THE MODULAR INEQUALITIES FOR A CLASS OF CONVOLUTION OPERATORS ON MONOTONE FUNCTIONS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Numbe 8, Augus 997, Pages S -9939( THE MODULAR INEQUALITIES FOR A CLASS OF CONVOLUTION OPERATORS ON MONOTONE FUNCTIONS JIM QILE SUN (Communicaed by Palle E. T. Jogensen Absac. This pape is devoed o he sudy of modula inequaliy Φ (a(xkf(xw(xdx Φ (Cf(xv(xdx whee Φ Φ and K is a class of Volea convoluion opeaos esiced o he monoone funcions. When Φ (x =x p /p, Φ (x =x q /q wih <p q<+ and he kenel k(x, ou esuls will exend hose fo he Hady opeao on monoone funcions on Lebesgue spaces.. Inoducion Le K be a class of Volea convoluion opeaos given by Kf(x= k(x f(d, whee he kenel k saisfies he following condiions: (a k(x is nondeceasing on (, + ; (b k(x + y D(k(x+k(y fo all x, y (, +. Conside he poblem of chaaceizing he weighs a, w, v fo which he modula inequaliy of he fom Φ (akf(xw(xdx Φ (Cf(xv(xdx holds fo all nonnegaive monoone funcions f. This so of poblem on Lebesgue spaces fo he Hady opeao has been widely sudied in [ES], [VS] and [HS]. In his pape, we will chaaceize he weighs w, v fo he above modula inequaliy when Φ Φ and he opeao K is esiced o he monoone funcions. The esuls ae even new on Olicz spaces when K is he Hady opeao. I is woh menioning ha when Φ, he above inequaliy is equivalen o Φ (akfw C Φ (fv We begin wih a bief summay of he noaions on he Olicz space seing.. Received by he edios July, 995 and, in evised fom, Januay 3, Mahemaics Subjec Classificaion. Pimay 6D5, 4B5. 93 c 997 Ameican Mahemaical Sociey

2 94 JIM QILE SUN An N-funcion Φ is a coninuous Young s funcion such ha Φ( = whee φ( is a nondeceasing igh coninuous funcion defined on [, + wih φ(+ =, φ(+ =+. Le φ be he igh coninuous invese funcion of φ; hen Ψ( = φ, φ is called he complemenay funcion of Φ(. Definiion. a An N-funcion Φ is said o saisfy he condiion (we wie Φ if hee is a consan B>, such ha Φ( BΦ( >. b We wie Φ Φ if hee is a consan L >, such ha ( Φ (a i L Φ ai holds fo evey sequence {a i } wih a i. c Le w be a nonnegaive, measuable weigh funcion and Φ an N-funcion. The Olicz space L Φ (w consiss of all nonnegaive measuable funcions f (modulo he equivalence elaion almos eveywhee such ha f Φ(w =inf{λ>: Φ(f/λ w } is finie. We call Φ(w he Luxembug nom. Fo moe sandad heoy of Olicz spaces, see [KR] and [RR]. Fo he poofs of ou coming heoems, we need he following special esuls fom Chape of [JS]. Poposiion. Le Φ Φ, and le a, b, w, v be weigh funcions. Then hee exiss a consan C such ha Φ (akf w Φ (Cbfv holds fo all nonnegaive funcions f if and only if hee exiss a consan B such ha boh ( ( + ( a(x Φ k( χ (, B w(xdx vb and Ψ(v ( a(x Φ χ (, k(x B vb Ψ(v hold fo all, >. We define he dual opeao of Kf(x= k(x f(d by K g( = w(xdx ( k(x g(xdx.

