac NORM INEQUALITIES FOR OFF-CENTERED MAXIMAL OPERATORS RiCHARD L . WHEEDEN I Su icien condi ions a e de i ed in o de ha he e exis s ong- ype weigh ed no m inequali ies o some o -cen e ed maximal unc ions . The maximal unc ions a e o Ha dy-Li lewood and ac ional ypes aken o e s a like se s in Rn . The su icien con- di ions a e close o necessa y and ex end some p e iously known weak- ype esul s . 1 . In oduc ion In [CWW], weigh ed no m inequali ies a e de i ed o some in eg al and maximal ope a o s associa ed wi h s a like se s in Euclidean space 1[8''x . Ou aim now is o ex end he esul s which deal wi h analogues o he Ha dy-Li lewood and ac ional maximal unc ions . The si ua ion we will conside is closely adap ed o bo h he geome y o he se s used in he de ini ions o hese unc ions and o he ela ionship be ween hese se s and he poin a which he maximal unc ions a e o med . The s udy o a e ages o unc ions o e se s o he han balls o cubes has a long his o y, and o some o he weigh ed esul s we e e o [J] and [P3], and he e e en es ci ed he e . To be mo e p ecise, gi en 0 < p, < n and wo (possibly un ela ed) se s S and E in Rn, we conside he maximal unc ion MS,E,w (x) =  sup  , ` -n  l . (y) ¡ dy, >0, ZER n  _ , + S xEz+ E whe e z+ E deno es he se {z+ ~ : ~E E}, and simila ly o z+ S . We a e mos in e es ed in he case when S is s a like abou he o igin and E is bounded . I M = 0, (1 .1) is a special case o an ope a o conside ed 1 Suppo ed in pa by NSF g an DMS91-04195 . 43 0  R . L . WHEEDEN in [Co ], al hough he no maliza ions a e di e en , and o 0 < , <n i was s udied in [CWW] . A simple example occu s by w i ing x = (x', xn) wi h x' = (x1, . . . , x n _1) and choosing E o be he uni cube Q1 cen e ed a 0 and S o be he unbounded se (1 .2) S,={x :Ixnl<min{1, Ix l7 }}, -y>0 . I we pick co so ha Q1 C S . y , hen he equi emen in (1 .1) ha x E z+ E simply amoun s o equi ing ha x belong o he cube-like cen al po ion o z+ S . y as opposed o lying a he ou in he "ski " o z+ S, y . The weak- ype beha io o (1 .1) was desc ibed in [CWW], and we now wan o s udy i s s ong- ype beha io . Fo pu poses o compa ison, we de ine he cen e ed maximal unc ion (1 .3)  Ms,/, (x) = sup " -n 1 (y) ¡ dy,  0 < p < n, >o .+ s which co esponds o choosing E o be he emp y se in (1 .1) . Bo hweak and s ong- ype esul s o (1 .3) a e de i ed in [CWW], and o pu ou esul s in pe spec i e, we ecall hese o he model case S = S . y gi en in (1 .2) . Fo a >_ 1 and ixed -y > 0, we associa e he linea ope a o s S Q x = (ax1, . . . . axn _1, a-7xn) and he ec angles R a = Sa Ql . These ec angles a e na u ally adap ed o S . y since UR,,,CS,C UKR 2 ; a>1 o some geome ic cons an . Gi en a ec angle R, we deno e by B(R) he collec ion o all ansla es and dila es o R, Le ., B(R) = {z + R : z E R n , > 0} . Le 1 <p < q < co, p' = p/(p - 1), and w(x), (x) be nonnega i e locally in eg able weigh unc ions . Deno e o, = -1 y(P -1 ) . I is p o ed in [CWW] ha i he cen e ed maximal ope a o MS,,, sa is ies he weak- ype es ima e w{x : MS ' , m (x) > A} <- ( C A II IIP, ) q wi h c independen o and A, A > 0, hen IRIñ-1w(R)'u(R)PL < CIRa1ñ_1 INEQUALITIES FOR OFF-CENTERED OPERATORS  431 o all R E B(R a ) and all a > 1 . He e we ha e used he s anda