Some weighted sum and product inequalities in L^ p spaces and ...

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Some weighted sum and product inequalities in L^ p spaces and ...

Banach J. Math. Anal. 2 (2008), no. 2, 42–58Banach Journal of Mathematical AnalysisISSN: 1735-8787 (electronic)http://www.math-analysis.orgSOME WEIGHTED SUM AND PRODUCT INEQUALITIES INL p SPACES AND THEIR APPLICATIONSR. C. BROWNThis paper is dedicated to Professor Joseph E. PečarićSubmitted by Th. M. RassiasAbstract. We survey some old and new results concerning weighted norminequalities of sum and product form and apply the theory to obtain limitpointconditions for second order differential operators of Sturm-Liouville formdefined in L p spaces. We also extend results of Anderson and Hinton by givingnecessary and sufficient criteria that perturbations of such operators be relativelybounded. Our work is in part a generalization of the classical Hilbertspace theory of Sturm-Liouville operators to a Banach space setting.1. IntroductionLet w, v 0 , v 1 be positive a.e. measurable or “weights” on the interval I a =[a, ∞), a > −∞. We are interested in obtaining conditions which guarantee thevalidity of the weightedsuminequality:∫∫ ∫w|y (j) | p ≤ K 1 (ɛ) v 0 |y| p + ɛ v 1 |y (n) | p (1.1)I a I a I afor 0 ≤ j < n where 1 ≤ p ≤ ∞ and ɛ ∈ (0, ɛ 0 ). The space of functionsD p (v 0 , v 1 ; I a ) on which (1.1) holds is defined by{∫ ∫}D p (v 0 , v 1 ; I a ) := y : y ∈ AC n−1 (I a ); v 0 |y| p , v 1 |y (n) | p < ∞I a I aDate: Received: 12 April 2008; Accepted 21 April 2008.2000 Mathematics Subject Classification. Primary: 26D10, 47A30, 34B24; Secondary 47E05.Key words and phrases. Weighted sum inequalities, weighted product inequalities, SturmLiouville operators, limit-point conditions, relatively bounded perturbations.42


SUM INEQUALITIES IN L p SPACES 43where AC j (I a ) denotes the class of functions whose j-th derivative is locallyabsolutely continuous on I a . We shall also show the inequality (1.1) often impliesthe “productinequality∫I av 1 |y (j) | p ≤ K 2(∫I av 1 |y| p ) n−jn(∫ I av 1 |y (n) | p ) jn.It will turn out that (1.1) has a number of interesting applications to problems inconcerning second-order differential operators determined by symmetric expressionsof the form −(ry ′ ) ′ + qy and defined in L p spaces. The results generalizeboth some aspects of the Hilbert theory presented in the book of Naimark [23]and criteria obtained by Anderson and Hinton in [1] that perturbations of suchoperators in the L 2 setting be relatively bounded. We close this section with a fewremarks on notation. Upper case letters such as K or C denote constants whosevalue may change from line to line. We distinguish between different constants bywriting K 1 , K 2 , C, C 1 , . . . , etc. K(·) indicates dependence on a parameter, e.g.,K 1 (ɛ). L p (I a ) signifies the (complex) L p space on I a having the norm(∫ ) 1/p||u|| p,Ia := |u| p .I aC ∞ (I a ) and C j (I a ) respectively denote the infinitely or j-fold differentiable functionshaving continuous j-th derivative on I a and C ∞ 0 (I a ) or C j 0(I a ) consists ofthe subspace of C ∞ (I a ) or C j (I a ) having compact support. A local property isindicated by the subscript “loc”, e.g., f ∈ L p loc (I), etc. Also, we write AC0 (I a )as AC(I a ) and L 1 (I a ) as L(I a ), and when the context is clear we abbreviateD p (v 0 , v 1 ; I a ) by “D p .” If T : X → Y is an operator where X and Y are Banachspaces D(T ), R(T ), G(T ), and N (T ) respectively denote the domain, range,graph, and null space of T . Finally, if f and g are two functions the notationf ≈ g means that there are constants C 1 and C 2 such that f ≤ C 1 g and g ≤ C 2 f.2. Some weighted norm inequalities of sum formSuppose that f is a positive continuous function on I a . Let J t,ɛ := [t, t + ɛf(t)].For 1 < p < ∞ set[] ∫ ] p/p ′S 1 (t) := f −jp (ɛf)∫J −1 w[(ɛf) −1 (2.1)t,ɛS 2 (t) := f (n−j)p [(ɛf) −1 ∫J t,ɛw] [(ɛf) −1 ∫v −p′ /p0J t,ɛv −p′ /p1J t,ɛ] p/p ′(2.2)where p ′ = p/(p − 1). In the case that p = 1 or ∞ some modifications in thesedefinitions are required. If p = 1 we substitute the L ∞ norm of v −1i , i = 0, 1, onJ t,ɛ for the integral term. For instance,[]S 1 (t) := f −j (ɛf)∫J −1 w ‖v −1 ‖ ∞,Jt,ɛ .t,ɛ


44 R.C. BROWNAnd when p = ∞ we writeSimilar changes apply to S 2 (t).S 1 (t) := f −j ‖w‖ ∞,Jt,ɛ[(ɛf) −1 ∫v0−1J t,ɛTheorem A. Suppose 1 ≤ p ≤ ∞. If there exists f and ɛ 0 ∈ (0, ∞] such thatS 1 (ɛ 0 ) := sup S 1 (t) < ∞t∈I a, 0