3 TWO WEIGHT INEQUALITIES 95 Fo he opeao K,wehave Poposiion. Le Φ Φ. Then hee exiss a consan C such ha Φ (ak g w Φ (Cbgv holds fo all nonnegaive measuable funcions g if and only if hee exiss a consan B such ha boh ( ( ( a( Φ k(,χ (,+ B w( d vb and Ψ(v ( ( a( Φ χ (,+ k(, B vb Ψ(v hold fo any, >. w( d ( Thoughou his pape, we use C o denoe consans which may be diffeen a diffeen places, alhough in some insances we wie C,C, o indicae diffeen consans. Also, we wie V (x = v and V (x =. The main esuls We have he following main esuls. Theoem. Le boh Φ and is complemenay funcion Ψ saisfy he condiion and suppose ha Φ Φ.IfKis a class of Volea convoluion opeaos, hen ( Φ (akfw Φ (Cfv holds fo all nonnegaive, noninceasing funcions f if and only if hee is a consan B such ha all of he following inequaliies hold fo all, > : ( (3 (4 ( Φ (a(x ( Φ ( a(x B k(y dy w(x dx g k( χ (, V Ψ(v ( a(x Φ g χ (, k(x B V Ψ(v ( a(x Φ g χ (, (5 B V Hee g (s =sand g (s = s k(d. Ψ(v x v. (Φ (BV ( ; w(x dx w(x dx w(x dx (. ( ; ( ;

4 96 JIM QILE SUN Fo he nex heoem, we need he following addiional condiion on he kenel k. Thee is a consan D such ha k(x + y D ( k(x+ k(y, whee k(x = k. Theoem. Le boh Φ and is complemenay funcion Ψ saisfy he condiion and suppose Φ Φ.IfKis a Volea convoluion opeao whose kenel k saisfies he above condiion, hen ( holds fo all nonnegaive, nondeceasing funcions f if and only if hee is a consan B such ha (6 and (7 Φ a(x B k( χ (, V Ψ(v w(x dx ( ( a(x Φ B Φ k(x V w(x dx ( ( ( hold fo any, >. Thee ae simila conclusions fo he dual opeao K ; we leave i o he eade as an easy execise. In Theoems and, when k(x, K is he Hady opeao Hf(x= f and s k = s. Thenwehave Coollay. Le boh Φ and is complemenay funcion Ψ saisfy he condiion, Φ Φ.Then Φ (ahfw Φ (Cfv holds fo all nonnegaive noninceasing funcions f if and only if hee exiss a consan B such ha boh ( Φ (xa(x w(xdx and ( a(x Φ g χ (, B V hold fo all, >. Ψ(v (Φ (BV ( w(xdx ( Thee is a simila esul when H is esiced o nondeceasing funcions by ou Theoem. We omi he deails. When a(x =/x, a(xhf(x = x x f, i is he Hady aveaging opeao. Moeove, if Φ (x =x p /p, Φ (x =x q /q wih <p q<+, hen Coollay ecoves some of he esuls in [ES], [VS] and [HS]. 3. Poof of Theoem We need he following esuls fom Chape 3 of [JS]. Lemma. Le Φ Φ ;hen Φ (a(xf(xw(xdx Φ (Cf(xv(xdx

5 TWO WEIGHT INEQUALITIES 97 holds fo all noninceasing f if and only if hee is a consan B such ha ( Φ (a(xw(xdx (Φ (BV ( holds fo any, >. Lemma. Suppose f is a nonnegaive, nondeceasing funcion on (, +. Then hee exiss a sequence {h n }, of nonnegaive funcions, each compacly suppoed in (, +, such ha fo almos each x>, h n inceases o f(x as n +. Also if f is a nonnegaive, noninceasing funcion on (, +, hen hee exiss asequence{h n }, of nonnegaive funcions, each compacly suppoed in (, +, such ha fo almos each x>, x h n inceases o f(x as n +. Fo he poofs of hese wo lemmas, see [JS], pages 39-4 and page 48 especively. We also need he following lemmas fom Chape 4 of [JS]. Lemma 3. Le Φ and is complemenay funcion Ψ be in.then ( ( Φ f(v(d v(xdx Φ(Cf(xv(xdx V (x holds fo any f wih he consan C independen of f. The poof of he lemma can be found eihe in [HK] o [JS], pages 6-6. I can also be deduced fom Poposiion wih k(x. Lemma 4. Le Φ and is complemenay funcion Ψ be in.then ( f(v( + Φ d v(xdx Φ(Cf(xv(xdx x V( holds fo any f wih he consan C independen of f. Poof. By Poposiion, i is easy o see ha he inequaliy in he lemma is valid povided ha hee exiss a consan B such ha ( ( Φ χ ( (,+ v( d B V Ψ(v holds fo all, >. This can be poved by he simila echnique used in [HK]. 3. Poof of sufficiency. Lemma shows ha wihou loss of genealiy, we may suppose f(x = h, wih h compacly suppoed in (, + and Φ x (fv< +.Wehave Kf(x= k(x y h(d dy = y ( s k(x y dy h(s ds + f(x k(y dy K + K. Fo K, apply Lemma wih a(x eplaced by a(x k( d and use condiion ( o ge Φ (a(x k( df(x w(x dx Φ (C f(xv(x dx.

