Bessel-Type Operators and a refinement of Hardy's inequality
aa r X i v : . [ m a t h . SP ] F e b A REFINEMENT OF HARDY’S INEQUALITY
FRITZ GESZTESY, MICHAEL M. H. PANG, AND JONATHAN STANFILL
Dedicated with great pleasure to Lance Littlejohn on the occasion of his 70th birthday.
Abstract.
The principal aim of this paper is to prove the inequality ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | sin ( x ) + 14 ˆ π dx | f ( x ) | , f ∈ H ((0 , π )) , where both constants 1 / f ≡
0. This inequality is derived with the help of the exactly solvable, stronglysingular, Dirichlet-type Schr¨odinger operator associated with the differentialexpression τ s = − d dx + s − (1 / ( x ) , s ∈ [0 , ∞ ) , x ∈ (0 , π ) . The new inequality represents a refinement of Hardy’s classical inequality ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | x , f ∈ H ((0 , π )) , it also improves upon one of its well-known extensions in the form ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | d (0 ,π ) ( x ) , f ∈ H ((0 , π )) , where d (0 ,π ) ( x ) represents the distance from x ∈ (0 , π ) to the boundary { , π } of (0 , π ). Contents
1. Introduction 22. An Exactly Solvable, Strongly Singular,Periodic Schr¨odinger Operator 33. A Refinement of Hardy’s Inequality 7Appendix A. The Weyl–Titchmarsh–Kodaira m -FunctionAssociated with T s,F Date : February 23, 2021.2020
Mathematics Subject Classification.
Primary: 26D10, 34A40, 34B20, 34B30; Secondary:34L10, 34B24, 47A07.
Key words and phrases.
Hardy-type inequality, strongly singular differential operators,Friedrichs extension.To appear in
From Operator Theory to Orthogonal Polynomials, Combinatorics, and Num-ber Theory. A Festschrift in honor of Lance L. Littlejohn’s 70th birthday , F. Gesztesy and A.Martinez-Finkelshtein (eds.), Operator Theory: Advances and Applications, Birkh¨auser, Springer. Introduction
Happy Birthday, Lance! We hope this modest contribution to Hardy-type inequal-ities will cause some joy.
In a nutshell, the aim of this note is to derive the Hardy-type inequality ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | sin ( x ) + 14 ˆ π dx | f ( x ) | , f ∈ H ((0 , π )) . (1.1)As is readily verified, (1.1) indeed represents an improvement over the classicalHardy inequality ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | x , f ∈ H ((0 , π )) , (1.2)while also improving upon one of its well-known extensions in the form ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | d (0 ,π ) ( x ) , f ∈ H ((0 , π )) . (1.3)Here d (0 ,π ) ( x ) represents the distance from x ∈ (0 , π ) to the boundary { , π } of theinterval (0 , π ), that is, d (0 ,π ) ( x ) = ( x, x ∈ (0 , π/ ,π − x, x ∈ [ π/ , π ) . (1.4)We emphasize that all constants 1 / f ≡ / q s , s ∈ [0 , ∞ ), given by q s ( x ) = s − (1 / ( x ) , x ∈ (0 , π ) , (1.5)as illustrated by Rosen and Morse [55] in 1932, P¨oschl and Teller [53] in 1933,and Lotmar [45] in 1935. These authors are either concerned with the followingextension of (1.5) c sin ( x ) + c cos ( x ) , x ∈ (0 , π/ , (1.6)or its hyperbolic analog of the form c sinh ( x ) + c cosh ( x ) , x ∈ R (or x ∈ (0 , ∞ )) . (1.7)The upshot of these investigations for the purpose at hand was the realizationthat such problems are exactly solvable in terms of the hypergeometric function F ( a, b ; c ; · ) (frequently denoted by F ( a, b ; c ; · )). These types of problems are fur-ther discussed by Infeld and Hull [35] and summarized in [20, Sect. 38, 39, 93], andmore recently in [16]. A discussion of the underlying singular periodic problem (1.5)on R , including the associated Floquet (Bloch) theory, was presented by Scarf [58].These investigations focus on aspects of ordinary differential equations as opposedto operator theory even though Dirichlet problems associated with singular end-points were formally discussed (in this context see also [57]). An operator theoreticapproach for (1.5) and (1.6) over a finite interval bounded by singularities and avariety of associated self-adjoint boundary conditions including coupled boundary REFINEMENT OF HARDY’S INEQUALITY 3 conditions leading to energy bands (Floquet–Bloch theory) in the periodic problemon R , was discussed in in [22] and [26]. Finally, we briefly mention that the case of n -soliton potentials q (1 / n ( x ) = n ( n + 1) / cosh ( x ) , n ∈ N , x ∈ R , (1.8)has received special attention as it represents a solution of infinitely many equa-tions in the stationary Korteweg–de Vries (KdV) hierarchy (starting from level n upward).Introducing the differential expression τ s = − d dx + q s ( x ) = − d dx + s − (1 / ( x ) , x ∈ (0 , π ) , (1.9)the exact solvability of the differential equation τ s y = zy , z ∈ C , or a comparisonwith the well-known Bessel operator case − ( d /dx ) + (cid:2) s − (1 / (cid:3) x − near x = 0and − ( d /dx ) + (cid:2) s − (1 / (cid:3) ( x − π ) − near x = π then yields the nonoscillatoryproperty of τ s if and only if s ∈ [0 , ∞ ). Very roughly speaking, nonnegativity of theFriedrichs extension associated with the differential expression τ − (1 / H ((0 , π )), implies therefinement (1.1) of Hardy’s inequality.In Section 2 we briefly discuss (principal and nonprincipal) solutions of the ex-actly solvable Schr¨odinger equation τ s y = 0 (solutions of the general equation τ s y = zy , z ∈ C , are discussed in Appendix A), and introduce minimal T s,min andmaximal T s,max = T ∗ s,min operators corresponding to τ s as well as the Friedrichsextension T s,F of T s,min and the boundary values associated with T s,max , followingrecent treatments in [24], [25]. Section 3 contains the bulk of this paper and is de-voted to a derivation of inequality (1.1). We also indicate how two related resultsby Avkhadiev and Wirths [7], [8], involving Drichlet boundary conditions on bothends and a mixture of Dirichlet and Neumann boundary conditions naturally fitsinto the framework discussed in this paper. In Appendix A we study solutions of τ s y = zy , z ∈ C , in more detail and also derive the singular Weyl–Titchmarsh–Kodaira m -function associated with T s,F . Finally, Appendix B collects some factson Hardy-type inequalities.2. An Exactly Solvable, Strongly Singular,Periodic Schr¨odinger Operator