SUM INEQUALITIES IN L p SPACES 45Theorem 2.1. Let f and J t,ɛ be as above. Suppose that there is a constant Knot depending on t (but possibly on ɛ) such thata.e. on J t,ɛ . If p = 1, ∞ assume also thatv 0 (s)v 0 (t) , v 1(s)v 1 (t) ≥ K (2.7)w(s)w(t) ≤ K 1 (2.8)a.e. on J t,ɛ . Then the sum inequality (2.5) holds on D p for 1 ≤ p < ∞ ifT 1 (ɛ 0 ) :=T 2 (ɛ 0 ) :=sup f −jp wv0 −1 < ∞ (2.9)t∈I a,0


46 R.C. BROWNand S 2 (t) by replacing the intervals J t,ɛ by ∆ t,ɛ , then (2.5) is readily seen to holdon each Γ j , 1 ≤ j ≤ l. Hence,∫∫ }w|y∫Γ (j) | p ≤ K 1{ɛ −jη v 0 |y| p + ɛ (n−j)η v 1 |y (n) | p ,jso that∫Iw|y (j) | p ≤ lK 1{ɛ −jη ∫IIv 0 |y| p + ɛ (n−j)η ∫IIv 1 |y (n) | p }.How can conditions like (2.3), (2.4), (2.9), (2.10), or (2.11), (2.12) be verified?Essentially, as we have already in part done in Theorem 2.1, we will want tochoose f so that v i (s) ≈ v i (t), i = 0, 1, and w(s) ≈ w(t) on J t,ɛ . In a very generalcase this can always be done as we now demonstrate.Proposition 2.4. Suppose that w, v 0 , and v 1 are continuous on I a . Then thereexists a positive function f ∗ depending on t and possibly ɛ such12 ≤ w(s)w(t) , v 0(s)v 0 (t) , v 1(s)v 1 (t) ≤ 3 2(2.13)on J t,ɛ . Moreover, the sequence {t j } defined as in Theorem A using f ∗ has nofinite limit point.Proof. Given t ≥ a, ɛ > 0, and for i = 0, 1 let f i (t, ɛ) := (s i (t) − t)/ɛ wheres i (t) = min{t + ɛ, sup{z > t : 3v i (t)/2 ≥ v i (u) ≥ v i (t)/2 for u ∈ (t, z]}. (2.14)Define s 2 (t) and f 2 (t, ɛ) similarly for w and set f ∗ (t, ɛ) = min{f i (t, ɛ)}, i = 0, 1, 2.With this construction of f ∗ (2.13) follows. To prove the second assertion Sets ∗,0 := a < · · · < s ∗,j+1 := s ∗,j + ɛf ∗ (s ∗,j , ɛ) ≡ s 0 (s ∗,j )and suppose that {s ∗,j } converges to ¯s ∗ < ∞. We show that for all sufficientlylarge j and for u ∈ (s ∗,j , ¯s ∗ ] that 3v i (s ∗,j )/2 ≥ v i (u) ≥ v i (s ∗,j )/2. If this is notso then for every j there is a k > j and a u ∗ ∈ [s ∗,k , ¯s ∗ ] such that for one of theweights, say, v 0 either (i) v 0 (s ∗,k )/2 > v 0 (u ∗ ) or (ii) 3v 0 (s ∗,k )/2 < v 0 (u ∗ ). Butfrom the continuity and positivity of v 0 , given, say, 1/10 > µ > 0 there is a jsuch that for any k > j and all u ∈ [s ∗,k , ¯s ∗ ] we have thatIf (i) is true then(1 − µ)v 0 (¯s ∗ ) < v 0 (u) < (1 + µ)v 0 (¯s ∗ ).(1 − µ)v 0 (¯s ∗ ) < v 0 (u ∗ ) < v 0 (s ∗,k )/2 < (1 + µ)v 0 (¯s ∗ )/2so that 9/10 < (1/2)(11/10) which is false. Similarly, if (ii) holds we have(3/2)(1 − µ)v 0 (¯s ∗ ) < (3/2)v 0 (s ∗,k ) < v 0 (u ∗ ) < (1 + µ)v 0 )(¯s ∗ )so that 27/20 < 11/10 which is also false. This argument shows that¯s ∗ ≤ s ∗ (s ∗,k ) = s ∗,k+1 < s ∗ (s ∗,k+1 ),and so ¯s ∗ cannot be a limit point of the sequence {s ∗,j }.□


