Weighted inequalities through factorization

Publicacions Ma emá iques, Vol 35 (1991), 141-153 . WEIGHTED INEQUALITIES THROUGH FACTORIZATION EUGENIO HERNÁNDEZ 1 . In oduc ion and esul s In [4] P . Jones sol ed he ques ion posed by B . Muckenhoup in [7] con- ce ning he ac o iza ion o A p weigh s . We ecall ha a non-nega i e mea- su able unc ion w on R" is in he class A p , 1 < p < oo i and only i he Ha dy-Li lewood maximal ope a o is bounded on LP(R',w) . In wha ol- lows LP(X, w) deno es he class o all measu able unc ions de ined on X o which Il w l/P II Lo(X) < oo, whe e X is a measu e space and w is a non-nega i e measu able unc ion on X . I has ecen ly been p o ed ha he ac o iza ion o A p weigh s is a pa icula case o a gene al ac o iza ion heo em conce ning posi i e sublinea ope a o s . The case in which he ope a o is bounded om LP(X, ) o LP(Y, u),1 < p < oo, o u and non-nega i e measu able unc ions on X and Y espec i ely, is ea ed in [8] . The case in which he ope a o is bounded om LP(X, ) o L9(X, u), 1 <p < q < oo is ea ed in [3] . Ou i s esul is a ac o iza ion heo em o weigh s u and associa ed o ope a o s bounded om LP(X, ) o L9 (Y, u), whe e X and Y a e wo, possibly di e en , measu e spaces, and p and q a e any index be ween 1 and oo . Le X and Y be wo measu e spaces and le M(X ), M(Y) be he class o measu able unc ions de ined in X and Y espec i ely . An ope a o T de ined on a subse o M(X) wi h alues in M(Y) is called sublinea i I T ( + g) I < IT( )I + IT(g)I and is called posi i e i I 1 < g ~--> IT( )I < T(g), o all , g E M(X) which belong o he domain o T . Theo em 1 (Fac o iza ion) . Le T and T' be wo posi i e sublinea op- e a o s de ined on subse s o M(X) and M(Y) espec i ely . Le E M(X) and u E M(Y) be non-nega i e unc ions and 1 < p, q < oo . Suppose ha T is bounded om LP(X, ) o L9(Y, u) wi h no m IITII and T' is bounded om L9'(Y,u-9'/q) o LP'(X, -p'/p) wi h no m IIT'II . Then he e exis non-nega i e unc ions uo E M(X), o E M(Y), ul E M(Y) and i E M(X) such ha = uO p/P,VI, u =  O q/q'ul, IIuo IIILI(X) < 1, IIVOUIIIL~(Y) < 1, T(u o ) _< IITIl o and T'(u l ) < 2p/p'IIT'Il l . This heo em can be applied o a la ge class o ope a o s o ob ain he ac- o iza ion o he associa ed weigh s . The eade can ind se e al examples in [8] and [3] . 14 2  E . HERNÁNDEZ Fo in eg al ope a o s wi h non-nega i e ke nel, he ac o iza ion heo em has a con e se o some pa icula cases o p and q . Le k(x, y) be a measu able non- nega i e unc ion on X x Y . Le us deno e by K and K* he ans o ma ions : (K )(y) = k(x,y) (x)dx,  (K*g)(x) = k(x,y)g(y)dy, X  Y he domain o K being he se o all unc ions E M(X) such ha he i s in eg al exis s and is ini e o almos all y, and he domain o K* being anal- ogously de ined . Theo em 2 . Leí 1 < q <p < oo and E M(X ), u EM(Y) be non-nega i e . A necessa y and su cien condi ion o K ío be bounded om LP(X, ) o Lq(Y, u) is hai Me e exis non-nega i e unc ions u o E M(X), o E M(Y), u l E M(Y), i EM(X) and ini e cons an s Co, Cl such ha Iluo 1IILl(X) < 1, = u0PIP' l, u = o g1q' u,, K(u 0 ) < Co o and K*(ul) < C l l . Mo eo e IIKII < Colq'Cilq . The case p = q is simple : Theo em 3 . Leí E M(X) and uEM(Y) be non-nega i e . A necessa y and su cien condi ion o K lo