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

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)‖)