6 98 JIM QILE SUN Since V (s = V (x + s V ( v( d and h(sv (s ds f(sv(s ds, we have ( s ( K = k(x y dy h(sv (s V (x + V ( v( d ds V (x k(x y h(sv (s ds dy y ( s + k(x y dy h(sv (s V ( v( d ds + + f(v( d k(x y dy V (x ( s k(x y dy h(sv (s dsv ( v( d k(x y dy ( K 3 + K 4, f(v( d V (x k(x y dy s f(sv(s ds V ( v( d whee we have used he popey (a of he kenel k fo he las inequaliy. Since f(sv(s ds/v (x is also a noninceasing funcion, Lemma again shows ha fo K 3, ( f(sv(s ds Φ (a(x k(y dy w(x dx V (x ( ( + f(sv(s ds Φ C v(x dx. V (x Now, accoding o Lemma 3, we see ha ( ( Φ fv v(xdx Φ (C f(xv(x dx V(x which complees he esimae fo K 3. Fo K 4, noice ha ( k(x y dy D k(x + We have K 4 D ( D(K 5 + K 6. k(x f (sv(s ds V ( v( d + s k(y dy, fo <y<<x. ( k(y dy fv V( v(d

7 TWO WEIGHT INEQUALITIES 99 Fo K 5, conside he inequaliy Φ (a(x k(x sg(s ds w(x dx (8 Φ (C 3 g(b( v( d, whee g( =v(v ( fv and b( =V(/(v(. Poposiion shows ha (8 holds fo any g povided condiions (3 and (4 hold. Since g(b( = f(sv(sds/v (, Lemma 3 again shows ha he igh hand side of (8 is bounded by Φ (Cf(xv(x dx. Fo K 6, conside he inequaliy Φ (a(x G( d w(x dx (9 Φ (C 4 G(b( v( d, whee G( =(v( k(ydy fv/v ( and b( =V(/(v( k(ydy. Applying Poposiion once moe fo he Hady opeao, we see ha (9 holds fo G( povided condiion (5 holds. Combining all of he above and using he convexiy of Φ, he esul follows. Sufficiency is poved. 3. Poof of necessiy. Since Kf(x ( k(y dy f(x fo any noninceasing funcion f, we may obain condiion ( by aking f = χ (, fo each, >. Noicing he Olicz nom dominaes he Luxembug nom (see [KR], page 8, we have, fo any fixed, >, s g χ (, k(y dyh(sv(s sup ds V Ψ(v Φ(hv V (s s k(s dh(sv(s = sup ds Φ(hv V (s ( = sup k(s h(sv(s Φ(hv V (s sup Φ(hv k( h(sv(s V (s ds d ds d. Then fo any η<, we can choose a funcion h such ha Φ (h(sv(s ds and η g χ (, k( f( d, V Ψ(v wih f( = h(xv(x V(x dx.