In this section we examine a slight variation of the example found in Section 4of [22] by implementing the methods found in [24].Let a = 0, b = π,p ( x ) = r ( x ) = 1 , q s ( x ) = s − (1 / ( x ) , s ∈ [0 , ∞ ) , x ∈ (0 , π ) . (2.1)We now study the Sturm–Liouville operators associated with the correspondingdifferential expression given by τ s = − d dx + q s ( x ) = − d dx + s − (1 / ( x ) , s ∈ [0 , ∞ ) , x ∈ (0 , π ) , (2.2)which is in the limit circle case at the endpoints x = 0 , π for s ∈ [0 , s ∈ [1 , ∞ ). The maximal and preminimal operators, F. GESZTESY, M. M. H. PANG, AND J. STANFILL T s,max and . T s,min , associated to τ s in L ((0 , π ); dx ) are then given by T s,max f = τ s f, s ∈ [0 , ∞ ) ,f ∈ dom( T s,max ) = (cid:8) g ∈ L ((0 , π ); dx ) (cid:12)(cid:12) g, g [1] ∈ AC loc ((0 , π )); (2.3) τ s g ∈ L ((0 , π ); dx ) (cid:9) , and . T s,min f = τ s f, s ∈ [0 , ∞ ) ,f ∈ dom (cid:0) . T s,min (cid:1) = (cid:8) g ∈ L ((0 , π ); dx ) (cid:12)(cid:12) g, g [1] ∈ AC loc ((0 , π )); (2.4)supp ( g ) ⊂ (0 , π ) is compact; τ s g ∈ L ((0 , π ); dx ) (cid:9) . Since q s ∈ L loc ((0 , π ); dx ) , s ∈ [0 , ∞ ) , (2.5)one can replace . T s,min by .. T s,min = τ s (cid:12)(cid:12) C ∞ ((0 ,π )) , s ∈ [0 , ∞ ) . (2.6)For s ∈ [0 , u ,s (0 , · ) and b u ,s (0 , · ) of τ s u = 0 at x = 0 by u ,s (0 , x ) = [sin( x )] (1+2 s ) / F (cid:0) (1 /
4) + ( s/ , (1 /
4) + ( s/ s ; sin ( x ) (cid:1) ,s ∈ [0 , , b u ,s (0 , x ) = (2 s ) − [sin( x )] (1 − s ) / × F (cid:0) (1 / − ( s/ , (1 / − ( s/ − s ; sin ( x ) (cid:1) , s ∈ (0 , , [sin( x )] / F (cid:0) / , /
4; 1; sin ( x ) (cid:1) × ˆ cx dx ′ [sin( x ′ )] − (cid:2) F (cid:0) / , /
4; 1; sin ( x ′ ) (cid:1)(cid:3) − , s = 0 , (2.7)and principal and nonprincipal solutions u π,s (0 , · ) and b u π,s (0 , · ) of τ s u = 0 at x = π by u π,s (0 , x ) = [sin( x )] (1+2 s ) / F (cid:0) (1 /
4) + ( s/ , (1 /
4) + ( s/ s ; sin ( x ) (cid:1) ,s ∈ [0 , , b u π,s (0 , x ) = − (2 s ) − [sin( x )] (1 − s ) / × F (cid:0) (1 / − ( s/ , (1 / − ( s/ − s ; sin ( x ) (cid:1) , s ∈ (0 , , − [sin( x )] / F (cid:0) / , /
4; 1; sin ( x ) (cid:1) × ˆ xc dx ′ [sin( x ′ )] − (cid:2) F (cid:0) / , /
4; 1; sin ( x ′ ) (cid:1)(cid:3) − , s = 0 . (2.8)Here F ( · , · , · ; · ) (frequently written as F ( · , · ; · ; · )) denotes the hypergeometricfunction (see, e.g., [1, Ch. 15]). Remark . We note that the case c = 1 in F ( a, b ; c ; ξ ), corresponding to thecase s = 0 in (2.7), (2.8), is a special one in the sense that linearly independent REFINEMENT OF HARDY’S INEQUALITY 5 solutions of the hypergeometric differential equation are then of the form (see, e.g.,[1, Nos. 15.5.16, 15.5.17]) y ( ξ ) = F ( a, b ; 1; ξ ) ,y ( ξ ) = F ( a, b ; 1; ξ )ln( ξ ) (2.9)+ X n ∈ N ( a ) n ( b ) n ( n !) [ ψ ( a + n ) − ψ ( a ) + ψ ( b + n ) − ψ ( b ) + 2 ψ (1) − ψ ( n + 1)] ξ n . Here ( d ) n , n ∈ N , represents Pochhammer’s symbol (see, (A.8)), and ψ ( · ) denotesthe Digamma function. Since we wanted to ensure (2.12) and our principal aimin connection with the boundary values (2.14)–(2.17) was the derivation of theasymptotic relations (2.10) and (2.11), our choice of b u , and b u π, in (2.7) and (2.8)is to be preferred over the use of the pair of functions in (2.9). For more details inthis connection see Appendix A. ⋄ Since u ,s (0 , x ) = x ↓ x (1+2 s ) / (cid:8) (cid:2)(cid:0) s − (cid:1) / (cid:0)
48 + 48 s (cid:1)(cid:3) x + O (cid:0) x (cid:1)(cid:9) , s ∈ [0 , , b u ,s (0 , x ) = x ↓ (2 s ) − x (1 − s ) / (cid:8) (cid:2)(cid:0) s − (cid:1) / (cid:0) − s (cid:1)(cid:3) x + O (cid:0) x (cid:1)(cid:9) ,s ∈ (0 , , ln(1 /x ) x / (cid:8) (cid:2)(cid:0) [ln( x )] − − (cid:1) / (cid:3) x + O (cid:0) x (cid:1)(cid:9) , s = 0 , (2.10) u π,s (0 , x ) = x ↑ π ( π − x ) (1+2 s ) / (cid:8) (cid:2)(cid:0) s − (cid:1) / (cid:0)
48 + 48 s (cid:1)(cid:3) ( π − x ) + O (cid:0) ( π − x ) (cid:1)(cid:9) , s ∈ [0 , , b u π,s (0 , x ) = x ↑ π − (2 s ) − ( π − x ) (1 − s ) / (cid:8) (cid:2)(cid:0) s − (cid:1) / (cid:0) − s (cid:1)(cid:3) ( π − x ) + O (cid:0) ( π − x ) (cid:1)(cid:9) , s ∈ (0 , , ln( π − x )( π − x ) / (cid:8) (cid:2)(cid:0) [ln( π − x )] − − (cid:1) / (cid:3) ( π − x ) + O (cid:0) ( π − x ) (cid:1)(cid:9) , s = 0 , (2.11)one deduces that W ( b u ,s (0 , · ) , u ,s (0 , · ))(0) = 1 = W ( b u π,s (0 , · ) , u π,s (0 , · ))( π ) , s ∈ [0 , , (2.12)and lim x ↓ u ,s (0 , x ) b u ,s (0 , x ) = 0 , lim x ↑ π u π,s (0 , x ) b u π,s (0 , x ) = 0 , s ∈ [0 , . (2.13)The generalized boundary values for g ∈ dom( T s,max ) (the maximal operatorassociated with τ s ) are then of the form e g (0) = ( lim x ↓ g ( x ) / (cid:2) (2 s ) − x (1 − s ) / (cid:3) , s ∈ (0 , , lim x ↓ g ( x ) / (cid:2) x / ln(1 /x ) (cid:3) , s = 0 , (2.14) e g ′ (0) = ( lim x ↓ (cid:2) g ( x ) − e g (0)(2 s ) − x (1 − s ) / (cid:3)(cid:14) x (1+2 s ) / , s ∈ (0 , , lim x ↓ (cid:2) g ( x ) − e g (0) x / ln(1 /x ) (cid:3)(cid:14) x / , s = 0 , (2.15) F. GESZTESY, M. M. H. PANG, AND J. STANFILL e g ( π ) = ( lim x ↑ π g ( x ) / (cid:2) − (2 s ) − ( π − x ) (1 − s ) / (cid:3) , s ∈ (0 , , lim x ↑ π g ( x ) / (cid:2) ( π − x ) / ln( π − x ) (cid:3) , s = 0 , (2.16) e g ′ ( π ) = ( lim x ↑ π (cid:2) g ( x ) + e g ( π )(2 s ) − ( π − x ) (1 − s ) / (cid:3)(cid:14) ( π − x ) (1+2 s ) / , s ∈ (0 , , lim x ↑ π (cid:2) g ( x ) − e g (0)( π − x ) / ln( π − x ) (cid:3)(cid:14) ( π − x ) / , s = 0 . (2.17)As a result, the minimal operator T s,min associated to τ s , that is, T s,min = . T s,min = .. T s,min , s ∈ [0 , ∞ ) , (2.18)is thus given by T s,min f = τ s f,f ∈ dom( T s,min ) = (cid:8) g ∈ L ((0 , π ); dx ) (cid:12)(cid:12) g, g [1] ∈ AC loc ((0 , π )); (2.19) e g (0) = e g ′ (0) = e g ( π ) = e g ′ ( π ) = 0; τ s g ∈ L ((0 , π ); dx ) (cid:9) , s ∈ [0 , , and satisfies T ∗ s,min = T s,max , T ∗ s,max = T s,min , s ∈ [0 , ∞ ). Due to the limit pointproperty of τ s at x = 0 and x = π if and only if s ∈ [1 , ∞ ), one concludes that T s,min = T s,max if and only if s ∈ [1 , ∞ ) . (2.20)The Friedrichs extension T s,F of T s,min , s ∈ [0 , T s,F f = τ s f, f ∈ dom( T s,F ) = (cid:8) g ∈ dom( T s,max ) (cid:12)(cid:12) e g (0) = e g ( π ) = 0 (cid:9) , s ∈ [0 , , (2.21)moreover, T s,F = T s,min = T s,max , s ∈ [1 , ∞ ) , (2.22)is self-adjoint (resp., . T min,s and .. T min,s , s ∈ [1 , ∞ ), are essentially self-adjoint)in L ((0 , π ); dx ). In this case the Friedrichs boundary conditions in (2.21) areautomatically satisfied and hence can be omitted.By (A.22) one hasinf( σ ( T s,F )) = [(1 /
2) + s ] , s ∈ [0 , ∞ ) , (2.23)in particular, T s,F ≥ [(1 /
2) + s ] I (0 ,π ) , s ∈ [0 , ∞ ) , (2.24)with I (0 ,π ) abbreviating the identity operator in L ((0 , π ); dx ).All results on 2nd order differential operators employed in this section can befound in classical sources such as [2, Sect. 129], [13, Chs. 8, 9], [17, Sects. 13.6, 13.9,13.0], [36, Ch. III], [46, Ch. V], [48], [51, Ch. 6], [59, Ch. 9], [60, Sect. 8.3], [61,Ch. 13], [62, Chs. 4, 6–8]. In addition, [24] and [30, Ch. 13] contain very detailedlists of references in this context. REFINEMENT OF HARDY’S INEQUALITY 7 A Refinement of Hardy’s Inequality
The principal purpose of this section is to derive a refinement of the classicalHardy inequality ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | x , f ∈ H ((0 , π )) , (3.1)as well as of one of its well-known extensions in the form ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | d (0 ,π ) ( x ) , f ∈ H ((0 , π )) , (3.2)were d (0 ,π ) ( x ) represents the distance from x ∈ (0 , π ) to the boundary { , π } of theinterval (0 , π ), that is, d (0 ,π ) ( x ) = ( x, x ∈ (0 , π/ ,π − x, x ∈ [ π/ , π ) . (3.3)The constant 1 / f ≡ Theorem 3.1.
Let f ∈ H ((0 , π )) . Then, ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | sin ( x ) + 14 ˆ π dx | f ( x ) | , (3.4) where both constants / in (3.4) are optimal. In addition, the inequality is strictin the sense that equality holds in (3.4) if and only if f ≡ .Proof. By Section 2 for s = 0 and by [22, Sect. 4] for s ∈ (0 , ∞ ), one has (cid:18) − d dx + s − (1 / ( x ) − [(1 /
2) + s ] I (0 ,π ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) C ∞ ((0 ,π )) ≥ , s ∈ [0 , ∞ ) . (3.5)Thus, setting s = 0 in (3.5) yields ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | sin ( x ) + 14 ˆ π dx | f ( x ) | , f ∈ C ∞ ((0 , π )) . (3.6)Now denote by H ((0 , π )) the standard Sobolev space on (0 , π ) obtained uponcompletion of C ∞ ((0 , π )) in the norm of H ((0 , π )). Since C ∞ ((0 , π )) is dense in H ((0 , π )), given f ∈ H ((0 , π )), there exists a sequence { f n } n ∈ N ⊂ C ∞ ((0 , π )) suchthat lim n →∞ k f n − f k H ((0 ,π )) = 0. Hence, one can find a subsequence { f n p } p ∈ N of { f n } n ∈ N such that f n p converges to f n pointwise almost everywhere on (0 , π ).Thus an application of Fatou’s lemma (cf., e.g., [21, Corollary 2.19]) yields that(3.6) extends to f ∈ H ((0 , π )), namely,14 ˆ π dx | f ( x ) | sin ( x ) + 14 ˆ π dx | f ( x ) | ≤ lim inf p →∞ ˆ π dx | f n p ( x ) | sin ( x ) + lim inf p →∞ ˆ π dx | f n p ( x ) | (by Fatou’s lemma) F. GESZTESY, M. M. H. PANG, AND J. STANFILL ≤ lim inf p →∞ (cid:26) ˆ π dx | f n p ( x ) | sin ( x ) + 14 ˆ π dx | f n p ( x ) | (cid:27) ≤ lim inf p →∞ ˆ π dx | f ′ n p ( x ) | (by (3.6))= lim p →∞ ˆ π dx | f ′ n p ( x ) | = ˆ π dx | f ′ ( x ) | . (3.7)The substitution s is in (2.7) results in solutions that have oscillatory behaviordue to the factor [sin( x )] ± is in (A.3), (A.7), rendering all solutions of τ s y ( λ, · ) = λy ( λ, · ) oscillatory for each λ ∈ R if and only if s <
0. Classical oscillationtheory results (see, e.g., [24, Theorem 4.2]) prove that . T s,min , and hence T s,min arebounded from below if and only if s ∈ [0 , ∞ ). This proves that the first constant1 / / ε > ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | sin ( x ) + (cid:18)
14 + ε (cid:19) ˆ π dx | f ( x ) | , f ∈ dom( T ,min ) . (3.8)Upon integrating by parts in the left-hand side of (3.8) this implies T ,min ≥ (cid:18)
14 + ε (cid:19) I (0 ,π ) , (3.9)implying T ,F ≥ (cid:18)
14 + ε (cid:19) I (0 ,π ) (3.10)(as T ,min and T ,F share the same lower bound by general principles), contradicting(2.23) for s = 0. Hence also the 2nd constant 1 / f
0: Arguing again bycontradiction, we suppose there exists 0 = f ∈ H ((0 , π )) such that ˆ π dx | f ′ ( x ) | = 14 ˆ π dx | f ( x ) | sin ( x ) + 14 ˆ π dx | f ( x ) | . (3.11)Since H ((0 , π )) ⊆ dom (cid:0) T / s,F (cid:1) , s ∈ [0 , ∞ ) (in fact, one even has the equality H ((0 , π )) = dom (cid:0) T / s,F (cid:1) for all s ∈ (0 , ∞ ), see, e.g., [4], [11], [15], [28], [38], [40]),one concludes via (3.11) that (cid:0) T / ,F f , T / ,F f (cid:1) L ((0 ,π ); dx ) = (1 / k f k L ((0 ,π ); dx ) . (3.12)Moreover, since T ,F is self-adjoint with purely discrete and necessarily simple spec-trum, T ,F has the spectral representation T ,F = X n ∈ N λ n P n , λ = 1 / < λ < λ < · · · , (3.13) REFINEMENT OF HARDY’S INEQUALITY 9 where σ ( T ,F ) = { λ n } n ∈ N and P n are the one-dimensional projections onto theeigenvectors associated with the eigenvalues λ n , n ∈ N , explicitly listed in (A.22),in particular, λ = 1 /
