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The generalized hardy operator with kernel and variable integral limits in banach function spaces

The generalized hardy operator with kernel and variable integral limits in banach function spaces aRazmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidze st., Tbilisi 380093, Georgia; b University of Missouri-Columbia, Department of Mathematics, 202 Mathematical Sciences Building, Columbia, MO 65211, USA (Received 18 December 1997; Revised 15 September 1998) Let we have an integral operator b k(x, y)u(y)f(y)dy for x > 0 a b are nondecreasing functions, u v are non-negative finite functions, k(x, y) >_ 0 is nondecreasing in x, nonincreasing in y k(x, z) < D[k(x, b( y)) + k( y, z)] for y <_ x a < z <_ b(y). We show that the integral operator K: X Y X Y are Banach functions spaces with/-condition is bounded if only if A < o0. A Ao + A1 A0 A1 Keywords: Banach function spaces; Hardy operator; Integral operator;/-condition 1991 Mathematics Subject Classification: Primary 46E30; Secondary 47B38 * Corresponding author. (On leave from the Mathematical Institute of the Academy of Science, Prague, Czech Republic) E-mail: langjan@aqua.math.missouri.edu. A. GOGATISHVILI J. LANG INTRODUCTION Let X Ybe two Banach function spaces on (c, d) respectively. We define the general Hardy operator [b k(x, y)u(y)f(y) dy for x (1.1) a,b are nondecreasing functions on -<c<a< v u are non-negative measurable finite b < d< functions a.e. on/R (c, d). The kernel k(x, y) > 0 is defined a.e. on {(x, y); x E IR, a < y < b} satisfies the following conditions" (i) it is nondecreasing in x nonincreasing in y; (ii) k(x, z) < D[k(x, b(y))+k(y, z)] for every y < x a < z < b(y), the constant D > is independent of x,y,z. (1.2) In this paper we describe the necessary sufficient condition for the boundedness of the operator (1.1) in Banach function spaces. This paper extends results of Lomakina Stepanov [3] Opic Kufner [4]. In these papers the operator (1.1) was characterized for a 0 b x. Sections 2 3 contain the definitions, formulations of the main results some comments. In Section 4 we treat the simpler case when the kernel k(x,y) is equal to the spaces X, Y satisfy the /-condition. We use this result in Section 5 to deal with the general kernel satisfng (1.2). DEFINITIONS In this section we recall the definition some basic properties of the Banach function spaces. In what follows A4(f) will be the set of all measurable functions on f, f is any measurable subset of. DEFINITION 2.1 A normed linear space (X, II.llx)onfiscalledaBanach function space (BFS) on f if the following conditions are satisfied: (2.1) the norm Ilfllx is defined for allfE M(f) fe Xif only if Ilfllx<; (2.2) Ilfll= Ifl I1 for everyfE l(f); (2.3) if 0 <fn/fa.e. in f then Itf=ll/llfllx; (2.4) if IEI < o, E C f, then XE E x; (2.5) for every set E, IEI < c, E C f, there exists a positive constant CE such that flJldx <_ Cllfll. By we denote a Banach sequence space (BSS), which means that the axioms (2.1)-(2.5) are fulfilled with respect to the counting measure {ek} denotes the stard basis in l. Recall that the condition (2.3) immediately elds the following property: (2.6) if 0 <f< g then [[fllx < Ilgllx. DEFINITION 2.2 The set X’={f’,fafgv < equipped with the norm for every g E X}, "f"e ={ fafgv ;,,g[[x < 1}, is called the associate space of X. It is own from Bennett Sharpley [1] that X" X that X’ is