8 3 JIM QILE SUN Le C be a consan o be deemined lae. Then we have ( ηa(x g χ (, Φ w(x dx C V Ψ(v ( a(x Φ k( f( d w(x dx C Φ (a(x k(x f( d w(x dx. C Since f/c is noninceasing, he hypohesis of he heoem shows ha he igh hand side is no geae han Now by Lemma 4, we see ha Φ x ( C Φ f(x v(x dx. C Ch(sv(s C V (s ds v(x dx Φ ((CC /C h(x v(x dx holds fo all h. Fo C sufficienly lage, CC /C <, so his is dominaed by (/. This poves he necessiy of condiion (5. I emains o pove he necessiy of condiions (3 and (4. Le g (s =s.wehave g k( χ (, sk( s V sup h(sv(s ds. Ψ(v V (s Since s = s Φ(hv d, he igh hand side is equal o ( k( sh(sv(s V (s sup Φ(hv sup Φ(hv k( ds d h(sv(s V (s ds d. Hee we have used he fac ha k( s k( fo any s. Now using he inequaliy ( and poceeding as in he poof of he necessiy of condiion (5, we can pove he necessiy of condiion (3. Similaly, we can pove (4. Thus we have poved he heoem. 4. Poof of Theoem Befoe he poof, we sae a few lemmas fom Chape 4 of [JS]. Lemma 5. Le Φ, Φ. Then hee is a consan C such ha fo all, >, χ ( (, V Ψ(v C V. (

9 TWO WEIGHT INEQUALITIES 3 On he ohe hand, if V ( = +, hen hee is a consan C such ha χ ( (, V Ψ(v C V ( fo any, >. The deailed poof of he fis inequaliy can be found in [HK]. The second one can be poved similaly. Lemma 6. Le Φ and is complemenay funcion Ψ ; hen hee is a consan C such ha ( ( g(yv(y + Φ V (y dy v(xdx Φ (Cg(yv(ydy holds fo all g. The poof can be found in [HK]. In fac, by applying Poposiion o he Hady opeao, he inequaliy is valid povided hee exiss a consan B such ha ( χ ( (, Φ V Ψ(v v(xdx B holds fo all, >. Now his inequaliy follows fom Lemma 5 easily. Lemma 7. Le Φ and is complemenay funcion Ψ ; hen hee is a consan C such ha ( holds fo all g. x Φ g(yv(ydy V v(xdx (x Φ (Cg(yv(ydy Poof. Poposiion shows ha he inequaliy is valid povided hee is a consan B such ha ( ( χ ( (,+ Φ V v( d B Ψ(v holds fo any, >. The validiy of his inequaliy is poved in [JS], page 54. Lemma 8. Le Φ, Ψ.If(7 in Theoem holds, hen [ ( ] a(x Φ k(xφ B V w(xdx ( ( holds fo all >. Poof. Suppose V ( < + ; ohewise i is obvious. Since V ( V ( fo any >, we have ( ( ( a(x Φ B Φ k(x V w(xdx. ( Since he lef hand side inceases as ends o, he lemma follows easily fom he Monoone Convegence Theoem. Now we begin o pove Theoem.

10 3 JIM QILE SUN 4. Poof of sufficiency. By Lemma, i is enough o pove ( fo funcions f of he fom f(x = hwih h nonnegaive and compacly suppoed in (, +. Thus we have ( s Kf(x= k(x s h ds = h(y k(x s ds dy = ( y k(s ds h(yv (yv (y dy. Since V (y = V ( + y V ( v( d and k(x y = y k(sds, wege Kf(x V ( k(s ds h(yv (y dy ( y + k(x yh(yv (y V ( v( d dy V ( k(s ds f(sv(s ds + k(x yh(yv (y dy V ( v( d V ( k(s ds f(sv(s ds + k(x f(sv(s ds V ( v( d I + I. By Lemma 5, condiion (7 implies ( ( a(x Φ B χ (, k(x V Ψ(v w(x dx y (. Fo I, if we apply Poposiion wih k(x eplaced by k(x, b(x =V (x/v(x and g( =v( f(sv(sds/(v (, we see ha he inequaliy Φ (a(x k(x g( d w(x dx ( Φ (C g(b(v(x dx holds povided condiions ( (6 and ( ae valid. + Since g(b( = fv /V,Lemma7shows ( y Φ V fv v(ydy which complees he esimae fo I. Φ (Cf(yv(y dy,