4. Thus, (cid:0) T / ,F f , T / ,F f (cid:1) L ((0 ,π ); dx ) = X n ∈ N λ n ( f , P n f ) L ((0 ,π ); dx ) > λ X n ∈ N ( f , P n f ) L ((0 ,π ); dx ) = λ k f k L ((0 ,π ); dx ) = (1 / k f k L ((0 ,π ); dx ) (3.14)contradicting (3.12) unless P n f = 0 , n ∈ N , and hence, P f = f , (3.15)that is, f ∈ dom( T ,F ) and T ,F f = (1 / f , (3.16)employing λ = 1 /
4. However, (3.16) implies that f ( x ) = x ↓ cx / (cid:2) O (cid:0) x (cid:1)(cid:3) and hence, f / ∈ H ((0 , π )) , (3.17)a contradiction. (cid:3) Remark . ( i ) That inequality (3.4) represents an improvement over the previ-ously well-known cases (3.1) and (3.2) can be shown as follows: Since triviallysin( x ) ≤ x, x ∈ [0 , π ] , (3.18)inequality (3.4) is obviously an improvement over the classical Hardy inequality(3.1). On the other hand, since alsosin( x ) ≤ ( x, x ∈ [0 , π/ ,π − x, x ∈ [ π/ , π ] , (3.19)that is (cf. (3.3)), sin( x ) ≤ d (0 ,π ) ( x ) , x ∈ [0 , π ] , (3.20)inequality (3.4) also improves upon the refinement (3.2).( ii ) Assuming a, b ∈ R , a < b , the elementary change of variables(0 , π ) ∋ x ξ ( x ) = [( b − a ) x + aπ ] /π ∈ ( a, b ) ,f ( x ) = F ( ξ ) , (3.21)yields ˆ ba dξ | F ′ ( ξ ) | ≥ π b − a ) ˆ ba dξ | F ( ξ ) | sin ( π ( ξ − a ) / ( b − a ))+ π b − a ) ˆ ba dξ | F ( ξ ) | , F ∈ H (( a, b )) . (3.22)These scaling arguments apply to all Hardy-type inequalities considered in thispaper and hence it suffices to restrict ourselves to convenient fixed intervals suchas (0 , π ), etc. ⋄ A closer inspection of the proof of Theorem 3.1 reveals that [sin( x )] − is justa very convenient choice for a function that has inverse square singularities at theinterval endpoints as it leads to explicit optimal constants 1 / ω = − d dx − x , α = ddx − x , α +0 = − ddx − x , x ∈ (0 , π ) , (3.23)such that α +0 α = ω . (3.24)The minimal and maximal L ((0 , π ); dx )-realizations associated with ω are thengiven by S ,min f = ω f,f ∈ dom( S ,min ) = (cid:8) g ∈ L ((0 , π ); dx ) (cid:12)(cid:12) g, g [1] ∈ AC loc ((0 , π )); (3.25)supp ( g ) ⊂ (0 , π ) is compact; ω g ∈ L ((0 , π ); dx ) (cid:9) .S ,max f = ω f,f ∈ dom( S ,max ) = (cid:8) g ∈ L ((0 , π ); dx ) (cid:12)(cid:12) g, g [1] ∈ AC loc ((0 , π )); (3.26) ω g ∈ L ((0 , π ); dx ) (cid:9) , implying S ∗ ,min = S ,max , S ∗ ,max = S ,min , and we also introduce the followingself-adjoint extensions of S ,min , respectively, restrictions of S ,max (see, e.g., [3],[5], [4], [11], [15], [19], [24], [28], [38], [40], [56]), S ,D,N f = ω f,f ∈ dom( S ,D,N ) = { g ∈ dom( S ,max ) | e g (0) = g ′ ( π ) = 0 } (3.27)= (cid:8) g ∈ dom( S ,max ) (cid:12)(cid:12) g ′ ( π ) = 0; α g ∈ L ((0 , π ); dx ) (cid:9) ,S ,F f = ω f,f ∈ dom( S ,F ) = { g ∈ dom( S ,max ) | e g (0) = g ( π ) = 0 } (3.28)= (cid:8) g ∈ dom( S ,max ) (cid:12)(cid:12) g ( π ) = 0; α g ∈ L ((0 , π ); dx ) (cid:9) , with S ,F the Friedrichs extension of S ,min . The quadratic forms correspondingto S ,D,N and S ,F are of the form Q S ,D,N ( f, g ) = ( α f, α g ) L ((0 ,π ); dx ) ,f, g ∈ dom( Q S ,D,N ) = (cid:8) g ∈ L ((0 , π ); dx ) (cid:12)(cid:12) g ∈ AC loc ((0 , π )); g ′ ( π ) = 0 , (3.29) α g ∈ L ((0 , π ); dx ) (cid:9) ,Q S ,F ( f, g ) = ( α f, α g ) L ((0 ,π ); dx ) ,f, g ∈ dom( Q S ,F ) = (cid:8) g ∈ L ((0 , π ); dx ) (cid:12)(cid:12) g ∈ AC loc ((0 , π )); g ( π ) = 0 , (3.30) α g ∈ L ((0 , π ); dx ) (cid:9) . One verifies (see (B.6)) that for all ε > g ∈ AC loc ((0 , ε )), α g ∈ L ((0 , ε ); dx ) implies e g (0) = 0 . (3.31)By inspection, f ( λ, x ) = x / J (cid:0) λ / x (cid:1) , x ∈ (0 , π ) , = x ↓ x / (cid:2) O (cid:0) x (cid:1)(cid:3) (3.32) REFINEMENT OF HARDY’S INEQUALITY 11 (where J ν ( · ) denotes the standard Bessel function of order ν ∈ C , cf. [1, Ch. 9])),satisfies (cf. (3.27), (3.28)) e f (0) = 0 . (3.33)Thus, introducing Lamb’s constant, now denoted by λ / D,N, , as the first positivezero of (0 , ∞ ) ∋ x J ( x ) + 2 xJ ( x ) (3.34)(see the brief discussion in [7]), one infers that λ D,N, /π is the first positive zeroof f ′ ( · , π ), that is, f ′ ( λ D,N, /π , π ) = 0 . (3.35)In addition denoting by λ F, /π the first strictly positive zero of f ( · , π ) one has f ( λ F, /π , π ) = 0 , (3.36)and hence λ D,N, /π and λ F, /π are the first eigenvalue of the mixed Dirich-let/Neumann operator S ,D,N and the Dirichlet operator (the Friedrichs extensionof S ,min ) S ,F , respectively . Equivalently,inf( σ ( S ,D,N )) = λ D,N, π − , inf( σ ( S ,F )) = λ F, π − , (3.37)in particular, S ,D,N ≥ λ D,N, π − I L ((0 ,π ); dx ) , S ,F ≥ λ F, π − I L ((0 ,π ); dx ) , (3.38) Q S ,D,N ( f, f ) ≥ λ D,N, π − k f k L ((0 ,π ); dx ) , f ∈ dom( Q S ,D,N ) , (3.39) Q S ,F ( f, f ) ≥ λ F, π − k f k L ((0 ,π ); dx ) , f ∈ dom( Q S ,F ) . (3.40)Numerically, one confirms that λ D,N, = 0 . ..., λ F, = 5 . ... . (3.41)Thus, arguments analogous to the ones in the proof of Theorem 3.1 yield thefollowing variants of (3.1), (3.2), ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | x + λ D,N, π ˆ π dx | f ( x ) | ,f ∈ dom( Q S ,D,N ) ∩ H ((0 , π )) , (3.42) ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | x + λ F, π ˆ π dx | f ( x ) | ,f ∈ H ((0 , π )) , (3.43)as well as, ˆ π dx | f ′ ( x ) | ≥ ˆ π dx | f ( x ) | d (0 ,π ) ( x ) + 4 λ D,N, π ˆ π dx | f ( x ) | ,f ∈ H ((0 , π )) . (3.44)All constants in (3.42)–(3.44) are optimal and the inequalities are all strict (for f A in the complex, separable Hilbert space H is a In particular, λ D,N, and λ F, are the first eigenvalue of the mixed Dirichlet/Neumann andDirichlet operator