SUM INEQUALITIES IN L p SPACES 47Remark 2.5. With this definition of f ∗ we see thatS 1 (t) ≈ f ∗−jp wv −10 (2.15)S 2 (t) ≈ f ∗(n−j)p wv −11 . (2.16)It is even simpler to define f ∗ so that (2.7) holds in Theorem 2.1 since we can omitw and only consider the lower bounds in (2.14). We omit the details. In particular,this means that if the weights are continuous then the integral expressions (2.1)and (2.2) in Theorem A can in theory always be replaced by the point evaluationexpressions (2.15) and (2.16). Also, in Theorem 2.1 if the weights are continuousthen the conditions (2.7) will be satisfied if f ∗ is chosen. But while Proposition2.4 is of some theoretical interest it is usually not of much practical use since it isdifficult to characterize f ∗ in a convenient fashion from its definition. Fortunately,a satisfactory substitute for f ∗ is often suggested by the particular weights w, v 0 ,and v 1 .Example 2.6. Let a = 1, w(t) = t β , v 0 (t) = t γ , v 1 (t) = t α and f(t) = t δ whereδ ≤ 1. Then1 ≤ sups∈J t,ɛst −1 ≤ 1 + ɛt δ−1 ≤ 1 + ɛ.A calculation shows that S 1 (ɛ 0 ), S 2 (ɛ 0 ) are finite ifβ ≤ min {δpj + γ, −δ(n − j)p + α} , (2.17)and any fixed ɛ 0 (say ɛ 0 = 1). In (2.17) β will be as large as possible relative toα and γ if δ is chosen by “equality”, i.e.,δ = (α − γ)/np ≤ 1. (2.18)With this choice of δ( ) n − jβ ≤ γn( j+ α .n)Example 2.7. Let w(t) = v 0 (t) = v 1 (t) = e t , a ≥ 0, and f(t) = 1.1 ≤ e s /e t ≤ ɛ on J t,ɛ . (2.5) follows.ThenIt was demonstrated in [7, Theorem 3.2] that either of the conditions (2.3) or(2.4) is necessary as well as sufficient for (2.5) provided the weights are chosenso that S 1 (t) ≈ S 2 (t). The choice of δ according to (2.18) forces this in Example2.6.Example 2.8. Let v 0 = v 1 = w and take f(t) = 1. Then (2.5) is true if and onlyif( ∫ t+ɛ) ∫ t+ɛ) p/p ′sup ɛ −1 w(ɛ −1 w −p′ /p< ∞.t∈I a,0


48 R.C. BROWNTheorem 2.9. Let w, v 0 , v 1 be weights such that w, v −p′ /p0 , v −p′ /p1 ∈ L loc (I a ) andthe following conditions are satisfied:|v ′ 1| ≤ Kv 1−1/np1 v 1/np0 ,|v ′ 0| ≤ npv −1/np1 v 1+1/np0 (2.19)on I a . Then the sum inequality (2.5) holds for 1 < p < ∞ and j = 0, . . . , n − 1if and only ifS(ɛ 0 ) :=where f(t) = (v 1 /v 0 ) 1/np .1supt∈I a,ɛ∈(0,ɛ 0 ) ɛf(t)∫ t+ɛf(t)twv −j/n1 v (j−n)/n0 < ∞ (2.20)Before proving this we need a lemma showing that our choice of f works likef ∗ in Proposition 2.4.Lemma 2.10. There are constants K 2 , K 3 > 0 possibly depending on ɛ so thatK 2 ≤ f(s)f(t) , v 1(s)v 1 (t) , v 0(s)v 0 (t) ≤ K 3 (2.21)on the intervals J t,ɛ for sufficiently small ɛ > 0.Proof. First, we note that|f ′ | = ∣ ( (v 1 /v 0 ) 1/np) ∣ ′ ∣∣ = (1/np)(v 1 /v 0 ) 1/np−1 v0 −2 (v 0 v 1 ′ − v 1 v 0) ∣ ′ ≤ (1/np)v 1/np−11 v −1/np0 |v ′ 1| + (1/np)(v 1 /v 0 ) 1/np−1 v −20 v 1 |v ′ 0|≤ K/np + 1.Hence for t < s ≤ t + ɛf(t),|f(s) − f(t)| =∣∫ stf ′ ∣ ∣∣∣≤∫ st|f ′ | ≤ ɛf(t)(K/np + 1),so that f(s)/f(t) ≤ 1 + ɛ(K/np + 1) and f(s)/f(t) ≥ 1 − ɛ(K/np + 1). Next,consider v 0 . Since|fv ′ 0| ≤ (v 1 /v 0 ) 1/np npv 1+1/np0 v −1/np1 ,= npv 0and by the previous result we obtain thatv ′ 0/v 0 ≤ |v ′ 0/v 0 | ≤ np(1 − ɛ(K/np + 1))f(t)) −1 .Integrating this over J t,ɛ implies thatBut also sincev 0 (s)/v 0 (t) ≤ exp(K 4 (ɛ)(s − t)/f(t))≤ exp(K 4 (ɛ)). (2.22)−v ′ 0/v 0 ≤ |v ′ 0/v 0 | ≤ np(1 − ɛ(K/np + 1))f(t)) −1 ,