d no a ions w(A) = A w dx, II llp, = (  I (x)Ip (x)dx l p , and c o a cons an which may be di e en a di e en occu ences . Con- e sely, he weak- ype es ima e holds i he e exis s a mono one unc ion C(a), a > 1, such ha (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 Mo eo e , we ha e he s ong- ype es ima e II MS,,jil q,w :5 CII p, , 1 < p < q < oo, i he e exis s > 1 so ha (1 .6)  IR¡ 1  p w ( R ) 1 (  <' IR¡ IR o all R E B(R a ), a _> 1, and C(a) is a mono one unc ion which sa is ies (1 .5) . O cou se, (1 .6) is s onge han (1 .4) due o H51de 's inequali y . Fo he uncen e ed ope a o (1 .1), only a weak- ype es ima e is p o ed in [CWW] . To desc ibe i , le 6* be de ined by S*(x) _ (ax1, . . . , axn_1, xn) o a _> 1, and le R* = 6*Q1 . No e o u u e e - e ence ha R* is he smalles ec angle which con ains bo h R a and Q1 . To each R E B(R a ), associa e a ec angle R* as ollows : i R= z + Ra, hen R* = z + R* .  Thus he pai (R, R*) is a join ansla ion and dila ion o (R a , Ra*) by he same z, .  I is p o ed in [CWW] ha i 1<p<q<ooand (1 .7)  w{x : MS 7 ,Q,m (x) > A} <_ C Cli . llp,  q hen IR1 ñ -1 w(R*)9Q(R)P' < CIRaiñ-1 43 2  R . L . WHEEDEN o all pai s (R, R*), R E 13(R), and all a > 1 . Con e sely, suppose 1 < p <- q < oo and he e is a mono one unc ion C(a) such ha {RI ñ -1 w(R*)9Q(R)P' < C(a)IRalñ-1 o all R E B(R a ) and all a >_ 1 . I C(a) also sa is ies (1 .5) hen he weak- ype es ima e (1 .7) holds . E en in he unweigh ed case w = = 1, i ollows ha he esul s o he cen e ed and uncen e ed maximal ope a o s associa ed wi h S . y a e di e en . In ac , i is easy o check ha he condi ions hen equi e 1/q = 1/p - p,/n, ha he cen e ed maximal unc ion is s ong- ype o y >n- 1 i 1 < p < n/p , , bu ha e en weak- ype o he uncen e ed maximal unc ion equi es p >  y  (> y - (n - 1) 1 1 - ñ) a posi i e esul being gua an eed when s ic inequali y holds . Fo he model case S = S y , we will p o e he ollowing s ong- ype esul o (1 .1) . Theo em 1 . Le y > 0 and S y be de ined by (1 .2) . Le 0 < < n, 1 < p <_ q < oo and assume he e exis s > 1 so ha ( .8)  IR¡!-lw(R*)1  1  u'dx p, <-C(a)IR .I!-1 1  p  9  IR¡ IR o all R E 13(R a ) and all a >_ 1, whe e C(a) is a mono one which sa is ies (1 .5) . Then ~IMS,,Q1,w liq,w < CII IIp, . unc ion A esul o gene al s a like S is gi en in Sec ion 3 . Condi ion (1 .8) is analogous o (1 .6) o he cen e ed maximal unc ion . To p o e Theo em 1, we use a co e ing echnique gi en by C . P . Calde ón in [Ca] oge he wi h a esul we now desc ibe . Le 13 be he amily o all ansla es and dila es o a ixed ec angle Rz3 (Le ., 13 = 13(R,3) in ou p e ious no a ion) . O cou se R13 is no uniquely de e mined by 13 bu i s eccen ici ies ( a ios o edgeleng hs) a e, and we may assume wi hou loss o gene ali y ha i s i s edgeleng h is 1 . Thus, o example, we may iew he basic ec angle in 13(R a ) as ha ing edgeleng hs 1, . . . , 1, a -- í -1 a he han a, . . . , a, a -- í .  