again a BFS. Let T be a linear operator from a BFS X into a BFS Y. Then T’ is an associate operator to the operator Tif fn(Tf)g faf(T’g) for allf X g Y. LEMMA 2.3 (Bennett Sharpley [1]) Let T be a linear operator from a BFS X into a BFS Y. Then TTII <_ CII f xfor allf X with afinite positive constant C, if only/f T’gllx’ _< Cllgll for all g Y’. Moreover I Tllx r I Z’ll r-. x,. DEFINITION 2.4 (Lomakina Stepanov [3]) Given a BFS X a BSS l, X is/-concave, if for any sequence of disjoint intervals (Jk} such that t_J Jk f, for allfE x A. GOGATISHVILI J. LANG is dl a finite positive constant independent on fE X {Jk}. Analogously, a BFS Y is said to be/-convex, if for any sequence of disjoint intervals {Ik}, t3 Ik f for all g Y Ilgllr d21[ekllxzgllrk with a finite positive constant d2 independent on g E Y {Ik). We say, that BFS X, Ysatisfy the/-condition, if there exist a BSS 1 such that X is/-concave Y is/-convex simultaneously. LEMMA 2.5 (Lomakina Stepanov [3]) Let Y be a l-convex BFS. Then Y’ is an l’-concave BFS for allf Y’ (Ik}, tO Ik Ft. MAIN RESULTS Assume X Y are two BFS on (c, d) denote respectively. Then we Ao A :: x<y,a(y) 0 satisfng (1.2). Then K: X-- Y is bounded, if only if, A is finite. Moreover To prove Theorem 3.1 we need a corresponding result for the general Hardy operator with kernel k(x, y) 1. Let b Hf u(y)J(y) dy (3.1) -o < c < a < b < d < o are nondecreasing functions on v u are real measurable finite functions a.e. on I (a,/3), respectively. THEOREM 3.2 Let X Y be two BFS on (c, d) 1t, respectively, satisfng the l-condition, let H be the operator defined by (3.1). Then H" X-. Y is bounded, if only if, A. x<y,a(y)<_b BOUNDEDNESS OF THE OPERATOR H In this section we prove Theorem 3.2. At first we prove a lemma. DEFINITION 4.1 Let v be a non-negative measurable function on an Let c E let interval (a, fl) o < a </3 < < a < c < < be nondecreasing functions, let u be a non-negative meab surable function on (e,d) eliminfx_a d lim x b. Then we define b . , Hbf for every measurable functionfon (c, d), Haf for every measurable function fon (e, c). A. GOGATISHVILI J. LANG LEMMA 4.2 Let 3( Y be two BFS on (e, d) (a,/3), respectively, satisfng the l-condition. Then Hb X-+ Y is bounded, if only if, Ab Ilvx(x,llrllux(c,b(xllll, < c<x</ Moreover Ilnbllx-+r Ab. Also Ha:X--> Y is bounded, if only if, ha Moreover :-’- a<x<fl 11VX(x,)II yl]ux(a,c)I1, < Proof We will give the proof only for Hb. The proof for Ha is similar. Necessity Given x E (a, fl) fE X such thatfu > 0, we have b(.) v(.) ,+c >_ V(.)X(x,Z)(.) b( x) b >_ v(.)X(x,Z)(.) VX(x,l r lu(t)f(t)l dt. Taking the remum over all suchf x E (a, fl) we obtain Ilnbllx+ y IlvX(x,lllrlluxc,b Ab. Sufficiency If Ab--Cxz then )lHllx-+ r Ab. If A b- 0 then II/-/,,11,,- ,-= o. Let O<Ab< . Choose fEX such that [[fllx 1. Define C/= {t; E (Ce,/), fbc(t) [fu[ 2i}, O Ci\ C + Ei {x; x (c, d), b C}, Bg E\E+ 1. Then I(c, d)\ t.J z Bil 0 [(a,/3)\ I..J EZ De[--0 we have gHbf < iz 2i+lgv <_ iZ, IDI>O 2e+lllvx,llrllgx,llr, iz, IDil>0 _< _< ’ 2 g/allvxc,llllgX,l[ 2i+1 hb [fXBi-1 Ilul (using 2 i-1 _< <- IIfi-, IIllux(c,up,_,/ll,) iZ, IDil>0 Ab2i+l (using Holders inequality /-condition) <_4Ab ieZ, lDl>0 eillfx(.i_,)[[x iez, [ai[>0 eillgxo, l[y, l’ <_ 4dld2Ab[[f[[x[[g[[r. Then we have lig[Ir,<l glib f<_ 4dld2Abllf[[x. Now we prove Theorem 3.2. Proof of Theorem 3.2 Necessity Let f X be such that fu >_ 0 1, let x, y be such that a _< x _< y </3 b >_ a(y). Then