we get by integration thatSUM INEQUALITIES IN L p SPACES 49v 0 (s)/v 0 (t) ≥ (exp(K 4 (ɛ))) −1 . (2.23)(2.21) for v 0 follows from (2.22) and (2.23). That (2.21) holds also for v 1 is aconsequence of the identityv 1 (s)/v 1 (t) = (v 0 (s)/v 0 (t)(f(s)/f(t)) npand (2.21) for v 0 and f.□Proof of Theorem 2.9. We know by Theorem A that the sum inequality (2.5)holds if (2.3) and (2.4) hold. However because of the conditions (2.21) allowingf, v 0 , and v 1 to be taken in and out of integrals over the interval J t,ɛ both conditionsare found to be equivalent to (2.20). Since the assumptions on the weightsguarantee that S 1 (t) = S 2 (t) we could apply [7, Theorem 3.2] to conclude that(2.20) is a necessary condition; but we choose to give an explicit argument. Letφ ≥ 0 be a C ∞ 0 function such that φ(t) = 1 on [0, 1] and φ has support on (−2, 2).DefineH j (t) = φ(t)(t j /j!).Then H (j)j (t) = 1 on [0, 1] and H j (t) has support on (−2, 2). Set u = (s−t)/ɛf(t)for t − 2ɛf(t) ≤ s ≤ t + 2ɛf(t) and defineNote thatH j,t (s) = (ɛf(t)) j H j (u).H (k)j,t(s) = H(k) (u)(ɛf(t)) j−kjso that in particular H (j)j,t (s) = 1. Next, choose t and sufficiently small ɛ suchthat t − 2ɛf(t) > a and consider the functionS(t) := (ɛf(t)) −1 ∫ t+ɛf(t)= (ɛf(t)) −1 ∫ t+ɛf(t)≈ (ɛf(t)) −1 (v −j/n×∫ t+ɛf(t)Therefore if (2.5) holds we havettt1 v j/n−1w|H (j)j,t |p .wv −j/n1 v j/n−10wv −j/n1 v j/n−10 |H (j)j,t |p0 )(t){∫ t+2ɛf(t)S(t) = O (ɛf(t)) −1 (v −j/n1 v j/n−10 )(t)[ɛ −jp+ɛ (n−j)p ∫ t+2ɛf(t)t−2ɛf(t)v 1 |H (n)j,t |p ]}.t−2ɛf(t)v 0 |H j,t | p


50 R.C. BROWNHoweverand( ∫ )t+2ɛf(t)(ɛf(t)) −1 (v −j/n1 v 0 (t) j/n−1 )(t) v 0 |H j,t | pt−2ɛf(t)( ∫ )t+2ɛf(t)≈ f(t) −jp (ɛf(t)) −1 |H j,t | p=∫ 2−2|H j (u)| p ,(ɛf(t)) −1 (v −j/n1 v j/n−10 )(t)t−2ɛf(t)≈ f(t) −jp (v 1 /v 0 )(ɛf(t)) −1 )=∫ 2−2|H (n)j (u)| p .( ∫ t+2ɛf(t)t−2ɛf(t)( ∫ t+2ɛf(t))v 1 |H (n)j,t |pt−2ɛf(t))|H (n)j,t |pPutting these two estimates together shows that S(t) is uniformly bounded fort ∈ I a and ɛ ∈ (0, ɛ 0 ] which is equivalent to (2.20) as was to be proved. □Our next result extends an inequality of Anderson and Hinton [1, Theorem 3.1]from L 2 (I a ) to the L p case.Corollary 2.11. Suppose v 0 , v 1 ∈ L p loc (I a), v 1/21 |v ′ 0| ≤ 2v 3/20 , and |v ′ 1| ≤ (K/p) √ v 0 v 1then the sum inequalityholds for all 1 < p < ∞ and ɛ > 0.‖v ′ 1y ′ ‖ p,Ia ≤ K(ɛ)‖v 0 y‖ p,Ia + ɛ‖v 1 y ′′ ‖ p,Ia (2.24)Proof. Here the weights (v ′ 1) p , v p 0, and v p 1 replace the weights w, v 0 , and v 1 . Theconditions on v ′ 1 and v ′ 0 are equivalent to (2.19) for this choice of weights. f(t) isgiven by (v p 1/v p 0) 1/2p ≡ √ v 1 /v 0 . We have also thatS 1 (t) ≡ S 2 (t) = ɛ −1 √v0v 1∫ t+ɛ√v1 /v 0t|v ′ 1| p (v 1 v 0 ) −p/2 ≤ (K/p) p .(2.24) follows by Theorem 2.9. □Example 2.12. Suppose w = v 0 = v 1 and |v ′ 1| ≤ npv 1 . Then f = 1 andS 1 (∞) = S 2 (∞) = 1. By Theorem 2.9 we have the inequality∫I aw|y (j) | p ≤ K 1(ɛ −jp ∫I aw|y| p + ɛ (n−j)p ∫I aw|y (n) | p ). (2.25)In this example unlike Examples 2.7 and 2.8 since the inequality holds for allɛ > 0.


