REVERSE HÖLDER INEQUALITIES AND INTERPOLATION

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1 REVERSE HÖLDER INEQUALITIES AND INTERPOLATION J. BASTERO, M. MILMAN, AND F. J. RUIZ Abtract. We preent new method to derive end point verion of Gehring Lemma uing interpolation theory. We connect revere Hölder inequalitie with Maurey-Piier extrapolation and extrapolation theory. 1. Introduction It ha been known for a long time that there exit a trong connection between the theory of weighted norm inequalitie for claical operator and interpolation theory. However, one feel that there are till many baic quetion that remain open. In recent work we have been exploring the interaction between weighted norm inequalitie and interpolation theory (cf. [1], [2], [3], [16]). In our work we have found that idea and method from one field often lead to new idea and reult in the other. In particular in [1] we have hown a verion of the extrapolation theorem of Rubio de Francia (an important reult from the theory of weighted norm inequalitie cf. [1]) in the context of the real method of interpolation, while at ame time oberving that the extrapolation method of [15] yield extrapolation theorem in the theory of weighted norm inequalitie. Moreover, thee development led to the tudy of certain clae of weight in connection with interpolation theory and function pace (cf. [1], [2]). In [3] and [1] we develop new technique to prove revere type Hölder inequalitie for certain type 1991 Mathematic Subject Claification. Primary 46M35, 42B25. Key word and phrae. Revere Hölder, Interpolation, Extrapolation. The firt author wa partially upported by DGICYT PB , the econd author participation at the workhop wa partially upported by Intitute of Mathematic, Technion, the third author wa partially upported by DGICYT and UR. Thi paper i in final form and no verion of it will be ubmitted for publication elewhere. 1

2 2 J. BASTERO, M. MILMAN, AND F. J. RUIZ of weight and in [16] we how an extenion of Gehring Lemma via Holmtedt formula and differential inequalitie. In thi paper we continue to develop thee connection. Firt we conider ome limiting cae of revere Hölder inequalitie which are important in PDE and have been recently tudied by R. Fefferman [7] and R. Fefferman, C. Kenig and J. Pipher [8]. More preciely we give new approache to work by R. Fefferman [7] on a limiting cae of Gehring Lemma. Our method i baed on the idea of uing maximal function and rearrangement inequalitie to reformulate the problem a an invere reiteration theorem (cf. [16]). Once the problem ha been reformulated in thi fahion we can alo prove Fefferman reult through the ue the iteration method developed in [3] and [1]. Indeed, uing thi method we obtain a omewhat harper reult in a much a we get a more precie etimate of the improvement obtained in the Gehring type Lemma. (We note, however, that neither of our method can be expected to give harp etimate on the improvement in the index of integrability due to the contant that we accumulate to reformulate the problem.) A econd problem we treat in thi paper i motivated both by the application of weighted norm inequalitie to PDE and ome problem and method from operator theory. In the application to PDE it i important to conider revere Hölder inequalitie where the cube involved in the etimate maybe dilation one of the other. We how that if we reinterpret thee condition in term of probability meaure we are in a ituation that i alo conidered in functional analyi. In fact in thi fahion we connect revere Hölder condition, the extrapolation method of Maurey-Piier [19] and extrapolation theory in the ene of [15], [17]. Indeed we how that the Maurey-Piier method can be incorporated to the general theory of extrapolation of [15] through the introduction of a uitable extrapolation functor. Converely, the method alo applie in the realm of PDE and cube (cf. [13]). It ha not been our purpoe here to prove the mot general reult but to illutrate the new method ariing from interaction between interpolation theory and the theory of weighted norm inequalitie. We hall conider other application and interaction elewhere. Acknowledgement: We would like to thank the organizer, and in particular M. Cwikel and hi family, for their cordial hopitality. We are alo very grateful to the referee for hi/her very careful reading of the manucript and for many ueful uggetion to improve the preentation.