Ilfllx- I[Hf[[r >_ [Iv(.)X(x,y)(.) (.) l[ r VX(x,l fb( u( t)f( t) dr. Ja(y) A. GOGATISHVILI J. LANG Taking the remum over all such x, y fwe obtain IIHII _> IlVX(x,y)II Sufficiency Define M {(x, t); x a < < b}. M is a measurable set. If IMI 0 then it is easy to see that IIHIIx_ y O. pose that IMI > O. We set My {x; (x, t) M, y}, y ]R, P {y; (x, t) M, IMyl > 0}. Then P tAim=lPi, Pi are intervals, [Pi[ > O, rn < Let yoint Pi; then we have a set My co =a(infMyo ), do b( Myo). pose we have defined , ci, di My,. If i>O diEint Pi then we define +l=di, di+l ci-1 i+1 ----a(infM+l), b(M+l). If i<O ciEintPi then we define -1 =ci, a(inf My_), di-1 b( M_). ni ni n ( c)}j=m, {dtj}j=mi, {My "})=mi’ By using this method we can construct, for every P sequences f i’l ni tYj]j=mi’ We can rewrite all these sequences in the following way: {Ci)gL, {di}i= {My,}g= k Zi%l(ni Then we have {}ik=l, mi + 1). a.eo nf Z XM i=1 ? + X) jfgHf Z L gHf i=1 i=I :Mr(IJ fb f(t)u(t) dt + v(x )fay f(t)u(t) d g(x dx tl ) i=1 f(t)X(,di)U(t dt g dx () f(t)X(ci,y,)U(t) dt g d (Using Lemma 4.2 Aa + Ab <_ AH) gHf <_ 4d d2 i=1 Al-II]fX(e,,y,)Ilxllgxt, i=1 r, + 4dl d2 8dd2An Z AHIIIxy,,d,I IlxllgXM, IIfX(c,,d,llxllgx,ll i=1 (use H61der’s inequality /-condition) BOUNDEDNESS OF THE OPERATOR K In this section we prove Theorem 3.1. LEMMA 5.1 Let b be a nondecreasing right continuous function on (a, fl) let b(cO c, b(/) d. Let ko(x, y) > 0 be a kernel satisfng(1.2), ko(x, y) > 0 on set of positive measure. pose that ko(x, y) is right continuous with respect to x for all x E [a, ] for a.e. y (c, b). Let u, f be measurable functions on (c, d), fu > O, b Go go(x, y)u(y)f(y) dy. For a fixed number 6 > D ( D is a constant from(1.2)), we define A k {x G (a,/); Go >_ (t5 + 1)}, k Z, N {k; Ak ). Then there exist sequences {}, {"/k) such that a < the inequality < - (6 + 1) "- < b() ko(x, y)u(y)f(y) dy ,]b(-l) dy .16(_2) + Dko(, b(-1)) ab(x_)u(y)f(y) dy + Oko(, b(-2)) u(y)f(y) dy. holds for all k <_ N, Go <_ (1 + 6) "’-’+1 when x [- ). A. GOGATISHVILI J. LANG Proof By the Lebesgue Dominated Convergence Theorem Go is a nondecreasing right continuous function for all a < x </ limx Go =0. Set ak inf Ak, for k < N. Fix E Z such that IzXel > 0. We set x0 ai, 7o max{i; ai xo), ak "Yk max{ i; ai a,k_+ for k > 0 "Yk max { i; ai a,k+ } for k<0. It is obvious that {7k} is an increasing sequence of integers, therefore 7k, < k- 1, G()- G(a.,) > (1 + 6) "k. If x [, + 1), then we have a+ +l, therefore x < a+l G < (1 + 6) + 1. Next on using (1.2) we find that As DGo( 2) D(1 + 6)"k-2 + < D(1 + 6)"k t(1 + 6)k "k *- (1 + 6)e*- the lemma follows. 6(1 + 6) (1 + 6) THEOREM 5.2 Let X Y be two BFS on (c, d) (c, ), respectively, ( b() d b(a) c) satisfng the l-condition b Kbf k(x, y)f(y)u(y) dy, k(x, y) satisfies (1.2). Then A A Proof "= a<z<g II<z,)vllrllX<c,bz)l(.)k(z,.)u(.)ll,, IIX(z,(.)v(.)k(.,b(z))llllX(c,b(zlull,. "= a<z a. Then b> c. Since k(x,y) is nondecreasing in x nonincreasing in y, for every a < x < z < fl we have b >_ v(z) k(x, t)f(t)u(t) dt b Kf(z) > v(z)k(z,b) Hence, . IIKfllf >- IIX(x,g)(.)v(.) for allf X a < x < b k(x, t)f(t)u(t) dtl] r >_ IIx(x,l vll Yllx(c,b (.)k(x, .)u(.) IIx, IlfX(c,bl IIx b IIKfll r _> IIX(x,l(.)v(.)k(.,b) ll, _> IIx(x, (.)v(.)k(., b)II rllx(,b(xlul[ IIX(c,b(xll fllx for allfE X a < x < g. Sufficiency Let D be the constant from condition (1.2) let 6 > D be fixed. Without loss of generality we may assume that k(x, y) b satisfy the