SUM INEQUALITIES IN L p SPACES 51Remark 2.13. If a sum inequality of the form (2.5) holds for arbitrary ɛ > 0 wecan minimize the right-hand side of the inequality as a function of ɛ providedj ≠ 0 and ∫ I av 1 |y (n) | p ≠ 0. This procedure applied to (2.25) in the previousexample will yield the product inequality∫I aw|y (j) | p ≤ K 2(∫I aw|y| p ) n−jn(∫ I aw|y (n) | p ) jn.Note that v 1 can be taken as e ±bt where 0 < b ≤ np.Remark 2.14. So far we have supposed because of the applications we have inmind that each of the terms in our weighted sum or product inequalities have acommon L p norm. However, versions of these inequalities exist when the threenorms are different. One can have, for instance, an inequality of the form{ [∫] ) p/tw|y∫I (j) | p ≤ K 1 v 1 |ya[∫I (m) | tawhere 1 ≤ p, s, t < ∞,ɛ −µ 1I av 0 |y| s ] p/s+ ɛ µ 2µ 1 = p(j + s −1 − p −1 )µ 2 = p(n − j − t −1 + p −1 ),and n, j, p, s, t satisfy various relationships. Also generalizations exist in R n , n >1. For information on these more general cases see [5], [7], [6], [9], and [11]. Thereare additionally other approaches to weighted norm inequalities similar to (2.5).See for example Wojteczek-Laszczak [26] and Kwong and Zettl [22].3. Some Applications to Relative Boundedness and Limit-pointconditions for differential operators in L p spacesIn [11] we gave applications of sum and product inequalities to various spectraltheoretic problems involving Sturm-Liouville operators in L 2 (I a ). In this sectionwe look at applications to operators determined by expressions of Sturm-Liouvilleform but defined in L p spaces. We first require some preliminary definitions andabstract results. In what follows ‖(·)‖ will denote the norm in an arbitrary Banachspace.Definition 3.1. Suppose A and T are operators from a Banach space X to aBanach space Y . Then A is said to be T bounded if the domain of T is containedin the domain of A and the inequality‖A(x)‖ ≤ K(‖x‖ + ‖T (x)‖) (3.1)holds for all x in the domain of T . Furthermore A is said to have T bound 0 if Ais T bounded and the inequality (3.1) has the formfor all ɛ ∈ (0, ɛ 0 ) for some ɛ 0 ∈ (0, ∞).‖A(x)‖ ≤ K(ɛ)‖x‖ + ɛ‖T (x)‖)


52 R.C. BROWNLemma 3.2. Suppose A, B, C, and L are operators from a Banach space X toa Banach space Y . Suppose that A is L-bounded and B, C are L-bounded with Lbound 0. Then A is L + B + C bounded.Proof. By the triangle inequality‖x‖ + ‖L(x)‖ ≤ ‖x‖ + ‖(L + B + C)(x)‖ + ‖B(x)‖ + ‖C(x)‖. (3.2)¿From the hypotheses on B and C we also have the estimates‖B(x)‖ ≤ K 1 (ɛ/2)‖x‖ + (ɛ/2)‖L(x)‖ (3.3)‖C(x)‖ ≤ K 2 (ɛ/2)‖x‖ + (ɛ/2)‖L(x)‖. (3.4)Substituting (3.3) and (3.4) into (3.2 gives that(1 − ɛ)(‖x‖ + ‖L(x)‖) ≤ (1 + K 1 (ɛ/2) + K 2 (ɛ/2))‖x‖ + ‖(L + B + C)(x)‖.(3.5)But since A is L bounded ‖A(x)‖ ≤ K(‖x‖ + ‖L(x)‖). Combining this with (3.5)yields that‖A(x)‖ ≤ K(1 − ɛ) −1 [(1 + K 1 (ɛ/2) + K 2 (ɛ/2))‖x‖ + ‖(L + B + C)(x)‖].Lemma 3.3. Let A, B, C and L be operators from a Banach space X to aBanach space Y . Suppose the inequalities‖A(y)‖ ≤ K(ɛ)‖B(y)‖ + ɛ‖L(y)‖) (3.6)‖B(y)‖ ≤ K(ɛ)‖C(y)‖ + ɛ‖L(y)‖ (3.7)‖C(y)‖ ≤ K(‖y‖ + ‖T (y)‖) (3.8)where T (y) = (A + B + L)(y). Then A is T bounded with relative bound 0.Proof. By (3.7) and the triangle inequality‖B(y)‖ ≤ K(ɛ)‖C(y)‖ + ɛ‖(L + B)(y)‖ + ɛ‖B(y)‖.Hence,‖B(y)‖ ≤ K(ɛ)(1 − ɛ) −1 ‖C(y)‖ + ɛ(1 − ɛ) −1 ‖(L + B)(y)‖.Substituting this into (3.6) after noting again that ‖L(y)‖ ≤ ‖(L + B)(y)‖ +‖B(y)‖ gives the inequalityFinally,‖A(y)‖ ≤ K(ɛ)(1 + ɛ(1 − ɛ) −1 )‖C(y)‖ + ɛ 2 (1 − ɛ) −1 ‖(L + B)(y)‖. (3.9)‖(L + B)(y)‖ ≤ ‖T (y) + ‖C(y)‖≤ K‖y‖ + (K + 1)‖T (y)‖.Substitution into (3.9) now gives the desired conclusion.Given a Banach space X with dual X ∗ , [x, x ∗ ] signifies the complex conjugatex ∗ (x) for x ∈ X and x ∗ ∈ X ∗ . If T is an operator on X we consider the set ofpairs G(T ∗ ) := (z, z ′ ) ∈ X × X ∗ such that[T (y), z] = [y, z ′ ]. (3.10)□□