3 REVERSE HÖLDER INEQUALITIES 3 2. Revere Hölder Inequalitie and Reiteration Let u tart recalling ome well known fact from the theory of weighted norm inequalitie. A a general reference we ue [1]. Let Q be a fixed cube with ide parallel to the coordinate axe and let w be a poitive meaurable function defined on Q. We hall ay that w atifie a Gehring condition (or a revere Hölder inequality) if there exit p > 1, and a contant c >, uch that for every cube Q Q, with ide parallel to the coordinate axe, we have (2.1) { 1 } 1/p w p (x)dx c 1 w(x)dx. In thi cae we hall write w RH p. A well known reult obtained by Gehring [11] tate that if w RH p then w atifie a better integrability condition, namely for ufficiently mall ε >, and q = p + ε, we have for every cube Q Q, { 1/q { 1/p 1 1 w (x)dx} q c w (x)dx} p. In other word, Gehring Lemma tate that w RH p = ε >.t. w RH p+ε. It i not difficult to extend Gehring Lemma by mean of replacing dx by a meaure of the form dµ(x) = h(x)dx, a long a thi meaure atifie a doubling condition. In uch cae we hould, of coure, alo conider average with repect to thi meaure, and replace Q by µ(q). Let u denote the correponding clae RH p (dµ). Let u alo recall the definition of the Muckenhoupt A p clae, which control the weighted L p norm inequalitie for the maximal operator of Hardy and Littlewood, and play a central role in the theory of weight. For 1 < p <, we ay that w A p if we have up Q ( ) ( p w(x)dx w(x) dx) 1/(p 1) <. Note that p = p/(p 1), 1 p = 1/(p 1), therefore we ee that w A p implie that for all Q Q, ( ) p 1 Q p w(x)dx w(x) 1 p dx c. Q Q

4 4 J. BASTERO, M. MILMAN, AND F. J. RUIZ Conequently, if we write Q = Q w(x)w(x) 1 dx, dµ = w(x)dx, we get ( ) ) 1/p w(x) 1 p dµ µ(q) Q 1 c µ(q) Q w 1 (x)dµ. We have hown that: w A p w 1 RH p (dµ). Therefore, by the weighted verion of Gehring Lemma, we can find ε > uch that w 1 RH p +ε(dµ), and tranlating back in term of A p condition we readily get w A p = ε >.t. w A p ε. Thi lat property play a central role in the theory of A p weight. Let u alo define the A condition by A = p>1 A p. There i a detailed tudy of A and it relationhip to RH p condition in the literature. In particular it i known that (cf. [6]) A = p>1 RH p. The uual proof of Gehring Lemma involve the ue of Calderón- Zygmund decompoition and the cale tructure of L p pace. However, only recently we oberved in [16] an explicit connection of Gehring Lemma to interpolation theory. More preciely, it wa hown in [16] that Gehring Lemma can be interpreted a an invere type of reiteration theorem valid in the general context of real interpolation pace. In particular a new proof of Gehring Lemma wa then derived via Holmtedt formula! Here i the tatement of the reult Theorem 2.1. Let (A, A 1 ) be an ordered pair of Banach pace (i.e. A 1 A ) and uppoe that f A i uch that for ome contant c > 1, θ (, 1), 1 p <, we have for all t (, 1), (2.2) K(t, f; A θ,p;k, A 1 ) ct K(t1/(1 θ ), f; A, A 1 ) t 1/(1 θ ). Then, there exit θ 1 > θ, uch that for q p, < t < 1, we have (2.3) K(t, f; A θ1,q;k, A 1 ) t K(t1/(1 θ 1), f; A, A 1 ) t 1/(1 θ 1).

5 REVERSE HÖLDER INEQUALITIES 5 In order to formulate Gehring Lemma in thi fahion we oberve that if define the local maximal operator of Hardy-Littlewood by M q w(x) = up Q Q,x Q 1/q w(u) du) q, where q [1, ), then w RH q implie the exitence of c > uch that (2.4) M q w cmw. Taking rearrangement in (2.4), and uing the well known fact (cf. [14], [4]) that (2.5) (Mw) (t) 1 t t w ()d, < t < Q, we ee that (2.4) implie the following rearrangement inequality t t ) 1/q w () q d c t w ()d, < t < Q. t Next, a well known formula for the K functional for the pair (L p, L ), p (, ), (cf. [5]) can be ued to how that the previou inequality take the form K(t, w; A θ,q;k, A 1 ) ct K(t1/(1 θ ), w; A, A 1 ) t 1/(1 θ ), with A = L 1, A 1 = L, θ = 1 1/q. Gehring Lemma can be readily derived from thi etimate, in fact we hall conider in detail the mechanim involved in the proof in the next ection where, moreover, we conider an end point verion of thee reult. The proof of the theorem i given in [16] and i baed on Holmtedt formula and ome elementary differential inequalitie aociated with it. 3. An End Point Verion of Gehring Lemma In thi ection we conider a limiting cae of Gehring Lemma due to R. Fefferman which play an intereting role in ome problem in PDE (cf. [7] and [8] for more on thi). Our approach i entirely different and baed on interpolation theory. In fact our final reult will a limiting verion of Theorem 2.1. We would like to let q 1 in the aumption of Gehring Lemma: thi lead u to define the clae RH LLogL a follow. We hall ay that