To each R E C3, associa e a se (no necessa ily a ec angle) R* so ha he ollowing holds : (1 .9)  I R l , R Z E 13 and R 1 C R Z hen Ri C R2 . Fo example, he pai s (R, R*) o join ansla es and dila es o (R o , R*) de ined ea lie ha e his p ope y . Mo e gene ally, i R* is de ined o be any ec angle con aining R 13 , and gi en R E 13, R= z + R,3, we de ine R* = z + R* hen he pai s (R, R*) sa is y (1 .9) . Fo such a collec ion o pai s and 0 <a< 1, de ine (1 .10)  M a (x) = sup IR¡' - '  I (y)¡ dy . REC3 IR R *Bx O cou se his depends on 13 and on he choice o he se s R*, al hough o simplici y ou no a ion does no e lec his dependence . We will need he ollowing esul . Theo em 2 . Le 1 < p < q < co and 0 <_ a < 1, and le M, be de ined by (1 .10), assuming ha (1 .9) holds . I he e exis s > 1 such ha (1 .11)  IRI a  P w ( R* )° C IR¡  o,' dx/_  p  < C o all R E , 3, hen INEQUALITIES FOR OFF-CENTERED OPERATORS  433 wi h a cons an C which is a mul iple depending on a, n, p, and no on 13 o , o he cons an in (1 .11) . We no e ha he condi ion IIMa 1I9,w < CII IIp, I RI a-l w(R*) 9o,(R) p~ < c,  R E I3, q, bu is necessa y o (1 .12) (e en o he co esponding weak- ype esul ), as can be seen by choosing = XRU in (1 .12) and using a s anda d a gumen . In case R* = R and R is a cube, Theo em 2 is due o C . Pé ez [P1], [P2] . Ou p oo will be modeled on ideas in [SW] and is gi en in Sec ion 2 . 43 4  R . L . WHEEDEN The p oo o Theo em 2 uses some ideas om [SW] . The de ails which a e ei he he same o nea ly he same as ones he e will be omi ed . Le X3 = 13(R L3 ) be a amily a ec angles R as in he in oduc ion, wi h associa ed se s R* which sa is y (1 .9) .  Le e l , . . . , en , (el = 1, say) be he edgeleng hs o R13, and le B dy be he co esponding g id o dyadic ec angles o he o m  1 and o z ER', de ine 2 . P oo o Theo em 2 [ m,el (ml + 1)el  [m ¿ e 7¿ (m .,+ 1)e n , l 2i '  2i  X . . . X  2i '  2i o j, ml, . . . , m,, = 0, ±1, ±2, ...  .  Each ec angle in B d y is also in 1i . De ine M . d y (x) =  sup  IR1  1  I (y)¡ dy, REI3dy IR R'gx Máy,z (x) =  Sup  I R+ zI a -1  I (Y)¡ d . RE 13dy  R+z O cou se, IR + zi = IR¡ . (R+z)`Dx Lemma (2 .1) . I 1 < q < oo and w is a weigh , hen JIM« lIq, . < C Sup IIMa y ' z IIq,w ZERn wi h c depending on a and n bu no on B o . P oo .. We a gue as in [W] and [SW], and ea lie [FS] . The impo an pa o he a gumen is as ollows . Fix R E . 3 and conside he collec- ion o hose Rl E .13 dy whose edgeleng hs a e abou wice hose o R, espec i ely, and hink o Rn as pa ioned in o he union o such Rl . O cou se, IR,¡ Pz :~ IR¡ o each Rl wi h cons an s o equi alen e depending only on n . A simple geome ic a gumen using ansla ions shows ha o each Rl, I{zER n :RCR1+z}I>cIR11 wi h c > 0 depending only on n . Also, wi h R s ill ixed, he se s {z E Rn : R C Rl + z} a e essen ially disjoin o di e en (nono e lapping) Rl . Le E(R 1 ) = {z E Rn : R C R l + z, R* C (R l + z)*}, INEQUALITIES FOR OFF-CENTERED OPERATORS  435 and no e by (1 .9) ha E(R1) is he same as he se {z E R' : R C R, +z} abo e . Also i x E R* and z E E(R1), hen (2 .2)  IRIa -1 R I (y)I dy <_ cI R1 + zia -1 Rl+z I (y) I dy <cMá y, ' (x) since IR, +zi = IR,¡ z- IRI, R CR1+z and x E (R1+z)* . The cons an c depends only on n, a . The key poin s o obse e a e ha i we deno e SZ = UE(R 1 ), hen R 1 he inequali y be ween he i s and hi d e ms in (2 .2) holds i x E R* and zE 9, ha IE(R1)I >_ cIR1I o each R1, and ha he E(R1) a e essen ially disjoin o di e en Rl . The es o he p oo hen p oceeds as in [SW] o [W], and is omi ed . To p