on the interval (0 , form core for A , equivalently, a core for | A | / . In addition, we used (cf. [19, The-orem 7.1]) that for f ∈ dom( S ,D,N ) ∪ dom( S ,F ), there exists K ( f ) ∈ C suchthatlim x ↓ x − / f ( x ) = K ( f ) , lim x ↓ x / f ′ ( x ) = K ( f ) / , lim x ↓ f ( x ) f ′ ( x ) = K ( f ) / . (3.45)Moreover, if in addition f ′ ∈ L ((0 , dx ), f (0) = 0, combining (3.45) with esti-mate (B.10) yields K ( f ) = 0, and hence lim x ↓ f ( x ) f ′ ( x ) = 0. This permits oneto integrate by parts in Q S ,D,N ( f, f ) and Q S ,F ( f, f ) and in the process verify(3.42)–(3.44).We note that inequalities (3.42) and (3.44) were first derived by Avkhadiev andWirths [7] (also recorded in [8] and [9, Sect. 3.6.3]; see also [6], [34], [47]) followinga different approach applicable to the multi-dimensional case. We have not foundinequality (3.43) in the literature, but expect it to be known. Remark . The arguments presented thus far might seem to indicate that Hardy-type inequalities are naturally associated with underlying second-order differentialoperators satisfying boundary conditions of the Dirichlet and/or Neumann type atthe interval endpoints. However, this is not quite the case as the following result(borrowed, e.g., from [14, Lemma 5.3.1], [49, Sect. 1.1]) shows: Suppose b ∈ (0 , ∞ ), f ∈ AC loc ((0 , b )), f ′ ∈ L ((0 , b ); dx ), f (0) = 0, then (with f real-valued withoutloss of generality), ˆ b dx | f ′ ( x ) | = ˆ b dx (cid:12)(cid:12) x / (cid:2) x − / f ( x ) (cid:3) ′ + (2 x ) − f ( x ) (cid:12)(cid:12) = ˆ b dx n − x − f ( x ) + x − / f ( x ) (cid:2) x − / f ( x ) (cid:3) ′ + x (cid:2)(cid:0) x − / f ( x ) (cid:1) ′ (cid:3) o ≥ ˆ b dx n − x − f ( x ) + x − / f ( x ) (cid:2) x − / f ( x ) (cid:3) ′ o = ˆ b dx | f ( x ) | x + 2 − (cid:2) x − / f ( x ) (cid:3) (cid:12)(cid:12) bx =0 ≥ ˆ b dx | f ( x ) | x − − lim x ↓ (cid:2) x − / f ( x ) (cid:3) = ˆ b dx | f ( x ) | x , (3.46)employing the estimate (B.10) with f (0) = 0. In particular, no boundary conditionswhatsoever are needed at the right end point b . One notes that the hypotheses on f imply that f ∈ AC ([0 , b ]) and hence actually that f behaves like an H -functionin a right neighborhood of x = 0, equivalently, f e χ [0 ,b/ ∈ H ((0 , b )), where e χ [0 ,r/ ( x ) = ( , x ∈ [0 , r/ , , x ∈ [3 r/ , r ] , e χ [0 ,r/ ∈ C ∞ ([0 , r ]) , r ∈ (0 , ∞ ) . (3.47) ⋄ Remark . Employing locality of the operators involved, one can show (cf. [27])that all considerations in the bulk of this paper, extend to the situation where q ( x ) = s − (1 / x , respectively, q ( x ) = s − (1 / ( x ) , s ∈ [0 , ∞ ) , (3.48) REFINEMENT OF HARDY’S INEQUALITY 13 is replaced by a potential q satisfying q ∈ L loc ((0 , π ); dx ) and for some s j ∈ [0 , ∞ ), j = 1 ,
2, and some 0 < ε sufficiently small, q s ,s ( x ) = ((cid:2) s − (1 / (cid:3) x − , x ∈ (0 , ε ) , (cid:2) s − (1 / (cid:3) ( x − π ) − , x ∈ ( π − ε, π ) . (3.49)As discussed in [38], this can be replaced by q ≥ q s ,s a.e.In addition, we only presented the tip of an iceberg in this section as these consid-erations naturally extend to more general Sturm–Liouville operators in L (( a, b ); dx )generated by differential expressions of the type − ddx p ( x ) ddx + q ( x ) , x ∈ ( a, b ) , (3.50)as discussed to some extent in [29]. We will return to this and the general three-coefficient Sturm–Liouville operators in L (( a, b ); rdx ) generated by1 r ( x ) (cid:20) − ddx p ( x ) ddx + q ( x ) (cid:21) , x ∈ ( a, b ) , (3.51)elsewhere. ⋄ Appendix A. The Weyl–Titchmarsh–Kodaira m -FunctionAssociated with T s,F We start by introducing a normalized fundamental system of solutions φ ,s ( z, · )and θ ,s ( z, · ) of τ s u = zu , s ∈ [0 , z ∈ C , satisfying (cf. the generalized boundaryvalues introduced in (2.14), (2.15)) e θ ,s ( z,
0) = 1 , e θ ′ ,s ( z,
0) = 0 , e φ ,s ( z,
0) = 0 , e φ ′ ,s ( z,
0) = 1 , (A.1)with φ ,s ( · , x ) and θ ,s ( · , x ) entire for fixed x ∈ (0 , π ). To this end, we introducethe two linearly independent solutions to τ s y = zy (entire w.r.t. z for fixed x ∈ (0 , π )) given by y ,s ( z, x ) = [sin( x )] (1 − s ) / × F (cid:0)(cid:2) (1 / − s + z / (cid:3)(cid:14) , (cid:2) (1 / − s − z / (cid:3)(cid:14)
2; 1 /
2; cos ( x ) (cid:1) ,y ,s ( z, x ) = cos( x )[sin( x )] (1 − s ) / (A.2) × F (cid:0)(cid:2) (3 / − s + z / (cid:3)(cid:14) , (cid:2) (3 / − s − z / (cid:3)(cid:14)
2; 3 /
2; cos ( x ) (cid:1) ,s ∈ [0 , , z ∈ C , x ∈ (0 , π ) . Using the connection formula found in [1, Eq. 15.3.6] yields the behavior near x = 0 , π,y ,s ( z, x ) = [sin( x )] (1 − s ) / π / Γ( s )Γ (cid:0)(cid:2) (1 /
2) + s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 /
2) + s − z / (cid:3)(cid:14) (cid:1) × F (cid:0)(cid:2) (1 / − s + z / (cid:3)(cid:14) , (cid:2) (1 / − s − z / (cid:3)(cid:14)
2; 1 − s ; sin ( x ) (cid:1) + [sin( x )] (1+2 s ) / π / Γ( − s )Γ (cid:0)(cid:2) (1 / − s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 / − s − z / (cid:3)(cid:14) (cid:1) × F (cid:0)(cid:2) (1 /
2) + s + z / (cid:3)(cid:14) , (cid:2) (1 /
2) + s − z / (cid:3)(cid:14)
2; 1 + s ; sin ( x ) (cid:1) ,s ∈ (0 , , z ∈ C , x ∈ (0 , π ) , (A.3) y ,s ( z, x ) = cos( x )[sin( x )] (1 − s ) / π / Γ( s )2Γ (cid:0)(cid:2) (3 /
2) + s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 /
2) + s − z / (cid:3)(cid:14) (cid:1) × F (cid:0)(cid:2) (3 / − s + z / (cid:3)(cid:14) , (cid:2) (3 / − s − z / (cid:3)(cid:14)
2; 1 − s ; sin ( x ) (cid:1) + cos( x )[sin( x )] (1+2 s ) / × π / Γ( − s )2Γ (cid:0)(cid:2) (3 / − s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 / − s − z / (cid:3)(cid:14) (cid:1) × F (cid:0)(cid:2) (3 /
2) + s + z / (cid:3)(cid:14) , (cid:2) (3 /
2) + s − z / (cid:3)(cid:14)