6 6 J. BASTERO, M. MILMAN, AND F. J. RUIZ w RH LLogL if there exit a contant c >, uch that for every cube Q Q, with ide parallel to the coordinate axe, we have w L(LogL)(Q, dx Q ) c 1 (3.1) w(x)dx, where w L(LogL)(Q, dx Q ) = inf{λ > : 1 Q It follow readily that Q w(x) λ A = RH p RH LLogL. We now how (cf. [7]) that we actually have A = RH LLogL. (1 + w(x) log+ )dx 1}. λ At thi point we need to introduce a maximal function aociated with L(LogL). Define, M L(LogL) w(x) = up w L(LogL)(Q, dx Q Q,x Q Q ). It i known, and not difficult to ee (cf. [2]), that (3.2) M L(LogL) w M(Mw). Therefore, if w RH LLogL, we ee that (3.3) M(Mw)(x) cmw(x). We hall how that (3.3) implie the exitence of q > 1 uch that ) 1/q w q (x)dx c 1 (3.4) w(x)dx. Q Q Q Q Since thi argument can be applied to any fixed ubcube Q Q imply by conidering localized maximal function with repect to Q, we ee that w RH L(LogL) w p>1 RH p. We tart by rearranging the inequality (3.3), then, uing (2.5), we get (M(Mw)) (t) c (Mw) (t) 1 t (Mw) ()d c (Mw) (t) t (3.5) P (2) w (t) cp w (t),

7 REVERSE HÖLDER INEQUALITIES 7 where P f(t) = 1 t t f()d, and P (2) f(t) = P (P f)(t) = 1 t t f() log t d. Now if we take into account the fact that K(, f, L 1, L )/ = P f (), we can alo rewrite the previou etimate a t (3.6) K(, w; L 1, L ) d ck(t, w; L1, L ). Oberve that (3.6) implie that there exit γ (, 1) uch that d t dt log( K(, w; L 1, L ) d ) = K(t, w; L1, L )/t t K(, w; L1, L ) d Therefore, if x < y, we ee, integrating from x to y, that ( y log K(, w; L1, L ) d ) x K(, w; L1, L ) d log( y x )γ, which lead to ( y K(, w; L1, L ) d ) x K(, w; L1, L ) d ( y x )γ. Conequently we obtain y y γ K(, w; L 1, L ) d x x γ Note that by aumption γ t. K(, w; L 1, L ) d. y y γ K(, w; L 1, L ) d cy γ K(y, w; L 1, L ), moreover, ince K(, f; L 1, L ))/ decreae, we have x x γ K(, w; L 1, L ) d x γ K(x, w; L 1, L ). Thu, the function x x γ K(x, w; L 1, L ) i eentially increaing. We now claim that for ome θ (, 1) we have t (3.7) θ K(, w; L 1, L ) d ct θ K(t, w; L 1, L ). Auming the validity of our claim and combining (3.7) with the following form of Holmtedt formula t 1/(1 θ) (3.8) K(t, f; Āθ,1;K, A 1 ) θ K(, f; A, A 1 ) d, we ee that the aumption of Theorem 2.1 are verified, and conequently we deduce the exitence θ 1 (, 1) uch that for all q 1 we have