be bounded om LP(X, ) o LP(Y, u) is ha he e exisi non-nega i e unc ions u o E M(X), o E M(Y), u 1 E M(Y), i E M(X) and ini e cons an s Co, C l such ha = uo PIP' l, u = o PIP'ul K(u0) < Co o and K*(u l ) < Cl l . Mo eo e IIKII < Co lIP'C1IP, The case - u - 1 o heo ems 2 and 3 is p o ed in [1] . Ou p oo o hese heo ems is an adap a ion o he p oo o he co esponding esul s in [1] . In he case p < q he condi ions o heo em 2 a e no su icien o he boundedness o K om LP(X, ) o Lq(X, u) e en in he case - u - 1 (see [1]) . Obse e ha in heo em 2 we only need he condi ion Iluo 1IIL'(X) <_ 1 while he "symme ic" condi ion IIUI 0IIL1(Y) < 1 is no needed . Nei he o hese is needed in heo em 3 . Fo some applica ions i is be e o eplace he su icien condi ion o heo em 2 by he ollowing one, whose s a emen is a gene aliza ion o he su icien condi ion o heo em 3 : Theo em 4 . Le¡ 1 < q < p < oo and E M(X ), uE M(Y) be non-nega i e . Suppose ha he e exis non-nega i e mesu able unc ions u o , o , u,, i such ha = uo PIP ' l q ' IP' u = u, o - PIq ' K(uo) o 1 E L''(u) (wi h L'(u) no m equal o C o ) and K*(ul) 1 1 E L''( -P'IP) (wi h L''( -P'IP) no m equal o C l ), whe e - 9 - .  Then, K is a bounded ope a o om LP(X, ) o Lq(Y,U) wi h no m less han o equal o Co I q , C1 , IP . Fo he cases q = 1 o p = oo, which a e no co e ed by he abo e heo ems, we ha e he ollowing sa is ac o y esul : WEIGHTED INEQUALITIES THROUGH FACTORIZATION  143 Theo em 5 . (A) I 1 _< p < oo, a necessa y and su cien condi ion o K o be bounded om LP(X, ) o L 1 (Y,u) wi h no m IIKII is 11 JY k(x, y)u(y)dylI LP'(X, -n'ln) :5 IIKII (B) I 1 <_ q < oo, a necessa y and su cien condi ion o K o be bounded om L'(X, ) o L9(Y, u) wi h no m IIKII is 11 JX k(x,y) -1 (y)dylIL9(Y,«) ~ IIKII In his heo em L'(X, ) = { E M(X) : Il ll . < oo} . Examples o ope a o s o which hese heo ems can be applied a e he ol- lowing : he Ha dy ope a o and i s dual x he ac ional in eg al ope a o T (x)=101 (y)dy  ,x>0, T * (x) = J ~ (y)dy  ,x>0 ; (Ia )( x ) = I n ( x - y)l yl `dy,  xE Rn, 0 < a < n R which is sel -adjoin ; he Riemann-Liou ille ope a o 1  j'  (Y)  «dy, (Ta )(x) = I (-)  (x - Y) ,-C, Laplace ans o m 00 £ (x) =  e-y (y)dy 0 and he mul idimensional Ha dy ope a o x>0,a>0 ; x~ x n T .Ax11 ... , xn )  (yl, . . . , y n )dy n . .. dyl . The p oo s o heo ems 1 o 5 will be gi en in sec ion 2 . Applica ions will be gi en in sec ion 3 . These a e conce ned wi h weigh ed inequali ies o some o he abo e ope a o s . I would like o hank B . Jawe h o calling my a en ion o [1], which u ned ou o be he s a ing poin o his esea ch . 14 4  E . HERNÁNDEZ 2 . P oo s o Theo ems 1 o 5 To p o e heo em 1 we need he ollowing lemma which can be ound in [1] . Lemma 2 .1 . Le B be a Banach space and P a con ex cone in B . By calling his cone "posi i e", B will be aken as an o de ed Banach space . Le us suppose o B and P ha e e y bounded inc easing sequence in P con e ges, mo e p ecisely : { n} C P, n+l - n E P, Il nll < M < oo => n - E P Le¡ S be a ans o ma ion de ined in B such ha S(P) C P, S is nondec easing ( ha is, , g, g - E P => Sg - S E P), S is con inuous and 11 II < 1 => Ils ll<Co<oo . Then he e exis s a E P, a 7É 0, llall < 1 such ha 2C o a - Sce E P To p o e heo em 1 we ake