2; 1 + s ; sin ( x ) (cid:1) ,s ∈ (0 , , z ∈ C , x ∈ (0 , π ) \{ π/ } . Remark
A.1 . Before we turn to the case s = 0, we recall Gauss’s identity (cf. [1,no. 15.1.20]) F ( a, b ; c ; 1) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) , c ∈ C \{− N } , Re( c − a − b ) > , (A.4)and the differentiation formula (cf. [1, no. 15.2.1]) ddz F ( a, b ; c ; z ) = abc F ( a +1 , b +1; c +1; z ) , a, b, c ∈ C , z ∈ { ζ ∈ C | | ζ | < } , (A.5)which imply that for s ∈ (0 , F ( · , · ; · ; 1) exist in (A.2) (indeed, for j = 1 , s ∈ (0 ,
1) that c − a − b = s >
0) and hencethe asymptotic behavior of y j,s ( z, x ), j = 1 ,
2, as x ↓ x ↑ π is dominatedby x (1 − s ) / and ( π − x ) (1 − s ) / , respectively. However, the analogous statementfails for y ′ j,s ( z, x ), j = 1 ,
2, as, taking into account (A.5), the analog of the 2ndcondition in (A.4), namely, Re[ c + 1 − ( a + 1) − ( b + 1)] >
0, is not fulfilled (inthis case (A.2) with s ∈ (0 ,
1) yields [ c + 1 − ( a + 1) − ( b + 1)] = s − < F ( · , · ; · ; x ) for y ,s ( z, x ) in (A.3) as x → π/ F ( · , · ; · ; 1) exist. Even though for y ,s ( z, x ) in (A.3) thetwo F ( · , · ; · ; 1) do not exist individually, the limit of each term does exist due tothe multiplication by the factor cos( x ). To see this, one can instead consider thelimit (cf. [50, no. 15.4.23])lim z → − F ( a, b ; c ; z )(1 − z ) c − a − b = Γ( c )Γ( a + b − c )Γ( a )Γ( b ) , c ∈ C \{− N } , Re( c − a − b ) < , (A.6)which through the appropriate change of variable reveals that the connection for-mula for y ,s ( z, x ) as x → π/ y ,s ( z, π/ c + 1 − ( a + 1) − ( b + 1)] >
0, fails for the four F ′ ( · , · ; · ; x ) in (A.3) as x → π/ ⋄ Similarly, by [1, Eq. 15.3.10] one obtains for the remaining case s = 0, y , ( z, x ) = π / [sin( x )] / Γ (cid:0)(cid:2) (1 /
2) + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 / − z / (cid:3)(cid:14) (cid:1) × ∞ X n =0 (cid:0)(cid:2) (1 /
2) + z / (cid:3)(cid:14) (cid:1) n (cid:0)(cid:2) (1 / − z / (cid:3)(cid:14) (cid:1) n ( n !) (cid:2) ψ ( n + 1) − ψ (cid:0) n + (cid:2) (1 /
2) + z / (cid:3)(cid:14) (cid:1) − ψ (cid:0) n + (cid:2) (1 / − z / (cid:3)(cid:14) (cid:1) − ln(sin ( x )) (cid:3) [sin( x )] n , (A.7) REFINEMENT OF HARDY’S INEQUALITY 15 y , ( z, x ) = π / cos( x )[sin( x )] / (cid:0)(cid:2) (3 /
2) + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 / − z / (cid:3)(cid:14) (cid:1) × ∞ X n =0 (cid:0)(cid:2) (3 /
2) + z / (cid:3)(cid:14) (cid:1) n (cid:0)(cid:2) (3 / − z / (cid:3)(cid:14) (cid:1) n ( n !) (cid:2) ψ ( n + 1) − ψ (cid:0) n + (cid:2) (3 /
2) + z / (cid:3)(cid:14) (cid:1) − ψ (cid:0) n + (cid:2) (3 / − z / (cid:3)(cid:14) (cid:1) − ln (cid:0) sin ( x ) (cid:1)(cid:3) [sin( x )] n ,s = 0 , z ∈ C , x ∈ (0 , π ) . Here ψ ( · ) = Γ ′ ( · ) / Γ( · ) denotes the Digamma function, γ E = − ψ (1) = 0 . . . . represents Euler’s constant, and( ζ ) = 1 , ( ζ ) n = Γ( ζ + n ) / Γ( ζ ) , n ∈ N , ζ ∈ C \ ( − N ) , (A.8)abbreviates Pochhammer’s symbol (see, e.g., [1, Ch. 6]). Direct computation nowyields e y ,s ( z,
0) = − e y ,s ( z, π ) = 2 π / Γ(1 + s )Γ (cid:0)(cid:2) (1 /
2) + s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 /
2) + s − z / (cid:3)(cid:14) (cid:1) , e y ′ ,s ( z,
0) = e y ′ ,s ( z, π ) = π / Γ( − s )Γ (cid:0)(cid:2) (1 / − s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 / − s − z / (cid:3)(cid:14) (cid:1) , e y ,s ( z,
0) = e y ,s ( z, π ) = π / Γ(1 + s )Γ (cid:0)(cid:2) (3 /
2) + s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 /
2) + s − z / (cid:3)(cid:14) (cid:1) , e y ′ ,s ( z,
0) = − e y ′ ,s ( z, π ) = π / Γ( − s )2Γ (cid:0)(cid:2) (3 / − s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 / − s − z / (cid:3)(cid:14) (cid:1) ,s ∈ (0 , , z ∈ C , (A.9) e y , ( z,
0) = − e y , ( z, π ) = 2 π / Γ (cid:0)(cid:2) (1 /
2) + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 / − z / (cid:3)(cid:14) (cid:1) , e y ′ , ( z,
0) = e y ′ , ( z, π )= − π / (cid:2) γ E + ψ (cid:0)(cid:2) (1 /
2) + z / (cid:3)(cid:14) (cid:1) + ψ (cid:0)(cid:2) (1 / − z / (cid:3)(cid:14) (cid:1)(cid:3) Γ (cid:0)(cid:2) (1 /
2) + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 / − z / (cid:3)(cid:14) (cid:1) , e y , ( z,
0) = e y , ( z, π ) = π / Γ (cid:0)(cid:2) (3 /
2) + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 / − z / (cid:3)(cid:14) (cid:1) , e y ′ , ( z,
0) = − e y ′ , ( z, π )= − π / (cid:2) γ E + ψ (cid:0)(cid:2) (3 /
2) + z / (cid:3)(cid:14) (cid:1) + ψ (cid:0)(cid:2) (3 / − z / (cid:3)(cid:14) (cid:1)(cid:3) (cid:0)(cid:2) (3 /
2) + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 / − z / (cid:3)(cid:14) (cid:1) ,s = 0 , z ∈ C . (A.10)In particular, one obtains φ ,s ( z, x ) = e y ,s ( z, y ,s ( z, x ) − e y ,s ( z, y ,s ( z, x ) ,θ ,s ( z, x ) = e y ′ ,s ( z, y ,s ( z, x ) − e y ′ ,s ( z, y ,s ( z, x ) ,s ∈ [0 , , z ∈ C , x ∈ (0 , π ) , (A.11) since W ( y ,s ( z, · ) , y ,s ( z, · )) = e y ,s ( z, e y ′ ,s ( z, − e y ′ ,s ( z, e y ,s ( z,
0) = − , (A.12)with the generalized boundary values given by (A.9), (A.10). To prove (A.12) onerecalls Euler’s reflection formula (cf. [1, no. 6.1.17])Γ( z )Γ(1 − z ) = π sin( πz ) , z ∈ C \ Z , (A.13)and hence concludes thatΓ (cid:0)(cid:2) (1 /
2) + εs ± z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 / − εs ∓ z / (cid:3)(cid:14) (cid:1) (A.14)= π sin (cid:0) π (cid:2) (1 /
2) + ε ± z / (cid:3)(cid:14) (cid:1) , ε ∈ {− , } . (A.15)Thus one computes for s ∈ (0 , W ( y ,s ( z, · ) , y ,s ( z, · ))= − [sin( πs )] − (cid:8) sin (cid:0) π (cid:2) (1 /
2) + s + z / (cid:3)(cid:14) (cid:1) sin (cid:0) π (cid:2) (1 /
2) + s − z / (cid:3)(cid:14) (cid:1) − sin (cid:0) π (cid:2) (1 / − s + z / (cid:3)(cid:14) (cid:1) sin (cid:0) π (cid:2) (1 / − s − z / (cid:3)(cid:14) (cid:1)(cid:9) = − [2 sin( πs )] − {− cos( π [(1 /