SUM INEQUALITIES IN L p SPACES 53The density of D(T ) implies that (3.10) determines an operator T ∗ called theadjoint of T such that T ∗ (z) = z ′ . If T : X ∗ → X ∗ has a domain D(T ) which istotal over X (i.e., [x, x ∗ ] = 0 for a fixed x ∈ X and all x ∗ ∈ D(T ) ⇒ x = 0) theset G( ∗ T ) of pairs (y ′ , y) ∈ X × X satisfying[y ′ , z] = [y, T (z)]also determines an operator which we denote by ∗ T and call the adjoint of T inX. 1 Both T ∗ and ∗ T are closed and [T (y), z] = [y, T ∗ (z)] or [ ∗ T (y), z] = [y, T (z)]for all y ∈ D(T ) or D( ∗ T ) ⊂ X and z ∈ D(T ∗ ) or D(T ) ⊂ X ∗ . In the particularcase X = L p (I a ) and X ∗ = L p′ (I a ) where 1 ≤ p ≤ ∞ the pairing [(·), (·)] onL p (J) × L p′ (J) for some interval J is given by [y, z] J := ∫ y¯z. Consider now theJdifferential expression M[y] := −(ry ′ ) ′ + qy. Assume that r > 0, r ∈ C 1 (I a ), andq ∈ C(I a ). Define{y, z}(t) := r(t)(y(t)¯z ′ (t) − y ′ (t)¯z(t)).for y, z ∈ AC loc (I a ) and the following operators and domains in L p (I a ).Definition 3.4. For p ∈ [1, ∞] let let T ′ 0,p, T p , T 0,p , and be the operators withdomain and range in L p (I a ) determined by M onD ′ 0,p := {y ∈ C ∞ 0 (I a )},D p := {y ∈ L p (I a ) : y ′ ∈ AC loc (I a ); M[y] ∈ L p (I a )},D 0,p := {y ∈ D p : y(a) = y ′ (a) = 0; limt→∞{y, z}(t) = 0, ∀z ∈ D p ′}.We call T ′ 0,p, T 0,p respectively the “preminimal” and “minimal” operators, and T pthe “maximal” operator determined by M.Theorem B. If 1 ≤ p < ∞ the operators T ′ 0,p, T 0,p , and T p have the followingproperties:(i) T 0,p and T p are closed densely defined operators.(ii) [T p (y), z] [a,t] = {y, z}(t) − {y, z}(a) + [y, T p ′(z)] [a,t] .(iii) T ∗ p = T 0,p ′ and T ∗ 0,p = T p ′.(iv) T ′ 0,p is closable and T ′ 0,p = T 0,p .Moreover, T 0,∞ and T ∞ are closed, the closure of T ′ 0,∞ is T 0,∞ , ∗ T 0,∞ = T 1 , and∗ T ∞ = T 0,1 .Proofs of (i)–(iii), the last statement, as well as more general results may be foundin one of [18, Chapter VI], [25], or [13]). The L 2 theory is thoroughly treatedin Naimark [23, §17]. It is almost certain that by extending the procedure ofNaimark given in the L 2 case that q, r and r −p′ /p need only be locally integrablefor Theorem B to hold. However, the verification of this is technically complicatedand will be omitted here.Definition 3.5. The operators T 0,p or T p are separated if on D 0,p or D p (ry ′ ) ′ ∈L p (I a )1 Goldberg calls this operator the preconjugate of T . Its properties are studied in [18, VI.I].


54 R.C. BROWNRemark 3.6. By the triangle inequality the separation of T 0,p or T p is equivalentto qy ∈ L p (I a ) on the domains of these operators. The closed graph theorem inturn implies that separation is equivalent to the existence of an inequality of theform‖(ry ′ ) ′ ‖ p,Ia + ‖qy‖ p,Ia ≤ K{‖y‖ p,Ia + ‖M[y]‖ p,Ia }.Necessary and sufficient conditions for separation in L p when M[y] = −y ′′ + qyhave been given by Chernyavskaya and Shuster [14]. However their conditionscan be difficult to verify. For p = 2 various sufficient conditions may be found in[12], [10], [2], [16]. The simplest condition guaranteeing separation for all p is torequire that q be essentially bounded.Limit-point Results in L p spaces.Definition 3.7. We say that T p is p-limit-point (p-LP) at ∞ if lim t→∞ {y, z}(t) =0 for all y ∈ D p and z ∈ D p ′Theorem 3.8. Consider the following conditions:(i) T p is p-LP.(ii) There do not exist linearly independent solutions y p ∈ L p (I a ) and z p ′L p′ (I a ) of M[y] = 0.(iii) dim D p /D 0,p = 2.Then (i) and (iii) are equivalent conditions and (i) ⇒ (ii).Proof. Suppose that (i) is true and that there were linearly independent solutionsy p ∈ L p (I a ) and z p ′ ∈ L p′ (I a ). Let t ∈ I a By (ii) of Theorem B {y, z}(t) ={y, z}(a) and the fact that T p is p-LP we have0 = limt→∞{y p , z p ′}(t) = {y p , z p ′}(a) = r −1 (a){y p , z p ′}(a).But this is impossible since r −1 {y, z} is just the Wronskian of of the solutionsy p , z p ′ and its zero value at a or t contradicts their assumed linear independence.Turning now to (iii), let φ 1 and φ 2 be C0 ∞ (I a ) functions such that φ 1 (a) = 1,φ ′ 1(a) = 0 and φ 2 (a) = 0, φ ′ 2(a) = 1. Since φ 1 , φ 2 ∈ D p and are linearly independent,we see that dim D p /D 0,p ≥ 2. Suppose there exists u ∈ D p such that{φ 1 , φ 2 , u} is linearly independent mod D 0,p . Let h be a linear combination ofthese functions such that h(a) = h ′ (a) = 0. Let G p and G 0,p respectively be D por D 0,p endowed with the graph norm. Now the dual G ∗ p of G p can be identifiedwith the space of pairs ξ = (z, z ∗ ) in L p′ (I a ) × L p′ (I a ) such that ξ(y), y ∈ G p , isgiven by∫y¯z +I a∫M[y]¯z ∗ .I aSince h /∈ G 0,p and G 0,p is closed in G p there exists an element ξ ∈ G ∗ p such thatξ(h) = 1 and ξ(y) = 0 for all y ∈ G 0,p , implying that ξ = (−M[z ∗ ], z ∗ ) for somez ∈ D p . It follows that1 = ξ(h) =∫h(−M[z] +I a∫M[h]¯zI a= {h, z}(∞),∈