B= LP(X ), P such ha E P <=> (x) > 0a .e . and S  -  T'  T( -1/P)  qlq' u  ] P'/p -P1/P2 ( ) [ (C IITII  ) and apply lemma 2 .1 . Obse e ha he boundedness o T and T' implies IIS lILD(X) < IIT'IIP'/PIL IILP(X) so ha Il 1I < 1 => IIS 1I < JIT'l1P'/P . Hence he e exis s a qÉ 0, a >_ 0, a E LP(X) wi h no m less han o equal o 1 and S(a) < 2l1T'1IP /P a . The p oo o heo em 1 is inished by aking and l = uó /P' . uo = a -1 /P, o = T(uo)/IITII, ul = (T(uo)/IITII)1/1'u Theo ems 2 and 3 will be es ablished once we p o e he su iciency o he condi ions, since he necessi y ollows immedia ely om heo em 1 . To p o e heo em 2 ake E LP(X, ), g E Lq'(Y, u- q'/q) and use Holde 's inequali y wi h , p and q', whe e = 1 - o ob ain (2 .2) WEIGHTED INEQUALITIES THROUGH FACTORIZATION  145 1/T ~X Y k(x,y) (x)9(y)dxdyl C (L 1Y k(x,y)u1(y)uo(x)dxdyl 1/p . ( / k(x,y)u1(y)uo(x)-pl"I (x)Ipdxdy/ X Y 1/q' . ~ / k(x,y)uo(x)u1(y)-q'lglg(y)Iq~dxdy) X Y Using K*(u 1 ) _< C 1 1 and Iluo 1IIL1(X) < 1 he i s ac o en he igh hand side o he abo e inequali y is bounded by Cl / . Using K* op/p'  (u 1 ) < C 1 1 and = u  1 we deduce ha he second ac o is bounded by Cl /p ll JI L (X, )- Finally, using K(u0) _< Co o and u = o '/"u, he hi d ac o can be majo- a ed by Cól q II9I6'(Y, .-199) . Pu ing hese es ima es oge he we ob ain JX Yk(-,y) (x)g(y)dxdy <Có/q'Cl/gli lILP(X ; )IIgIIL F om he e he desi ed esul ollows . The same a gumen applies o p = q, ha is o heo em 3, excep ha in his case 1 -- 1 - ñ = 0, and hence he i s ac o on he igh hand side o (2 .2) does no appea . Thus heo em 3 does no equi e he condi ion IIuo 1IIL1(X) < 1 . To p e e heo em 4 le ELP(X, ) and g E L9 (Y, u - q'1q) . F om =g- we deduce , -{- ñ = , so ha we can apply Holde 's inequali y wi h indices p/ ' and q'/ ' o ob ain I k(x,y) (x)9(y)dxdy X Y T /p < CJxJYk(x,y)u1(y)uo(x)-plq'I (x)Ip/ 'dxdy) l '/q' . CJx /Y k(x, y)uo(x)u1(y)-q'lpl9(y)I q'l dxdy'  = (I) .(II) . Using Holde 's inequali y wi h index , oge he wi h = uoplp ' q ' lp' and K*(u 1 ) 1 1 E L ( - p ' 1 p) we ob ain '/ p (I) < C x_  [K*u1)(x)] 1 (x) -p'1p(x)dx) 146  E . HERNÁNDEZ 1/p '  I (x)Ip 1(x) '*)p, /p uo ( x )-p lq'dx) x Using again Holde 's inequali y wi h he same index, oge he wi h u u l o p/ q' and K(uo) o 1 E L'(u) we ob ain (B) The p oo is analogous . 1/p = C1'/p (L I (x)Ip (x)dx ) ' / g , (II) <( [K(uo)(y)] o(y)`u(y)dy) 1/q ' ' (l YIg(y)I9 u1(Y) -q, 'lp o(y) u(y)- / dy) 1 /q~ = Co'/q, (/ I g(y)I q' u(y) -q'lg dy) Y The desi ed esul ollows by pu ing hese wo es ima es oge he . This inishes he p oo o heo em 4 . We now p o e heo em 5 . (A) Su ciency . Fo E LP(X, ) and g E L'(Y, u -1 ) we ha e k(x, y) (x)g(y)dxdy _< IIglIL-(Y,u-1) ix  Y / k(x, y)I (x)I u(y)dxdy ~ x Y IIgIIL-(Y,u-1) { u k(x,y)u(y)dy) pl (x) -p'lp dx X Y Necessi y . The boundedness o K implies o all E LP(X, ), g E L'(Y, u -1 ) . Wi h g - u we ob ain (x) (/ k(x,y)u(y)dy) dx  <_ IIKIIII lILD(X, ) ~ X  Y o all E LP(X, ) . Thus he esul ollows . 