2) + s ]) + cos( π [(1 / − s ]) } = − . (A.16)For the case s = 0, one recalls the reflection formula for the Digamma function (cf.[1, no. 6.3.7]) ψ (1 − z ) − ψ ( z ) = π cot( πz ) , z ∈ C \ Z , (A.17)and applies trigonometric identities to obtain W ( y , ( z, · ) , y , ( z, · )) = − m , ,s ( z ) is then uniquely de-termined (cf. [25, Eq. (3.18)] and [24] for background on m -functions) to be m , ,s ( z ) = − e θ ,s ( z, π ) e φ ,s ( z, π ) , s ∈ [0 , , z ∈ ρ ( T s,F ) . (A.18)Direct calculation once again yields m , ( z ) = − e y ′ ,s ( z, e y ,s ( z, π ) − e y ′ ,s ( z, e y ,s ( z, π )2 e y ,s ( z, e y ,s ( z, π Γ( − s )4Γ(1 + s ) (cid:20) Γ (cid:0)(cid:2) (3 /
2) + s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 /
2) + s − z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 / − s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (3 / − s − z / (cid:3)(cid:14) (cid:1) + Γ (cid:0)(cid:2) (1 /
2) + s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 /
2) + s − z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 / − s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 / − s − z / (cid:3)(cid:14) (cid:1) (cid:21) ,s ∈ (0 , , − (cid:2) γ E + ψ (cid:0)(cid:2) (1 /
2) + z / (cid:3)(cid:14) (cid:1) + ψ (cid:0)(cid:2) (1 / − z / (cid:3)(cid:14) (cid:1) + ψ (cid:0)(cid:2) (3 /
2) + z / (cid:3)(cid:14) (cid:1) + ψ (cid:0)(cid:2) (3 / − z / (cid:3)(cid:14) (cid:1)(cid:3) / , s = 0 ,z ∈ ρ ( T s,F ) , (A.19) Here the subscript 0 , m , ,s indicate the Dirichlet (i.e., Friedrichs) boundary conditions at x = 0 , π , a special case of the m α,β -function discussed in [24] associated with separated boundaryconditions at x = 0 , π , indexed by boundary condition parameters α, β ∈ [0 , π ]. REFINEMENT OF HARDY’S INEQUALITY 17 which has simple poles precisely at the simple eigenvalues of T s,F given by σ ( T s,F ) = (cid:8) [(1 /
2) + s + n ] (cid:9) n ∈ N , s ∈ [0 , . (A.20) Remark
A.2 . For the limit point case at both endpoints, that is, for s ∈ [1 , ∞ ), thesolutions y j,s ( z, · ) in (A.2) remain linearly independent and also the connectionformulas (A.3) remain valid for s ∈ [1 , ∞ ) \{ N } . Moreover, employing once again(A.4) and (A.5) one verifies that that the two F ( · , · ; · ; 1) as well as F ′ ( · , · ; · ; 1)are well defined in (A.2) and hence for s ∈ [1 , ∞ ), the asymptotic behavior of y j,s ( z, x ) and y ′ j,s ( z, x ), j = 1 ,
2, as x ↓ x ↑ π is dominated by x (1 − s ) / and x − (1+2 s ) / and ( π − x ) (1 − s ) / and ( π − x ) − (1+2 s ) / , respectively. Since in connectionwith (A.3) one has c − a − b = ± /
2, independently of the value of s ∈ (0 , ∞ ), thesituation described in Remark A.1 for (A.3) and s ∈ (0 ,
1) applies without changeto the current case s ∈ [1 , ∞ ).Actually, some of these failures (as x → π/ y ′ j,s ( z, x ), j = 1 ,
2) are crucial forthe following elementary reason: The function[sin( x )] (1+2 s ) / π / Γ( − s )Γ (cid:0)(cid:2) (1 / − s + z / (cid:3)(cid:14) (cid:1) Γ (cid:0)(cid:2) (1 / − s − z / (cid:3)(cid:14) (cid:1) × F (cid:0)(cid:2) (1 /
2) + s + z / (cid:3)(cid:14) , (cid:2) (1 /
2) + s − z / (cid:3)(cid:14)
2; 1 + s ; sin ( x ) (cid:1) , (A.21) s ∈ [1 , ∞ ) , z ∈ C , x ∈ (0 , π ) , (i.e., the analog of the second part of y ,s ( z, · ) on the right-hand side in (A.3))generates an L ((0 , π ); dx )-element near x = 0 , π , and hence if this function andits x -derivative were locally absolutely continuous in a neighborhood of x = π/ , π )), the self-adjoint maximaloperator T s,max , s ∈ [1 , ∞ ), would have eigenvalues for all z ∈ C , an obviouscontradiction. ⋄ Because of the subtlety pointed out in Remark A.2 we omit further details onthe limit point case s ∈ [1 , ∞ ) and refer to [22, Sect. 4], instead. In particular, [22,Theorem 4.1 b)] extends (A.20) to s ∈ [1 , ∞ ) and hence one actually has σ ( T s,F ) = (cid:8) [(1 /
2) + s + n ] (cid:9) n ∈ N , s ∈ [0 , ∞ ) . (A.22) Appendix B. Remarks on Hardy-Type Inequalities
In this appendix we recall a Hardy-type inequality useful in Section 2.Introducing the differential expressions α s , α + s (cf. (3.23) for s = 0), α s = ddx − s + (1 / x , α + s = − ddx − s + (1 / x , s ∈ [0 , ∞ ) , x ∈ (0 , π ) , (B.1)one confirms that α + s α s = ω s = − d dx + s − (1 / x , s ∈ [0 , ∞ ) , x ∈ (0 , π ) . (B.2)Following the Hardy inequality considerations in [28], [37], [39], one obtains thefollowing basic facts. Lemma B.1.
Suppose f ∈ AC loc ((0 , π )) , α s f ∈ L ((0 , π ); dx ) for some s ∈ R , and < r < r < π < R < ∞ . Then, ˆ r r dx | ( α s f )( x ) | ≥ s ˆ r r dx | f ( x ) | x + 14 ˆ r r dx | f ( x ) | x [ln( R/x )] − s | f ( x ) | x (cid:12)(cid:12)(cid:12)(cid:12) r x = r − | f ( x ) | x [ln( R/x )] (cid:12)(cid:12)(cid:12)(cid:12) r x = r , (B.3) ˆ r r dx x ln( R/x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) f ( x ) x / [ln( R/x )] / (cid:21) ′ (cid:12)(cid:12)(cid:12)(cid:12) = ˆ r r dx (cid:20) | f ′ ( x ) | − | f ( x ) | x − | f ( x ) | x [ln( R/x )] (cid:21) (B.4) − | f ( x ) | x (cid:12)(cid:12)(cid:12)(cid:12) r x = r + | f ( x ) | x ln( R/x ) (cid:12)(cid:12)(cid:12)(cid:12) r x = r ≥ , ˆ r r dx | ( α s f )( x ) | = ˆ r r dx (cid:20) | f ′ ( x ) | + (cid:2) s − (1 / (cid:3) | f ( x ) | x (cid:21) − [ s + (1 / | f ( x ) | x (cid:12)(cid:12)(cid:12)(cid:12) r x = r ≥ . (B.5) If s = 0 , ˆ r dx | f ( x ) | x [ln( R/x )] < ∞ , lim x ↓ | f ( x ) | [ x ln( R/x )] / = 0 . (B.6) If s ∈ (0 , ∞ ) , then ˆ r dx | f ′ ( x ) | < ∞ , ˆ r dx | f ( x ) | x < ∞ , lim x ↓ | f ( x ) | x / = 0 , (B.7) in particular, f e χ [0 ,r / ∈ H ((0 , r )) , (B.8) where e χ [0 ,r/ ( x ) = ( , x ∈ [0 , r/ , , x ∈ [3 r/ , r ] , e χ [0 ,r/ ∈ C ∞ ([0 , r ]) , r ∈ (0 , ∞ ) . (B.9) Proof.