assumptions of Lemma 5.1. Otherwise we replace k(x, y) by k(x +, y) b by b(x +). A. GOGATISHVILI J. LANG By the principle of duality it is sufficient to show that v(t)G(t)g(t) dt < Allxllgllr,, for all f E X g E Y’, G(t) fbc(t)Ik(t,y)f(y)u(y)l dy. By Lemma 5.1 we obtain J< f+l Iv(t)G(t)g(t)l < -(1 + 6) f+ Iv(t)g(t)l dt , "k+l dt _< (1 + 6)[Jl + J + Jl + J], Jll :-- f ’-’D b() Ik(, t)u(t)f(t)l dt J12 :-- D J21 Jb Jb(_) rb(-) f+ Ig(t)v(t)l dt, Ik(_l, t)u(t)f(t)l dt (x-2) Ig(t)v(t)l dt, +l k(, b(-1)) fb(-l [u(t)f(t)ldt f . J b(-2) Ig(t)v(t)l dt, Applng the Holder inequality the/-condition we find IIx(x,x+lgllllx(x,x+lvllr) IIX(c,b)k(, .)ull IIx(x,)vll rllx(b(_,),b)fllxllX(x,x+,)gll r’ 1’ dd2Ailfi[xllgllr,. Analogously, we obtain J12 < dd2Abllxllgll. The estimate for J21 is similar to that for Jll on applng the owledge that k(x, y) is nondecreasing in x proceeding like for Jl1" we find that J2 <_ dd2Abllllgllr,. For J22 we write J.2 fc bl lu(t)f( t) dt k(x, b(xg_2 ))X,+,)Igldx _< Ilgtfll rllgll , Ktf X(,+) Theorem 3.2 we have that IIglt_y if o fbc’(Xl lu(t)f(t)ldt bl"= Ek<_Nb(Xg-2) -k<_Nk(xg, b(_2))X(,x/,). By C a<z< IIx(z,vlllix(,b(zull < z < o+l when bl (z) b(o-2) (t)X(z,)(t) < k =ko E k(, b(-2))X(x,x+,)(t) < k(t,b(o-2)). <_ Ab Therefore we have Thus J22 < C4bllfllxllgll,. Then we have that Ilgbil-y C(b + 40b). Necessity Letfu > 0 a.e., x < y b > a(y). Since k(x, y) is nondecreasing in x nonincreasing in y, for every x < z < y we have Proof of Theorem 3.1 gf(z) fb y(z)da(y) k(x, t)f( t)u( t) dt A. GOGATISHVILI J. LANG b Kf(z) > v(z)k(z, b) Therefore we get b a a(y) f(t)u(t) dt. y) k(x, t)llx(,ylll <_ Ilgfll r _< Ilgll-rllfll b() u(t)f(t)dtllvx(x,y)k(.,b)l[r Ilgfllr (y) Ilgll_rllfll for all f X such that fu >_ O. Then by duality we have Sufficiency We use the same technique as in the proof of Theorem as in 3.2. For a, bwe define {ci}ik=l, {di}ik=l, {}/k=l {My that proof. Then we have }/k=l E XM ( ib k(x, t)u(t)f(t)dt i=1 + i=1 k(x, t)u(t)f(t)dt) i=1 u(t)f(t)dtxM,. IICfll r < fb k(x, t)u(t)f(t)dtxMy, K jy, f.ay(gx) k(x, t) By the/-condition we obtain ellJ (f)Xy, I i=1 i=1 I1 +I2. By Theorem 5.2 we have IIgJ (f)xMy, I r therefore we obtain I1 <_ CAll Z eillx(,,a,lfllx]lz <- C411fllx. i=1 To estimate 12 we use the condition (1.2) for X --inf(My,) a < < b(xi), xi < x. Then k(x, t) < D[k(x, ) 4- k(xi, t)] we have XMy gf XMy k(x, t) <__ DXMy k(x, ) k(xi, t). 4- DXM, Theorem 3.2 elds XM,, k(x, ) AIIf<c,,dlllx XM, Therefore k(xi, t)u(t)f(t)dt () by the/-condition we obtain that 12 <_2cA i=1 A. GOGATISHVILI J. LANG Combining the estimates of I1 12 we arrive at Ilgfll Theorem 3.1 is proved. Afllfll. Remark When this paper was finished we learned (by oral communications) that this problem for Hardy operators in Lebesgue spaces was considered by Heinig Sinnamon [2]. Acowledgements The authors would like to express their gratitude to the Royal Society NATO for the possibility to visit the School of Mathematics at Cardiff during 1997/8, under their Postdoctoral Fellowship programme also to the School of Mathematics at Cardiff for its hospitality. A. Gogatishvili was also ported by grant No. 1.7 of the Georgian Academy of Sciences. J. Lang was also ported by grant No. 201/96/0431 of the Grant Agency of the Czech Republic. The authors also would like thank W.D. Evans J. Rkosnik for important hints helpful remarks. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Inequalities and Applications Hindawi Publishing Corporation