8 8 J. BASTERO, M. MILMAN, AND F. J. RUIZ K(t, w; (L 1, L ) θ1,q;k, L ) t K(t1/(1 θ 1), w; L 1, L ) t 1/(1 θ 1). Selecting 1/q = 1 θ 1, and tranlating back, we obtain (3.9) t t ) 1/q w () q d c 1 t w ()d, < t < Q. t The inequality (3.9) applied to t = Q give ) 1/q w(x) q Q dx = w () q d Q Q Q c Q w ()d Q = c w(x)dx Q Q ) 1/q a deired. It remain to etablih (3.7). We imply pick θ (, γ), then uing the fact that x γ K(x, w; L 1, L ) i eentially increaing we find that t θ K(, w; L 1, L ) d t = θ+γ γ K(, w; L 1, L ) d t ct γ K(t, w; L 1, L θ+γ d ) = ct γ K(t, w; L 1, L )t γ θ = ct θ K(t, w; L 1, L ), a we wihed to how. The only part of the argument where we ued pecific information about the pair (L 1, L ) wa in order to tranlate (back and forward) the original problem. We have thu etablihed an end point verion of Theorem 2.1 which correpond to θ =. Theorem 3.1. Let (A, A 1 ) be an ordered pair of Banach pace (i.e. A 1 A ) and uppoe that f A i uch that for ome contant c > 1 we have t (, 1), t K(, f; A, A 1 ) d ck(t, f; A, A 1 ).

9 REVERSE HÖLDER INEQUALITIES 9 Then there exit θ (, 1), uch that for q 1, < t < 1, we have K(t, f; A θ,q;k, A 1 ) t K(t1/(1 θ), f; A, A 1 ) t 1/(1 θ). It i intereting to note, in comparing the aumption in the previou theorem with the one in theorem 2.1, that the right formulation wa obtained by replacing the hypothei of theorem 2.1 uing Holmtedt formula K(t, f; Āθ,1;K, A 1 ) t 1/(1 θ ) θ K(, f; A, A 1 ) d, and then letting formally θ to derive the new hypothei. We note that in [12] there i alo an extenion of Holmtedt formula for the cae when θ = which in our context i related to a different way of defining maximal function uing Orlicz function. For more on the relationhip between reiteration formulae for extrapolation pace and generalized Revered Hölder inequalitie ee [18]. 4. The Iteration Method In thi ection we rederive the end point verion of Gehring Lemma dicued in the previou ection uing a completely different method. The method we employ here wa developed by the author to attack ome problem connected with the application of weighted norm inequalitie in interpolation theory in [3] and [1]. It i baed on the iteration of inequalitie and it probably ha it root in the early work of Gagliardo [9]. Let u then aume again that w i uch that for ome contant c we have M(Mw) cmw. Rearranging, we may take a our tarting point the etimate (cf. (3.5) above) P (2) w (t) cp w (t). Now we iterate applying repeatedly the operator P to both ide of the inequality and get P (n) w (t) c n 1 P w (t), n = 2, 3,... Pick ε > uch that εc < 1 and multiply the correponding n th inequality by ε n 1, n = 2, 3,... Summing we get (4.1) ( ) ε n P (n) w (t) ε n c n P w (t). n>1

10 1 J. BASTERO, M. MILMAN, AND F. J. RUIZ Since we ee that P (n) w (t) = n>1 ε n P (n) w (t) = n>1 = 1 t t 1 1 t ( w () log t n 1 d, (n 1)! t ) w ()ε n>1 = ε t t ε n 1 1 t ( w () log t n 1 d (n 1)! t ) ( ( ) 1 t ε ) n 1 log d (n 1)! ( ) t ε w ()[ 1]d. Inerting thi back in (4.1) we obtain, for a uitable contant C, 1 t ( ) t ε w () d CP w (t). t Combining the lat etimate with the elementary inequality (cf. [5]) t t we obtain, with q = 1/(1 ε), ) 1 ε t w () 1/(1 ε) d t 1 ε w () ε d, t t w () q d) 1/q cp w (t). The argument we gave in the previou ection (right at the point where (3.9) wa etablihed) can be now ued to obtain the deired reult: ) 1/q w(x) q dx c w(x)dx. Q Q Q Q We note that the argument jut preented i contained in [1] in connection with revered Hölder type condition for weight in the M 1 cla, which are preciely the weight that atify the condition (3.5). 5. Gehring lemma and Maurey-Piier Extrapolation It i alo natural to ak what happen if in (2.1) we fix p > 1 and conider the improvement to thi inequality that would reult from lowering the exponent on the right hand ide. Namely, we conider if