o e Theo em 2, i is enough by Lemma (2 .1) o p o e he ana- logue o (1 .12) o each Máy,z , wi h a cons an independen o z . I we eplace by o , , his amoun s o showing ha (2 .3)  IIM« y'z ( U)Ilq,w <ClI IIp,a wi h c equal o a mul iple depending only on a, n, p and q o he cons an in (1 .11) . To p o e (2 .3), ix z and > 0, and o k = 0, l, 2, . . . . le S2 k = {x E R' : M d ,, y,z ( o )(x) > 2k,} . Then x E SZ k i and only i heie exis s R E B d y such ha x E (R + z)* and (2 .4)  IRI"  dy > 2kn . R+z In pa icula , i R E 13d" and (2 .4) holds hen (R+z)* C 1? k . Le {R jk }j be he maximal (wi h espec o inclusion) ec angles in 13 d y which sa is y (2 .4) ; hei exis ence is assu ed i has compac suppo , which we may assume o be he case wi hou loss o gene ali y . By maximali y, he {R j k + z}j a e nono e lapping o each k . Mo eo e , i R~ is he nex la ges dyadic ec angle con aining R .~, hen IR~Ia-1  dy < 2kn R~ -I-z by maximali y, so ha since I R~ I = 2n I R~ I, we ha e (2 .5)  2kn < IR ; la-1  dy < 2n(1-n)2kn . Rk+z 436  R . L . WHEEDEN We claim ha (2 .6) SZk = U(R jk + z)* . j We ha e al eady obse ed ha each (R i + z)* mus lie in Qk . On he o he hand, i x E Qk he e is a dyadic R wi h x E (R + z)* such ha (2 .4) holds . Thus R C Rh o o some jo (since R is maximal o no ), and consequen ly (R + z)* C (R~ + z)* by (1 .9) . Hence, xE (R 3 ~ o + z)* and he claim ollows . By (2 .6), whe e E j' = (R j + z)* S2k + 1 . Thus II M a d ` ( o , )Ilq,w whe e o a ec angle R, II May,z( o,)Ilq,w < 2'l Qk Qk+l = U[(Rj + z) * 9 k+l1 = U E j [M .,,,( a)(x)14w(x) dx E 9 k k Qk+1 E2 (k+1)ngw(Ek) kj q < 2nq 1 : (IR j k j` l  Qdy~ 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 u A(R) = IR¡ T ( Q dy) T . R We es ima e he las sum by using hypo hesis (1 .11) o he ec angles R~k + z and he ac ha E j k C (R jk + z)*, ob aining ha (2 .7) q A(R~ k + z) P  k  I  u dy l  , A(R j + z)  R~+z INEQUALITIES FOR OFF-CENTERED OPERATORS  437 whe e c is he cons an in (1 .11) . The emainde o he p oo is based on using he nex lemma o es i- ma e he sum in (2 .7) . Lemma 2 .8 . Le {Ri}jEI be a collec ion o ec angles om a ixed dyadic g id (e .g ., om B dy + z o ixed z), le ~3 _> 1, and le {ai}iEI be posi i e numbe s which sa is y (i) a(Ri) <_ coa¡ (ii)  E  ¿ < coaQ j :RjCR ; o each i, wi h co independen o i . Then i 1 <p < oc and q = pp, II  1 II  aá (- I lo,dy)q LiEl al Ri 9 <_ ell llp,a, wi h c depending on co, p and q, bu no on o he pa icula g id . The p oo is i ually he same as ha o Lemma (2 .10) o [SW], which deals wi h he case o dyadic cubes, and is he e o e omi ed . I we apply Lemma (2 .8) o he sum in (2 .7) and no e ha o,(R) < A(R) by lldlde 's inequali y, we immedia ely ob ain (2 .3) om (2 .7) i we e i y (2 .9)  A(R y ~ + z )g1p < cA(R- + z)q/p k,j :R~ CR-1 o each R' and 0 <p< q < oc, wi h c independen o l, m and z . We a gue as in he p oo o (2 .11) in [SW] . Using he simple inequali y a ¡ < (~ ai)qlp, q > p, al > 0, we may p o e jus he case q = p . I Rk is a p ope subse o Rm hen whe e he second inequali y ollows om he maximali y o R~ . The e- o e, we mus ha e k >_ m in (2 .9), and we may ew i e he le side o (2 .9) (wi h q/p = 1) as (2 .10) 2 mn < IR¿ l a-1  Q dy < 2kn Rm+z A(R 7 ~ + z) . =m j :R ; CR-