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 concern**in**g **weighted** norm** inequalities** of

SUM INEQUALITIES IN L p SPACES 43where AC j (I a ) denotes the class of functions whose j-th derivative is locallyabsolutely cont**in**uous on I a . We shall also show the **in**equality (1.1) often impliesthe “**product**” **in**equality∫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 **in**terest**in**g applications to problems **in**concern**in**g second-order differential operators determ**in**ed by symmetric expressionsof the form −(ry ′ ) ′ + qy **and** def**in**ed **in** L p **spaces**. The results generalizeboth some aspects of the Hilbert theory presented **in** the book of Naimark [23]**and** criteria obta**in**ed by Anderson **and** H**in**ton **in** [1] that perturbations of suchoperators **in** the L 2 sett**in**g 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 l**in**e to l**in**e. We dist**in**guish between different constants bywrit**in**g K 1 , K 2 , C, C 1 , . . . , etc. K(·) **in**dicates dependence on a parameter, e.g.,K 1 (ɛ). L p (I a ) signifies the (complex) L p space on I a hav**in**g the norm(∫ ) 1/p||u|| p,Ia := |u| p .I aC ∞ (I a ) **and** C j (I a ) respectively denote the **in**f**in**itely or j-fold differentiable functionshav**in**g cont**in**uous 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 ) hav**in**g compact support. A local property is**in**dicated 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 Banach**spaces** D(T ), R(T ), G(T ), **and** N (T ) respectively denote the doma**in**, range,graph, **and** null space of T . F**in**ally, 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

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 depend**in**g on t (but possibly on ɛ) such thata.e. on J t,ɛ . If p = 1, ∞ as**sum**e 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** **in**equality (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. BROWN**and** S 2 (t) by replac**in**g the **in**tervals 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 cont**in**uous on I a . Then thereexists a positive function f ∗ depend**in**g 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 } def**in**ed as **in** Theorem A us**in**g f ∗ has nof**in**ite limit po**in**t.Proof. Given t ≥ a, ɛ > 0, **and** for i = 0, 1 let f i (t, ɛ) := (s i (t) − t)/ɛ wheres i (t) = m**in**{t + ɛ, sup{z > t : 3v i (t)/2 ≥ v i (u) ≥ v i (t)/2 for u ∈ (t, z]}. (2.14)Def**in**e s 2 (t) **and** f 2 (t, ɛ) similarly for w **and** set f ∗ (t, ɛ) = m**in**{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 cont**in**uity **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 po**in**t of the sequence {s ∗,j }.□

SUM INEQUALITIES IN L p SPACES 47Remark 2.5. With this def**in**ition 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 def**in**e f ∗ so that (2.7) holds **in** Theorem 2.1 s**in**ce 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 cont**in**uous then the **in**tegral expressions (2.1)**and** (2.2) **in** Theorem A can **in** theory always be replaced by the po**in**t evaluationexpressions (2.15) **and** (2.16). Also, **in** Theorem 2.1 if the weights are cont**in**uousthen the conditions (2.7) will be satisfied if f ∗ is chosen. But while Proposition2.4 is of some theoretical **in**terest it is usually not of much practical use s**in**ce it isdifficult to characterize f ∗ **in** a convenient fashion from its def**in**ition. 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 f**in**ite ifβ ≤ m**in** {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 accord**in**g 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 ) **and**the follow**in**g 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** **in**equality (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 prov**in**g this we need a lemma show**in**g that our choice of f works likef ∗ **in** Proposition 2.4.Lemma 2.10. There are constants K 2 , K 3 > 0 possibly depend**in**g 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 **in**tervals 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 . S**in**ce|fv ′ 0| ≤ (v 1 /v 0 ) 1/np npv 1+1/np0 v −1/np1 ,= npv 0**and** by the previous result we obta**in** thatv ′ 0/v 0 ≤ |v ′ 0/v 0 | ≤ np(1 − ɛ(K/np + 1))f(t)) −1 .Integrat**in**g this over J t,ɛ implies thatBut also s**in**cev 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 **in**tegration 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)) np**and** (2.21) for v 0 **and** f.□Proof of Theorem 2.9. We know by Theorem A that the **sum** **in**equality (2.5)holds if (2.3) **and** (2.4) hold. However because of the conditions (2.21) allow**in**gf, v 0 , **and** v 1 to be taken **in** **and** out of **in**tegrals over the **in**terval J t,ɛ both conditionsare found to be equivalent to (2.20). S**in**ce the as**sum**ptions 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).Def**in**eH 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** def**in**eNote 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. BROWNHowever**and**( ∫ )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 |pPutt**in**g 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 **in**equality of Anderson **and** H**in**ton [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** **in**equalityholds 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 **and**S 1 (∞) = S 2 (∞) = 1. By Theorem 2.9 we have the **in**equality∫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 s**in**ce the **in**equality holds for allɛ > 0.