The generalized hardy operator with kernel and variable integral limits in banach function spaces

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aRazmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidze st., Tbilisi 380093, Georgia; b University of Missouri-Columbia, Department of Mathematics, 202 Mathematical Sciences Building, Columbia, MO 65211, USA (Received 18 December 1997; Revised 15 September 1998) Let we have an integral operator b k(x, y)u(y)f(y)dy for x > 0 a b are nondecreasing functions, u v are non-negative finite functions, k(x, y) >_ 0 is nondecreasing in x, nonincreasing in y k(x, z) < D[k(x, b( y)) + k( y, z)] for y <_ x a < z <_ b(y). We show that the integral operator K: X Y X Y are Banach functions spaces with/-condition is bounded if only if A < o0. A Ao + A1 A0 A1 Keywords: Banach function spaces; Hardy operator; Integral operator;/-condition 1991 Mathematics Subject Classification: Primary 46E30; Secondary 47B38 * Corresponding author. (On leave from the Mathematical Institute of the Academy of Science, Prague, Czech Republic) E-mail: langjan@aqua.math.missouri.edu. A. GOGATISHVILI J. LANG INTRODUCTION Let X Ybe two Banach function spaces on (c, d) respectively. We define the general Hardy operator [b k(x, y)u(y)f(y) dy for x (1.1) a,b are nondecreasing functions on -<c<a< v u are non-negative measurable finite b < d< functions a.e. on/R (c, d). The kernel k(x, y) > 0 is defined a.e. on {(x, y); x E IR, a < y < b} satisfies the following conditions" (i) it is nondecreasing in x nonincreasing in y; (ii) k(x, z) < D[k(x, b(y))+k(y, z)] for every y < x a < z < b(y), the constant D > is independent of x,y,z. (1.2) In this paper we describe the necessary sufficient condition for the boundedness of the operator (1.1) in Banach function spaces. This paper extends results of Lomakina Stepanov [3] Opic Kufner [4]. In these papers the operator (1.1) was characterized for a 0 b x. Sections 2 3 contain the definitions, formulations of the main results some comments. In Section 4 we treat the simpler case when the kernel k(x,y) is equal to the spaces X, Y satisfy the /-condition. We use this result in Section 5 to deal with the general kernel satisfng (1.2). DEFINITIONS In this section we recall the definition some basic properties of the Banach function spaces. In what follows A4(f) will be the set of all measurable functions on f, f is any measurable subset of. DEFINITION 2.1 A normed linear space (X, II.llx)onfiscalledaBanach function space (BFS) on f if the following conditions are satisfied: (2.1) the norm Ilfllx is defined for allfE M(f) fe Xif only if Ilfllx<; (2.2) Ilfll= Ifl I1 for everyfE l(f); (2.3) if 0 <fn/fa.e. in f then Itf=ll/llfllx; (2.4) if IEI < o, E C f, then XE E x; (2.5) for every set E, IEI < c, E C f, there exists a positive constant CE such that flJldx <_ Cllfll. By we denote a Banach sequence space (BSS), which means that the axioms (2.1)-(2.5) are fulfilled with respect to the counting measure {ek} denotes the stard basis in l. Recall that the condition (2.3) immediately elds the following property: (2.6) if 0 <f< g then [[fllx < Ilgllx. DEFINITION 2.2 The set X’={f’,fafgv < equipped with the norm for every g E X}, "f"e ={ fafgv ;,,g[[x < 1}, is called the associate space of X. It is own from Bennett Sharpley [1] that X" X that X’ is again a BFS. Let T be a linear operator from a BFS X into a BFS Y. Then T’ is an associate operator to the operator Tif fn(Tf)g faf(T’g) for allf X g Y. LEMMA 2.3 (Bennett Sharpley [1]) Let T be a linear operator from a BFS X into a BFS Y. Then TTII <_ CII f xfor allf X with afinite positive constant C, if only/f T’gllx’ _< Cllgll for all g Y’. Moreover I Tllx r I Z’ll r-. x,. DEFINITION 2.4 (Lomakina Stepanov [3]) Given a BFS X a BSS l, X is/-concave, if for any sequence of disjoint intervals (Jk} such that t_J Jk f, for allfE x A. GOGATISHVILI J. LANG is dl a finite positive constant independent on fE X {Jk}. Analogously, a BFS Y is said to be/-convex, if for any sequence of disjoint intervals {Ik}, t3 Ik f for all g Y Ilgllr d21[ekllxzgllrk with a finite positive constant d2 independent on g E Y {Ik). We say, that BFS X, Ysatisfy the/-condition, if there exist a BSS 1 such that X is/-concave Y is/-convex simultaneously. LEMMA 2.5 (Lomakina Stepanov [3]) Let Y be a l-convex BFS. Then Y’ is an l’-concave BFS for allf Y’ (Ik}, tO Ik Ft. MAIN RESULTS Assume X Y are two BFS on (c, d) denote respectively. Then we Ao A :: x<y,a(y) 0 satisfng (1.2). Then K: X-- Y is bounded, if only if, A is finite. Moreover To prove Theorem 3.1 we need a corresponding result for the general Hardy operator with kernel k(x, y) 1. Let b Hf u(y)J(y) dy (3.1) -o < c < a < b < d < o are nondecreasing functions on v u are real measurable finite functions a.e. on I (a,/3), respectively. THEOREM 3.2 Let X Y be two BFS on (c, d) 1t, respectively, satisfng the l-condition, let H be the operator defined by (3.1). Then H" X-. Y is bounded, if only if, A. x<y,a(y)<_b BOUNDEDNESS OF THE OPERATOR H In this section we prove Theorem 3.2. At first we prove a lemma. DEFINITION 4.1 Let v be a non-negative measurable function on an Let c E let interval (a, fl) o < a </3 < < a < c < < be nondecreasing functions, let u be a non-negative meab surable function on (e,d) eliminfx_a d lim x b. Then we define b . , Hbf for every measurable functionfon (c, d), Haf for every measurable function fon (e, c). A. GOGATISHVILI J. LANG LEMMA 4.2 Let 3( Y be two BFS on (e, d) (a,/3), respectively, satisfng the l-condition. Then Hb X-+ Y is bounded, if only if, Ab Ilvx(x,llrllux(c,b(xllll, < c<x</ Moreover Ilnbllx-+r Ab. Also Ha:X--> Y is bounded, if only if, ha Moreover :-’- a<x<fl 11VX(x,)II yl]ux(a,c)I1, < Proof We will give the proof only for Hb. The proof for Ha is similar. Necessity Given x E (a, fl) fE X such thatfu > 0, we have b(.) v(.) ,+c >_ V(.)X(x,Z)(.) b( x) b >_ v(.)X(x,Z)(.) VX(x,l r lu(t)f(t)l dt. Taking the remum over all suchf x E (a, fl) we obtain Ilnbllx+ y IlvX(x,lllrlluxc,b Ab. Sufficiency If Ab--Cxz then )lHllx-+ r Ab. If A b- 0 then II/-/,,11,,- ,-= o. Let O<Ab< . Choose fEX such that [[fllx 1. Define C/= {t; E (Ce,/), fbc(t) [fu[ 2i}, O Ci\ C + Ei {x; x (c, d), b C}, Bg E\E+ 1. Then I(c, d)\ t.J z Bil 0 [(a,/3)\ I..J EZ De[--0 we have gHbf < iz 2i+lgv <_ iZ, IDI>O 2e+lllvx,llrllgx,llr, iz, IDil>0 _< _< ’ 2 g/allvxc,llllgX,l[ 2i+1 hb [fXBi-1 Ilul (using 2 i-1 _< <- IIfi-, IIllux(c,up,_,/ll,) iZ, IDil>0 Ab2i+l (using Holders inequality /-condition) <_4Ab ieZ, lDl>0 eillfx(.i_,)[[x iez, [ai[>0 eillgxo, l[y, l’ <_ 4dld2Ab[[f[[x[[g[[r. Then we have lig[Ir,<l glib f<_ 4dld2Abllf[[x. Now we prove Theorem 3.2. Proof of Theorem 3.2 Necessity Let f X be such that fu >_ 0 1, let x, y be such that a _< x _< y </3 b >_ a(y). Then Ilfllx- I[Hf[[r >_ [Iv(.)X(x,y)(.) (.) l[ r VX(x,l fb( u( t)f( t) dr. Ja(y) A. GOGATISHVILI