11 REVERSE HÖLDER INEQUALITIES 11 the validity of (2.1) implie that for ome r < 1, the following inequality hold { 1 1/p { 1 1/r (5.1) w (x)dx} p c w (x)dx} r. It turn out that thi i true, and well known, and in fact the proof i an immediate conequence of Hölder inequality. Lemma 5.1. Suppoe that w atifie the condition (2.1) then r (, 1), w atifie (5.1). Proof. Given r (, 1), chooe θ (, 1) uch that 1 = 1 θ p + θ r. Then, by Hölder inequality, we have { 1 } 1/p w p (x)dx c 1 Q c Q w(x)dx { 1 } (1 θ)/p { 1 θ/r w(x) p dx w(x) dx} r. Therefore, dividing, we find { 1 θ/p { 1 θ/r w (x)dx} p c (w(x)) dx} r, and the reult follow. Condition of the form (2.1) and (5.1) appear in other context in analyi. In thee application it i important to extrapolate, a indicated in Lemma 5.1, even though the cube appearing on both ide of the hypotheized inequalitie may be different. Thi i, for example, the cae that appear commonly in problem in PDE where cube on each ide of (2.1) are dilation one of the other. For a treatment of problem in thi direction we refer to [13]. However, to connect thi extrapolation proce with other problem in functional analyi it i important to reformulate the problem in a more general context. In fact in the context of (2.1) one could view the condition ariing from PDE application and conider condition of the form: λ probability meaure in there exit a probability meaure µ uch that { } 1/p (5.2) w(x) p dλ c w(x) dµ.

12 12 J. BASTERO, M. MILMAN, AND F. J. RUIZ The argument in the previou Lemma would work if we had the ame meaure on both ide of the inequality. In order to arrange to have uch a ituation we ue a method developed by Maurey and Piier which eentially allow one to replace (5.2) with uitable norm o that the method of Lemma 5.1 can be applied. The proce involved i via iteration and could be conidered a a fixed point theorem for functional pace. Let u then conider the Maurey-Piier method lightly rephraed o that it can be incorporated a a more general argument in the extrapolation theory of Jawerth-Milman. The appropriate extrapolation functor here i the 1 functor. Recall that if {X γ } γ I i a compatible family of Banach pace we formally let ( ) 1 {Xγ } γ I = {f : f Xγ < }, with γ I f 1({X γ } γ I ) = γ I f Xγ. Here i a typical example of uch a contruction. Let {(X j, X j 1)} j=1 be a family of Banach pair, and let u conider the correponding interpolation pace X j θ = F θ(x j, X j 1), where {F θ } θ (,1) i a family of interpolation functor. Let X(θ) = 1 ({2 n Xθ n } ), θ [, 1]. A natural aumption on the family of functor {F θ } θ (,1) i that for θ (, 1), we have (5.3) f Xθ f 1 θ X f θ X 1, or in other word that the functor are exact of type θ. Then (5.3) perit at the level of the extrapolation pace: f X(θ) = 2 n f Xθ 2 n(1 θ) f 1 θ X 2 nθ f θ n X1 n ( ) 1 θ ( ) 2 n θ f X n 2 n f X n 1 = f 1 θ X() f θ X(1). Thi contruction allow u to conider the argument of Maurey- Piier from an interpolation-extrapolation view point. In fact note

13 REVERSE HÖLDER INEQUALITIES 13 that f 1({2 n X n θ } n=2 ) 2 f 1({2 n X n θ } ). Therefore if we tart with the aumption (5.2) and are given a probability meaure λ we contruct a equence of probability meaure {λ n }, with λ 1 = λ, o that { } 1/p w(x) p dλ n c w(x) dλ n+1, n = 1, 2,... Applying the 1 functor we get w 1 ({2 n L p (dλ n )} ) c w 1 ({2 n L 1 (dλ n )} n=2 ) 2c w 1 ({2 n L 1 (dλ n )} ). At thi tage the argument of Lemma 5.1 can be applied verbatim in view of our previou dicuion. Indeed, for any r < 1, let u write 1 = 1 θ + θ, with θ (, 1), then p r w 1 ({2 n L 1 (dλ n)} ) ( w 1 ({2 n L p (dλ n)} ) ) 1 θ ( w 1 ({2 n L r (dλ n)} ) ) θ, and thu w 1 ({2 n L p (dλ n )} ) C w 1 ({2 n L r (dλ n )} ). We can rephrae thi etimate in term of the original aumption if we oberve that and { w(x) p dλ} 1/p 2 w 1 ({2 n L p (dλ n )} ), w 1 ({2 n L r (dλ n )} ) = { 2 n w(x) r dλ n } 1/r { } 1/r 2 nr w(x) r dλ n ( { = 2 nr)1/r w(x) r 2 nr } 1/r dλ n 2 nr ( 1/r = c w(x) dµ) r,