11 TWO WEIGHT INEQUALITIES 33 Fo I,ake> such ha Φ (fv =. Then f =and Φ(v Φ (a(x k(sdsv ( f(v( d w(x dx Φ (a(x k(s dsv ( f Φ(v / Ψ (v w(x dx. Since / Ψ(v =/(Ψ (/(V (, he igh hand side is no geae han Φ a(x x k(s ds ( w(x dx. V (Ψ V ( Noicing ha (Ψ ( in [KR], page 3, i is bounded by ( Φ (a(x k(s ds V w(x dx. ( Replacing f by C f wih some consan C, and using Φ and Lemma 8, i is easy o see ha he above esimae is bounded by ( ( + Φ (C f(xv(x dx by he choice of. This complees he poof of sufficiency. 4. Poof of necessiy. Fo he necessiy of condiion (6, fix any, >. Since he Olicz nom dominaes he Luxembug nom, we have k( χ(, y V k(s ds g(yv(y V dy. (y Ψ(v sup Φ(gv Fo any η<, ake a funcion g such ha Φ (gv, and k( χ y (, η V k(s ds g(yv(y V dy (y Ψ(v [ s ] g(yv(y = k( s V dy ds. (y Take s g(yv(y f (s= C V (y dy, whee C is a consan o be deemined lae. Then by (, we have Φ ηa(x k( χ (, C V Φ (a(xkf (x w(x dx Φ (Cf (s v(s ds. Ψ(v w(x dx

12 34 JIM QILE SUN On aking C sufficienly lage and by Lemma 6, he igh hand side is bounded by ( Φ (g(yv(y dy. The Monoone Convegence Theoem allows us o eplace η above by is limiing value of and hus condiion (6 is poved. Similaly, fo fixed, > andanyη<, we can choose a funcion g wih + Φ (gv and η χ (, g(yv(y V Ψ(v V dy. (y Le C be a consan o be deemined lae. Then ( ηa(x χ (, Φ k(x C V Ψ(v w(x dx ( a(x g(yv(y Φ C V dy k(x w(x dx (y Φ ( C x Φ ( a(x C k(x y g(yv(y V (y [ s k(x s dy w(x dx g(yv(y V (y ] dy ds w(x dx. By hypohesis (, he igh hand side is no geae han ( C x g(yv(y Φ C V dy v(x dx. (y Again, on aking C sufficienly lage and using Lemma 6, we have Φ (g(yv(y dy ( Replacing η by is limiing value of as befoe, we have ( a(x χ (, Φ k(x C V Ψ(v w(x dx. (. When V ( = +, (7 follows fom Lemma 5 easily. Ohewise, suppose V ( < +. Fo any δ>, eplacing v by v + δ/x in (, (7 is sill valid. Since by he above agumen, we have ( ( a(x Φ B Φ (v(x+δ/x dx =+, (V (+δ/ k(x w(x dx (. Leing δ end o, he Monoone Convegence Theoem shows ha (7 is valid.

13 TWO WEIGHT INEQUALITIES 35 Refeences [KR] M.A.Kasnosel skii and Ya.B.Ruickii, Convex funcions and Olicz Spaces, P.Noodhoff LTD., 96. [ES] E.Sawye, Boundedness of classical opeaos on classical Loenz spaces, Sudia Mah. 96 (99, MR 9d:66 [VS] V. Sepanov, The weighed Hady s inequaliy fo noninceasing funcions, Tans. Ame. Mah. Soc. 338( (993, MR 93j:6 [HS] H.P. Heinig and V.D. Sepanov, Weighed Hady inequaliies fo inceasing funcions, Canad. J. Mah. 45( (993, 4-6. MR 93j:6 [HK] H.P.Heinig and A. Kufne, Hady opeaos of monoone funcions and sequences in Olicz spaces, J. London Mah. Soc. 53 (996, MR 96m:65 [JS] Jim Qile Sun, Hady ype inequaliies on weighed Olicz spaces, Ph.D Thesis, The Univ. of Wesen Onaio, London, Canada, 995. [RR] M.M. Rao and Z.D. Ren, Theoy of Olicz spaces, A seies of Monogaphs and exbooks, vol. 46, Macel Dekke, Inc., 99. MR 9e:4659 Depamen of Mahemaics, Univesiy of Wesen Onaio, London, Onaio, Canada N6A 5B7 addess: jsuen@swichview.com