J. LANG Taking the remum over all such x, y fwe obtain IIHII _> IlVX(x,y)II Sufficiency Define M {(x, t); x a < < b}. M is a measurable set. If IMI 0 then it is easy to see that IIHIIx_ y O. pose that IMI > O. We set My {x; (x, t) M, y}, y ]R, P {y; (x, t) M, IMyl > 0}. Then P tAim=lPi, Pi are intervals, [Pi[ > O, rn < Let yoint Pi; then we have a set My co =a(infMyo ), do b( Myo). pose we have defined , ci, di My,. If i>O diEint Pi then we define +l=di, di+l ci-1 i+1 ----a(infM+l), b(M+l). If i<O ciEintPi then we define -1 =ci, a(inf My_), di-1 b( M_). ni ni n ( c)}j=m, {dtj}j=mi, {My "})=mi’ By using this method we can construct, for every P sequences f i’l ni tYj]j=mi’ We can rewrite all these sequences in the following way: {Ci)gL, {di}i= {My,}g= k Zi%l(ni Then we have {}ik=l, mi + 1). a.eo nf Z XM i=1 ? + X) jfgHf Z L gHf i=1 i=I :Mr(IJ fb f(t)u(t) dt + v(x )fay f(t)u(t) d g(x dx tl ) i=1 f(t)X(,di)U(t dt g dx () f(t)X(ci,y,)U(t) dt g d (Using Lemma 4.2 Aa + Ab <_ AH) gHf <_ 4d d2 i=1 Al-II]fX(e,,y,)Ilxllgxt, i=1 r, + 4dl d2 8dd2An Z AHIIIxy,,d,I IlxllgXM, IIfX(c,,d,llxllgx,ll i=1 (use H61der’s inequality /-condition) BOUNDEDNESS OF THE OPERATOR K In this section we prove Theorem 3.1. LEMMA 5.1 Let b be a nondecreasing right continuous function on (a, fl) let b(cO c, b(/) d. Let ko(x, y) > 0 be a kernel satisfng(1.2), ko(x, y) > 0 on set of positive measure. pose that ko(x, y) is right continuous with respect to x for all x E [a, ] for a.e. y (c, b). Let u, f be measurable functions on (c, d), fu > O, b Go go(x, y)u(y)f(y) dy. For a fixed number 6 > D ( D is a constant from(1.2)), we define A k {x G (a,/); Go >_ (t5 + 1)}, k Z, N {k; Ak ). Then there exist sequences {}, {"/k) such that a < the inequality < - (6 + 1) "- < b() ko(x, y)u(y)f(y) dy ,]b(-l) dy .16(_2) + Dko(, b(-1)) ab(x_)u(y)f(y) dy + Oko(, b(-2)) u(y)f(y) dy. holds for all k <_ N, Go <_ (1 + 6) "’-’+1 when x [- ). A. GOGATISHVILI J. LANG Proof By the Lebesgue Dominated Convergence Theorem Go is a nondecreasing right continuous function for all a < x </ limx Go =0. Set ak inf Ak, for k < N. Fix E Z such that IzXel > 0. We set x0 ai, 7o max{i; ai xo), ak "Yk max{ i; ai a,k_+ for k > 0 "Yk max { i; ai a,k+ } for k<0. It is obvious that {7k} is an increasing sequence of integers, therefore 7k, < k- 1, G()- G(a.,) > (1 + 6) "k. If x [, + 1), then we have a+ +l, therefore x < a+l G < (1 + 6) + 1. Next on using (1.2) we find that As DGo( 2) D(1 + 6)"k-2 + < D(1 + 6)"k t(1 + 6)k "k *- (1 + 6)e*- the lemma follows. 6(1 + 6) (1 + 6) THEOREM 5.2 Let X Y be two BFS on (c, d) (c, ), respectively, ( b() d b(a) c) satisfng the l-condition b Kbf k(x, y)f(y)u(y) dy, k(x, y) satisfies (1.2). Then A A Proof "= a<z<g II<z,)vllrllX<c,bz)l(.)k(z,.)u(.)ll,, IIX(z,(.)v(.)k(.,b(z))llllX(c,b(zlull,. "= a<z a. Then b> c. Since k(x,y) is nondecreasing in x nonincreasing in y, for every a < x < z < fl we have b >_ v(z) k(x, t)f(t)u(t) dt b Kf(z) > v(z)k(z,b) Hence, . IIKfllf >- IIX(x,g)(.)v(.) for allf X a < x < b k(x, t)f(t)u(t) dtl] r >_ IIx(x,l vll Yllx(c,b (.)k(x, .)u(.) IIx, IlfX(c,bl IIx b IIKfll r _> IIX(x,l(.)v(.)k(.,b) ll, _> IIx(x, (.)v(.)k(., b)II rllx(,b(xlul[ IIX(c,b(xll fllx for allfE X a < x < g. Sufficiency Let D be the constant from condition (1.2) let 6 > D be fixed. Without loss of generality we may assume that k(x, y) b satisfy the assumptions of Lemma 5.1. Otherwise we replace k(x, y) by k(x +, y) b by