1/p' 1/p I (x)Ip (x)dx}  _< IIsIIL~(Y,u-1)IIh~~II IILy(X, ) . X L lyk(x,y) (x)g(y)dydx1 `IIhIIII ilLD(X, )II9IIL-(Y,u-1) 3 . Applica ions Conside he in eg al ans o ma ions (3 .2) B l = (3 .3)  B z = WEIGHTED INEQUALITIES THROUGH FACTORIZATION  147 x (K )(x) =  k(x,y) (y)dy, (h * )(x) = J  k(y,x) (y)dy 0o  x de ined on he eal line, whe e k(x, y) is a nonnega i e measu able unc ion de ined on A = {(x, y) E R 2 : y < . x} . Gi en wo non-nega i e measu able unc ions u and de ined on he eal line, we w i e (u, ) E W l (K, p, q), 1 < q<p<ooi {1-00 I~ / OOk(x,y)u(x)dx)1/q 00 y -00 and (u, )EW 2 (K,p,q),1<q<p<ooi y  1/q  1/ , k(y, z) (z) - P 1Pdz )  ] (y)-PIPdy l 00 y 00  ~ k(x, y)u(x)dx  1/P  y J  CJ-  k(y, z) (z)-P'1Pdz  1/P  ~  u(y)dy 1/ j J < +oo whe e = 1 - P . Obse e ha in he limi ing case p = q, (3 .2) and (3 .3) become ~/P y  i/P , (3 .4)  B = sup o ~~  k(x, y)u(x)dx p  (  k(y, z) (z) - P '1 Pdz~  < +oo y Y>  00 P oposi ion 3 .1 . Le K be he in eg al ans o ma ion de ined by (3 .1), whe e k(x, y) >_ 0 is nondec easing in y and noninc easing in x . I (u, ) E W ; (Is, p, q), 1 < q <_ p < oo, j = 1, 2, hen ( ~~ I(h )(x)Iqu(x)dxlllq _< C  I (x)IPV(x)dx~1/P < +oo 148  E . HERNÁNDEZ whe e C <_ (gB1)TI /P(pT 2 ) / ', P oo . Since > y implies k(x, ) > k(x, y) he no m o he unc ion 00  00  1/9 ' 1 y  , 1/9' k ( , y) ~~  k(x, )u(x)dx~  u( )d ' ~~  k(y, z) (z)-P 1Pdz) y   j 0o in LT( - P'1P) is bounded by he no m o he unc ion ~o (Y) = 00  00  1 /9 '  1  y  , k( , y)  k(x, y)u(x)dx~  u( )d l ~~  k(y, z) (z)-P 1Pdzl y   /  l00 in he same space . In eg a ing by pa s we ob ain Taking he abo e inequali y can be w i en as K*(u l ) 1 1 E L ( - P'/P) wi h no m no exceeding qB l . Since < y implies k( , z) > k(y, z), he no m o he unc ion / w k(x, y)u(x)dx l 1/P (I-Y . k(y, ) ( )-P'1P ~,1 y in L''(u) is bounded by he no m o he unc ion WP11U( - 9P) < qBl . u l ( ) =  k(x, )u(x)dx~ -1/ u( ), y 1 (y) _ ~~  k(y, z) (z)_P'lndzl _ 1 /P k( , z) (z) - P '1 Pdzl  d 00  1 /P y   1/P k(x, y)u(x)dx l  k(y, ) ( ) - P '1 P k(y, z) (z) - P' 1Pdz'  d y  00 V00 in he same space . In eg a ing by pa s we ob ain 11011U(u) < p'B2 . Taking he abo e inequali y can be w i en as K(u o ) . -1 E L'(u) wi h no m no ex- ceeding p'B 2 . The p oo o p oposi ion 3 .1 is inished by applying heo em 4 i q < p and heo em 3 i q = p . Rema ks .  1 . Fo he case q = 1, (u, ) E W, (K, p,1) is an equi alen condi ion o he boundedness o he ope a o K de ined by (3 .1) om LP(X, ) o L 1 (Y,u) . This ollows om pa (A) o heo em 5 . 2 .  Fo he case p = oo, (u, ) E W 2 (K, oo, q) is an equi alen condi ion o he boundedness o K om L'(X, ) o L9(Y, u) . This ollows om pa (B) o heo em 5 . 3 . A esul simila o p oposi ion 3 .1 . can be ob ained o K* . De ails a e le o he in e es ed eade . 4 . The ope a o s de ined by (3 .1) ha e been s udied in [2] ; he condi ions imposed on he weigh s u and o ob ain weigh ed inequali ies o K a e di e en om hose used he e . Fo he pa icula case o he Ha dy ope a o T (x) = o (y)dy, N T i (T, p, q) becomes (3 .5) Bi  00 ~  y0  00 0 U) 1/9 (,y -P,/P  1/9' j  1/ < + 00 =  , ~  )  (y)-P./Pdy l 0 W2 (T, p, q) becomes WEIGHTED INEQUALITIES THROUGH FACTORIZATION  149 (3 .6)  B2 = { 00  1 /P  y -P, /P ) 1/P' j u(y)dy  1/ [(  (1  }  <+ oo, 0 y00  00 and when p = qwe ha e ( ) = (1-  k(y, z) (z) -P'1P dz)  ( ) - P , /P o( ) = (1 00 k (x, )u(x)dx) ilP Y> . (u)  y U In his case we shall show ha he condi ion W 2 (T, p, q) is implied by W 1 (T, p, q) and ha his condi ion is also necessa y o he boundedness o T .