SUM INEQUALITIES IN L p SPACES 55contradicting (i). We conclude that dim D/D 0 = 2. In the above argument wehave shown that (i)⇒ (ii) and that (i)⇒ (iii). It is clear that (iii) ⇒ (i). For ifD p is a two dimensional extension of D 0,p then span{φ 1 , φ 2 , D 0,p } = D p . Since φ 1and φ 2 have compact support and by the definition of D 0,p , (i) will hold. □Remark 3.9. For p ≠ 2 Theorem 3.8 represents an extension of the well-knownlimit point concept for differential operators in L 2 (I a ). In particular (ii) is ageneralization of the fact that if M is 2-LP at ∞ then M has at most one L 2solution. In the limiting case p = 1, p ′ = ∞, if there is a solution y in D 1in L(I a ), (ii) says that that there cannot be another solution independent of ywhich is bounded. As in the Hilbert space case we have also shown that T p isp-limit-point if and only if D p /D 0,p = 2.Remark 3.10. If any one of T 0,p , T p , T 0,p ′ and T p ′ has closed range then more can besaid. Specifically all the other minimal and maximal operators also have closedrange. In particular, the minimal operators are one-to-one and have boundedinverses while the maximal operators are surjective anddim (D p /D 0,p ) = dim N (T p ) + dim N (T p ′).For the proof in a considerably more general setting see Goldberg [18] TheoremsVI.2.7 and VI.2.11. Furthermore, since the dimensions of both null spaces do notexceed 2 and since as we have seen D p is at least a two dimensional extension ofD 0,p we have that2 ≤ dim (D p /D 0,p ) ≤ 4.Brown and Cook [13, Corollary 2.9] have shown that T 0,p defined on I a has abounded inverse and thus closed range for 1 ≤ p ≤ ∞ if both ∫ I ar −1 < ∞ and∫I aq −1 < ∞Remark 3.11. If q ≥ 0 then M is disconjugate and by by Corollary 6.4 and Theorem6.4 of Hartman [20] there is a fundamental set of of positive linearly independentsolutions y 1 and y 2 of M[y] = 0, called respectively the principal and nonprincipalsolutions, such that y 1 ′ ≤ 0 and y ′ > 0 on I a . Additionally, lim t→∞ y 1 /y 2 = 0.It follows at once in our setting that dim N (T p ) = dim N (T p ′) ≤ 1. In [3] it isfurthermore shown that if r = 1, q ≥ 0, and there exists b ∈ (0, ∞) such thatinfx∈I a,x−b>a∫ x+bx−bq > 0, (3.11)then(i) T p has closed range.(ii) dim (D p /D 0,p ) = 2 and dim (R(T p )/R(T 0,p ) = 1,(iii) dim N (T p ) = 1,(iv) If y 1 denotes the principal or “small” solution of M[y] = 0 then y 1 ∈ L p (I a )for all p ∈ [1, ∞].The condition (3.11) was shown by Chernyavskya and Shuster [15] to be necessaryand sufficient for T p defined on R to have a bounded inverse. In the case when q ≥k > 0 one can show that M[y] = 0 has exponentially growing and exponentially