SUM INEQUALITIES IN L p SPACES 51Remark 2.13. If a **sum** **in**equality of the form (2.5) holds for arbitrary ɛ > 0 wecan m**in**imize the right-h**and** side of the **in**equality 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** **in**equality∫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 **in**m**in**d that each of the terms **in** our **weighted** **sum** or **product** ** inequalities** have acommon L p norm. However, versions of these

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 **in**equality‖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)Substitut**in**g (3.3) **and** (3.4) **in**to (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 s**in**ce A is L bounded ‖A(x)‖ ≤ K(‖x‖ + ‖L(x)‖). Comb**in****in**g 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)

SUM INEQUALITIES IN L p SPACES 53The density of D(T ) implies that (3.10) determ**in**es an operator T ∗ called theadjo**in**t of T such that T ∗ (z) = z ′ . If T : X ∗ → X ∗ has a doma**in** 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 satisfy**in**g[y ′ , z] = [y, T (z)]also determ**in**es an operator which we denote by ∗ T **and** call the adjo**in**t of T **in**X. 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 pair**in**g [(·), (·)] onL p (J) × L p′ (J) for some **in**terval J is given by [y, z] J := ∫ y¯z. Consider now theJdifferential expression M[y] := −(ry ′ ) ′ + qy. As**sum**e that r > 0, r ∈ C 1 (I a ), **and**q ∈ C(I a ). Def**in**e{y, z}(t) := r(t)(y(t)¯z ′ (t) − y ′ (t)¯z(t)).for y, z ∈ AC loc (I a ) **and** the follow**in**g operators **and** doma**in**s **in** L p (I a ).Def**in**ition 3.4. For p ∈ [1, ∞] let let T ′ 0,p, T p , T 0,p , **and** be the operators withdoma**in** **and** range **in** L p (I a ) determ**in**ed 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 “prem**in**imal” **and** “m**in**imal” operators, **and** T pthe “maximal” operator determ**in**ed by M.Theorem B. If 1 ≤ p < ∞ the operators T ′ 0,p, T 0,p , **and** T p have the follow**in**gproperties:(i) T 0,p **and** T p are closed densely def**in**ed 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 found**in** one of [18, Chapter VI], [25], or [13]). The L 2 theory is thoroughly treated**in** Naimark [23, §17]. It is almost certa**in** that by extend**in**g the procedure ofNaimark given **in** the L 2 case that q, r **and** r −p′ /p need only be locally **in**tegrablefor Theorem B to hold. However, the verification of this is technically complicated**and** will be omitted here.Def**in**ition 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 **in**equality the separation of T 0,p or T p is equivalentto qy ∈ L p (I a ) on the doma**in**s of these operators. The closed graph theorem **in**turn implies that separation is equivalent to the existence of an **in**equality 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 guarantee**in**g separation for all p is torequire that q be essentially bounded.Limit-po**in**t Results **in** L p **spaces**.Def**in**ition 3.7. We say that T p is p-limit-po**in**t (p-LP) at ∞ if lim t→∞ {y, z}(t) =0 for all y ∈ D p **and** z ∈ D p ′Theorem 3.8. Consider the follow**in**g conditions:(i) T p is p-LP.(ii) There do not exist l**in**early **in**dependent 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 l**in**early **in**dependent 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 s**in**ce 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 as**sum**ed l**in**ear **in**dependence.Turn**in**g 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. S**in**ce φ 1 , φ 2 ∈ D p **and** are l**in**early **in**dependent,we see that dim D p /D 0,p ≥ 2. Suppose there exists u ∈ D p such that{φ 1 , φ 2 , u} is l**in**early **in**dependent mod D 0,p . Let h be a l**in**ear comb**in**ation 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 aS**in**ce 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 , imply**in**g 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 55contradict**in**g (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 . S**in**ce φ 1**and** φ 2 have compact support **and** by the def**in**ition of D 0,p , (i) will hold. □Remark 3.9. For p ≠ 2 Theorem 3.8 represents an extension of the well-knownlimit po**in**t 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 limit**in**g case p = 1, p ′ = ∞, if there is a solution y **in** D 1**in** L(I a ), (ii) says that that there cannot be another solution **in**dependent of ywhich is bounded. As **in** the Hilbert space case we have also shown that T p isp-limit-po**in**t 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 m**in**imal **and** maximal operators also have closedrange. In particular, the m**in**imal operators are one-to-one **and** have bounded**in**verses while the maximal operators are surjective **and**dim (D p /D 0,p ) = dim N (T p ) + dim N (T p ′).For the proof **in** a considerably more general sett**in**g see Goldberg [18] TheoremsVI.2.7 **and** VI.2.11. Furthermore, s**in**ce the dimensions of both null **spaces** do notexceed 2 **and** s**in**ce 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 def**in**ed on I a has abounded **in**verse **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 l**in**early **in**dependentsolutions y 1 **and** y 2 of M[y] = 0, called respectively the pr**in**cipal **and** nonpr**in**cipalsolutions, 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 sett**in**g 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 that**in**fx∈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 pr**in**cipal 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 necessary**and** sufficient for T p def**in**ed on R to have a bounded **in**verse. In the case when q ≥k > 0 one can show that M[y] = 0 has exponentially grow**in**g **and** exponentially

56 R.C. BROWNdecay**in**g solutions. Read [24] has extended this by show**in**g that the same is trueiffor some a > 0.lim **in**fx→∞∫ x+axq 1/2 dt > 0Relative Boundedness for perturbations of Differential Operators**in** L p **spaces**.The follow**in**g two results are generalizations to L p of relative boundedness criteriagiven by Anderson **and** H**in**ton [1] **in** the L 2 sett**in**g.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 **in**tegrable functions. Let T 0,p be the m**in**imal operator **in** L p (I a )determ**in**ed by M[y] = −(ry ′ ) ′ + qy **and** As**sum**e 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. Def**in**e 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 replac**in**g 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

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 beg**in** 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. H**in**ton, 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-po**in**t criterion for a class of Sturm-Liouville operators def**in**ed **in** L p **spaces**,Proc. Am. Math. Soc 132 (2004), 2273–2280.4. R. C. Brown **and** D. B. H**in**ton, Sufficient conditions for **weighted** ** inequalities** of