14 14 J. BASTERO, M. MILMAN, AND F. J. RUIZ where µ = 2 nr dλ n 2 nr obtained the following i a probability meaure on. We have thu Theorem 5.2. (Maurey-Piier extrapolation) Suppoe w i uch that for each probability meaure λ on there exit a probability meaure µ uch that (5.2) hold for a univeral contant c >. Then r (, 1) there exit a contant C > uch that for each probability meaure λ there exit a probability meaure µ uch that { } 1/p ( 1/r w(x) p dλ C w(x) dµ) r. Reference [1] J. Batero, M. Milman, and F. J. Ruiz, On the connection between weighted norm inequalitie commutator and real interpolation, ubmitted. [2] J. Batero, M. Milman, and F. J. Ruiz, Calderón weight and the real method of interpolation, Revita Matematica Univ. Complutene de Madrid 9 Suppl. (1996), [3] J. Batero and F. J. Ruiz, Elementary revere Hölder type inequalitie with application to interpolation of operator theory, Proc. Amer. Math. Soc. 124 (1996), [4] C. Bennett and R. Sharpley, Weak type inequalitie for H p and BMO, Proc. Sympo. Pure Math. 35, Part 1, S. Wainger and G. Wei, Editor, Amer. Math. Soc., Providence, Rhode Iland, [5] J. Bergh and J. Löftröm, Interpolation pace, An Introduction, Springer- Verlag, New York, [6] R. Coifman and C. Fefferman, Weighted norm inequalitie for maximal function and ingular integral, Studia Math. 51 (1974), [7] R. Fefferman, A criterion for the abolute continuity of the harmonic meaure aociated with an elliptic operator, J. Amer. Math. Soc. 2 (1989), [8] R. Fefferman, C. Kenig, and J. Pipher, The theory of weight and the Dirichlet problem for elliptic equation, Ann. Math. 134 (1991), [9] E. Gagliardo, On integral tranformation with poitive kernel, Proc. Amer. Math. Soc. 16 (1965), [1] J. Garcia Cuerva and J. Rubio de Francia, Weighted norm inequalitie and related topic, North Holland, [11] F. W. Gehring, The L p integrability of partial derivative of a quaiconformal mapping, Acta Math. 13 (1973), [12] M. Gomez and M. Milman, Extrapolation pace and a.e. convergence of ingular integral, J. London Math. Soc. 34 (1986), [13] J. T. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear potential theory of degenerate elliptic equation, Oxford Univ. Pre, [14] C. Herz, The Hardy-Littlewood maximal theorem, Sympoium on Harmonic Analyi, Univerity of Warwick, [15] B. Jawerth and M. Milman, Extrapolation theory with application, Memoir Amer. Math. Soc. 44 (1991).

15 REVERSE HÖLDER INEQUALITIES 15 [16] M. Milman, A note on Gehring lemma, Ann. Acad. Sci. Fenn. Math. 21 (1996), [17] M. Milman, Extrapolation and optimal decompoition, Lecture Note in Math. 158, Springer-Verlag, New York, [18] M. Milman and B. Opic, Real interpolation and two variant of Gehring Lemma, to appear in Journal of the London Math. Soc.. [19] G. Piier, Factorization of linear operator and geometry of Banach pace, Lecture, Univ of Miouri, Columbia, [2] C. Perez, Endpoint etimate for commutator of ingular integral, J. Funct. Anal. 128 (1995), Departamento de Matematica, Univeridad de Zaragoza, Zaragoza 59, Spain addre: jbatero@mf.unizar.e Department of Mathematic, Florida Atlantic Univerity, Boca Raton, Fl addre: milman@acc.fau.edu Departmento de Matematica, Univeridad de Zaragoza, Zaragoza 59, Spain addre: fruiz@mf.unizar.e

TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL

GLASNIK MATEMATIČKI Vol. 38583, 73 84 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian Haihen Lü, Donal O Regan and Ravi P. Agarwal Academy of Mathematic and Sytem Science, Beijing, China, National