b(x +). A. GOGATISHVILI J. LANG By the principle of duality it is sufficient to show that v(t)G(t)g(t) dt < Allxllgllr,, for all f E X g E Y’, G(t) fbc(t)Ik(t,y)f(y)u(y)l dy. By Lemma 5.1 we obtain J< f+l Iv(t)G(t)g(t)l < -(1 + 6) f+ Iv(t)g(t)l dt , "k+l dt _< (1 + 6)[Jl + J + Jl + J], Jll :-- f ’-’D b() Ik(, t)u(t)f(t)l dt J12 :-- D J21 Jb Jb(_) rb(-) f+ Ig(t)v(t)l dt, Ik(_l, t)u(t)f(t)l dt (x-2) Ig(t)v(t)l dt, +l k(, b(-1)) fb(-l [u(t)f(t)ldt f . J b(-2) Ig(t)v(t)l dt, Applng the Holder inequality the/-condition we find IIx(x,x+lgllllx(x,x+lvllr) IIX(c,b)k(, .)ull IIx(x,)vll rllx(b(_,),b)fllxllX(x,x+,)gll r’ 1’ dd2Ailfi[xllgllr,. Analogously, we obtain J12 < dd2Abllxllgll. The estimate for J21 is similar to that for Jll on applng the owledge that k(x, y) is nondecreasing in x proceeding like for Jl1" we find that J2 <_ dd2Abllllgllr,. For J22 we write J.2 fc bl lu(t)f( t) dt k(x, b(xg_2 ))X,+,)Igldx _< Ilgtfll rllgll , Ktf X(,+) Theorem 3.2 we have that IIglt_y if o fbc’(Xl lu(t)f(t)ldt bl"= Ek<_Nb(Xg-2) -k<_Nk(xg, b(_2))X(,x/,). By C a<z< IIx(z,vlllix(,b(zull < z < o+l when bl (z) b(o-2) (t)X(z,)(t) < k =ko E k(, b(-2))X(x,x+,)(t) < k(t,b(o-2)). <_ Ab Therefore we have Thus J22 < C4bllfllxllgll,. Then we have that Ilgbil-y C(b + 40b). Necessity Letfu > 0 a.e., x < y b > a(y). Since k(x, y) is nondecreasing in x nonincreasing in y, for every x < z < y we have Proof of Theorem 3.1 gf(z) fb y(z)da(y) k(x, t)f( t)u( t) dt A. GOGATISHVILI J. LANG b Kf(z) > v(z)k(z, b) Therefore we get b a a(y) f(t)u(t) dt. y) k(x, t)llx(,ylll <_ Ilgfll r _< Ilgll-rllfll b() u(t)f(t)dtllvx(x,y)k(.,b)l[r Ilgfllr (y) Ilgll_rllfll for all f X such that fu >_ O. Then by duality we have Sufficiency We use the same technique as in the proof of Theorem as in 3.2. For a, bwe define {ci}ik=l, {di}ik=l, {}/k=l {My that proof. Then we have }/k=l E XM ( ib k(x, t)u(t)f(t)dt i=1 + i=1 k(x, t)u(t)f(t)dt) i=1 u(t)f(t)dtxM,. IICfll r < fb k(x, t)u(t)f(t)dtxMy, K jy, f.ay(gx) k(x, t) By the/-condition we obtain ellJ (f)Xy, I i=1 i=1 I1 +I2. By Theorem 5.2 we have IIgJ (f)xMy, I r therefore we obtain I1 <_ CAll Z eillx(,,a,lfllx]lz <- C411fllx. i=1 To estimate 12 we use the condition (1.2) for X --inf(My,) a < < b(xi), xi < x. Then k(x, t) < D[k(x, ) 4- k(xi, t)] we have XMy gf XMy k(x, t) <__ DXMy k(x, ) k(xi, t). 4- DXM, Theorem 3.2 elds XM,, k(x, ) AIIf<c,,dlllx XM, Therefore k(xi, t)u(t)f(t)dt () by the/-condition we obtain that 12 <_2cA i=1 A. GOGATISHVILI J. LANG Combining the estimates of I1 12 we arrive at Ilgfll Theorem 3.1 is proved. Afllfll. Remark When this paper was finished we learned (by oral communications) that this problem for Hardy operators in Lebesgue spaces was considered by Heinig Sinnamon [2]. Acowledgements The authors would like to express their gratitude to the Royal Society NATO for the possibility to visit the School of Mathematics at Cardiff during 1997/8, under their Postdoctoral Fellowship programme also to the School of Mathematics at Cardiff for its hospitality. A. Gogatishvili was also ported by grant No. 1.7 of the Georgian Academy of Sciences. J. Lang was also ported by grant No. 201/96/0431 of the Grant Agency of the Czech Republic. The authors also would like thank W.D. Evans J. Rkosnik for important hints helpful remarks.

Journal

Journal of Inequalities and ApplicationsHindawi Publishing Corporation

Published: Jan 1, 1900

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