56 R.C. BROWNdecaying solutions. Read [24] has extended this by showing that the same is trueiffor some a > 0.lim infx→∞∫ x+axq 1/2 dt > 0Relative Boundedness for perturbations of Differential Operatorsin L p spaces.The following two results are generalizations to L p of relative boundedness criteriagiven by Anderson and Hinton [1] in the L 2 setting.Theorem 3.12. Let A j : L p → L p be given by ∫ I aa j y (j) , j = 0, 1, on D p wherethe a j are locally integrable functions. Let T 0,p be the minimal operator in L p (I a )determined by M[y] = −(ry ′ ) ′ + qy and Assume that |r ′ | ≤ (K/p) √ r and thatsupt∈I aThen the A j are T 0,p -bounded if and only if∫ √1 t+ɛ r√ rt∫ √1 t+ɛ r√r |q| p < ∞. (3.12)t|a j | p r −jp/2 < ∞, j = 0, 1, (3.13)is bounded on I a . If T p is p-LP at ∞ then the A j are also T p bounded.Proof. This is an application of Theorem 2.9 and Lemma 3.2. Define the operatorsL, B, C : L p (I a ) → L p (I a ) on C ∞ 0 (I a ) by L(y) = ry ′′ , B(y) = r ′ y ′ , and C(y) = qywhere y ∈ D 0,p . Then T ′ 0,r = L + B + C. Let f = √ r, and take v 0 = 1,The condition on r ′ is just the first condition in (2.19) with r p replacing v 1 . ByTheorem 2.9 the A j are L-bounded or equivalently the sum inequalities‖A j (y)‖ p,Ia ≤ K(‖y‖ p,Ia + ‖ry ′′ ‖ p,Ia ), j = 0, 1,hold if and only if (3.13) is true. By Corollary 2.11 (with v 0 = 1) B is L-boundedwith relative bound 0. Finally, another application of Theorem 2.9 using (3.12)gives that C is L-bounded with relative bound 0. The fact that the A j are T ′ 0,pbounded now now follows from Lemma 3.2. A closure argument shows that theA j are T 0,p bounded. Finally, if T p is p-LP at ∞, then T p is a two dimensionalextension of T 0,p via the functions φ 1 and φ 2 . Hence, if y ∈ D p , y = y 0 + z wherez = c 1 φ 1 +c 2 φ 2 . Since A j is T 0 bounded and by an elementary inequality we have‖A j (y 0 )‖ p p,I a≤ K p 2 p−1 {‖y 0 ‖ p p,I a+ ‖T (y 0 )‖ p p,I a}.The same inequality is true for z since the p-th root of each side defines twonorms on Z := span {φ 1 , φ 2 } and the mapping j : Z → Z given by j(z) = z iscontinuous with respect to these norms since Z is finite (2!) dimensional. Hence,‖A j (y 0 + z)‖ p p,I a≤ 2 p−1 [‖A j (y 0 )‖ p p,I a+ A j (z)‖ p p,I a≤ K 1 [‖y 0 ‖ p p,I a+ ‖z‖ p p,I a+ ‖T (y 0 )‖ p p,I a+ ‖T (z)‖ p p,I a]≤ K 1 [‖y 0 + z‖ p p,I a+ ‖T (y 0 + z)‖ p p,I a]= K 1 {‖y‖ p p,I a+ ‖T (y)‖ p p,I a}.


SUM INEQUALITIES IN L p SPACES 57Theorem 3.13. Suppose that T 0,p is separated Additionally suppose that|r ′ | ≤ (K/p) √ r|q||q ′ | ≤ 2|q| 3/2 r −1/2 .Let A j , j = 0, 1, be as in Theorem 3.12. Then A j is T 0,p -bounded with T p -bound0 if and only if√ ∫√|q|t+ɛ r/qS(t) =|a j | p r −jp/2 |q| p(j−2)/2 (3.14)ris bounded on I a .tProof. As before we begin with the C0 ∞ functions. Let C(y) = qy, B(y) = r ′ y ′and L(y) = ry ′′ . By the hypothesis of separation (3.8) holds. By Theorem 2.9where f(t) = (r p /q p ) 1/2p = √ r/q (3.6) holds if and only if (3.14) is true. ByCorollary 2.11 (3.7) is true. The conclusion that A j is T 0,p ′ bounded follows fromLemma 3.3.□References1. T. G. Anderson and D. B. Hinton, Relative boundedness and compactness for second orderdifferential operators, J. Ineq. and Appl. 1 (4) (1997), 375–400.2. R. C. Brown, Separation and discongugacy, JPAM 4 (3), Article 56, 2002 (Electronic).3. , A limit-point criterion for a class of Sturm-Liouville operators defined in L p spaces,Proc. Am. Math. Soc 132 (2004), 2273–2280.4. R. C. Brown and D. B. Hinton, Sufficient conditions for weighted inequalities of sum form,J. Math. Anal. Appl. 112 (1985), 563–578.5. , Sufficient conditions for weighted Gabushin inequalities, Casopis Pěst. Mat. 111(1986), 113–122.6. , Weighted interpolation inequalities of sum and product form in R n , J. LondonMath Soc. (3) 56 (1988), 261–280.7. , Weighted interpolation inequalities and embeddings in R n , Canad. J. Math. 47(1990), 959–980.8. , A Weighted Hardy’s inequality and nonoscillatory differential equations, Quaes.Mathematicae 115 (1992), 197–212.9. , An interpolation inequality and applications, Inequalities and Applications (R. P.Agarwal, ed.), World Scientific, Singapore-New Jersey-London-Hong Kong, 1994, 87–101.10. , Two separation criteria for second order ordinary or partial differential operators,Math. Bohem. 124 (1999), 273–292.11. , Some One Variable Weighted Norm Inequalities and Their Applications to Sturm-Liouville and other Differential Operators, to appear in Inequalities and Applications, (C.Bandle, A. Gilányi, L. Losonczi, Z. Pales, M. Plum, eds.), Vol. 157, International Series ofNumerical Mathematics, Birkhäuser, Boston, Basel, Stuttgart, 2008.12. R. C. Brown, D. B. Hinton, and M. F. Shaw, Some separation criteria and inequalitiesassociated with linear second order differential operators, in Function Spaces and Aplications(D. E. Edmunds, et al., eds.), Narosa Publishing House, New Delhi, 2000, 7–35.13. R. C. Brown and J. Cook, Continuous invertibility of minimal Sturm-Liouville operatorsin Lebesgue spaces, Proc. Roy. Soc. Edinburgh. A 136 (01) (2006), 53–70.14. N. Chernyavskaya and L. Schuster, Weight summability of solutions of the Sturm-Liouvilleequation, J. Diff. Eqs. 151 (1999), 456–473.□

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