Relations (B.4) and (B.5) are straightforward (yet somewhat tedious) iden-tities; together they yield (B.3). The 1st relation in (B.6) is an instant conse-quence of (B.3), so is the fact that lim x ↓ | f ( x ) | / [ x ln( R/x )] exists. Moreover,since [ x ln( R/x )] − is not integrable at x = 0, the 1st relation in (B.6) yieldslim inf x ↓ | f ( x ) | / [ x ln( R/x )] = 0, implying the 2nd relation in (B.6).Finally, if s ∈ (0 , ∞ ), then inequality (B.3) implies the 2nd relation in (B.7);together with α s f ∈ L ((0 , π ); dx ), this yields the 1st relation in (B.7). By in-equality (B.3), lim x ↓ | f ( x ) | /x exists, but then the second relation in (B.7) yieldslim inf x ↓ | f ( x ) | /x = 0 and hence also lim x ↓ | f ( x ) | /x = 0. (cid:3) We also recall the following elementary fact.
Lemma B.2.
Suppose f ∈ H ((0 , r )) for some r ∈ (0 , ∞ ) . Then, for all x ∈ (0 , r ) , | f ( x ) − f (0) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ x dt f ′ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ x / (cid:18) ˆ x dt | f ′ ( t ) | (cid:19) / ≤ x / k f ′ k L ((0 ,x ); dt ) = x ↓ o (cid:0) x / (cid:1) . (B.10) REFINEMENT OF HARDY’S INEQUALITY 19
Thus, if f ∈ H ((0 , r )) , then ´ r dx | f ( x ) | /x < ∞ if and only if f (0) = 0 , that is,if and only if f e χ [0 ,r/ ∈ H ((0 , r )) .In particular, if f ∈ H ((0 , r )) and f (0) = 0 , then actually, lim x ↓ | f ( x ) | x / = 0 . (B.11) Proof.
Since (B.10) is obvious, we briefly discuss the remaining assertions in LemmaB.2. If f ∈ H ((0 , r )) and ´ r dx | f ( x ) | /x < ∞ then identity (B.5) for s < − / ˆ r r dx | ( α s f )( x ) | = ˆ r r dx (cid:20) | f ′ ( x ) | + (cid:2) s − (1 / (cid:3) | f ( x ) | x (cid:21) − [ s + (1 / | f ( x ) | x (cid:12)(cid:12)(cid:12)(cid:12) r x = r ≥ , s < − / , (B.12)yields the existence of lim x ↓ | f ( x ) | /x . Since ´ r dx | f ( x ) | /x < ∞ implies thatlim inf x ↓ | f ( x ) | /x = 0, one concludes that lim x ↓ | f ( x ) | /x = 0 and hence f be-haves locally like an H -function in a right neighborhood of x = 0. Conversely, if f (0) = 0, then ´ r dx | f ( x ) | /x < ∞ by Hardy’s inequality as discussed in Remark3.3. Relation (B.11) is clear from (B.10) with f (0) = 0. (cid:3) Remark
B.3 . ( i ) If f ∈ AC loc ((0 , r )) and f ′ ∈ L p ((0 , r ); dx ) for some p ∈ [1 , ∞ ),the H¨older estimate analogous to (B.10), | f ( d ) − f ( c ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ dc dt f ′ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | d − c | /p ′ (cid:18) ˆ dc dt | f ′ ( t ) | p (cid:19) /p , ( c, d ) ⊂ (0 , r ) , p + 1 p ′ = 1 , (B.13)implies the existence of lim c ↓ f ( c ) = f (0) and lim d ↑ r f ( d ) = f ( r ) and hence yields f ∈ AC ([0 , r ]).( ii ) The fact that f ∈ H ((0 , r )) and ´ r dx | f ( x ) | /x < ∞ implies f e χ [0 ,r/ ∈ H ((0 , r )) is a special case of a multi-dimensional result recorded, for instance, in[18, Theorem 5.3.4].( iii ) When replacing x − , x ∈ (0 , r ), by [sin( x )] − , x ∈ (0 , π ), due to locality, theconsiderations in Lemmas B.1 and B.2 at the left endpoint x = 0 apply of courseto the right interval endpoint π . ⋄ Acknowledgments.
We are indebted to Jan Derezinski, Aleksey Kostenko, andGerald Teschl for very helpful discussions.
References [1] M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions , Dover, New York,1972.[2] N. I. Akhiezer and I. M. Glazman,
Theory of Linear Operators in Hilbert Space, Volume II ,Pitman, Boston, 1981.[3] V. S. Alekseeva and A. Yu. Ananieva,
On extensions of the Bessel operator on a finite intervaland a half-line , J. Math. Sci. , 1–8, 2012.[4] A. Yu. Anan’eva and V. S. Budyka,
On the spectral theory of the Bessel operator on a finiteinterval and the half-line , Diff. Eq. , 1517–1522 (2016).[5] A. Yu. Ananieva and V. S. Budyika, To the spectral theory of the Bessel operator on finiteinterval and half-line , J. Math. Sci. , 624–645 (2015). [6] F. G. Avkhadiev,
Brezis–Marcus problem and its generalizations , J. Math. Sci. , 291–301(2021).[7] F. G. Avkhadiev and K. J. Wirths,
Unified Poincar´e and Hardy inequalities with sharpconstants for convex domains , Angew. Math. Mech. , 632–642 (2007).[8] F. G. Avkhadiev and K. J. Wirths, Sharp Hardy-type inequalities with Lamb’s constant , Bull.Belg. Math. Soc. Simon Stevin , 723–736 (2011).[9] A. A. Balinsky, W. D. Evans, and R. T. Lewis, The Analysis and Geometry of Hardy’sInequality , Universitext, Springer, 2015.[10] H. Brezis and M. Marcus,
Hardy’s inequalities revisited , Ann. Scuola Norm. Sup. Pisa Cl.Sci. (4) , 217–237 (1997).[11] L. Bruneau, J. Derezi´nski, and V. Georgescu, Homogeneous Schr¨odinger operators on half-line , Ann. H. Poincar´e , 547–590 (2011).[12] R. S. Chisholm, W. N. Everitt, and L. L. Littlejohn, An integral operator inequality withapplications , J. of Inequal. & Applications , 245–266 (1999).[13] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations , Krieger Publ.,Malabar, FL, 1985.[14] E. B. Davies,
Spectral Theory and Differential Operators , Cambridge Studies in AdvancedMathematics, Vol. 42, Cambridge University Press, Cambridge, UK, 1995.[15] J. Derezi´nski and V. Georgescu,
On the domains of Bessel operators , preprint, 2020.[16] J. Derezi´nski and M. Wrochna,
Exactly solvable Schr¨odinger operators , Ann. H. Poincar´e ,397–418 (2011); see also the update at arXiv:1009.0541.[17] N. Dunford and J. T. Schwartz, Linear Operators. Part II: Spectral Theory , Wiley, Inter-science, New York, 1988.[18] D. E. Edmunds and W. D. Evans,
Spectral Theory and Differential Operators , 2nd ed., OxfordUniversity Press, 2018.[19] W. N. Everitt and H. Kalf,
The Bessel differential equation and the Hankel transform , J.Comp. Appl. Math. , 3–19 (2007).[20] S. Fl¨ugge,
Practical Quantum Mechanics , Vol. I, reprinted 2nd 1994 ed., Springer, Berlin,1999.[21] G. B. Folland,
Real Analysis. Modern Techniques and Their Applications , 2nd ed., Wiley-Interscience, New York, 1999.[22] F. Gesztesy and W. Kirsch,
One-dimensional Schr¨odinger operators with interactions singu-lar on a discrete set , J. Reine Angew. Math. , 28–50 (1985).[23] F. Gesztesy, L. L. Littlejohn, I. Michael, and R. Wellman,
On Birman’s sequence of Hardy–Rellich-type inequalities , J. Diff. Eq. , 2761–2801 (2018).[24] F. Gesztesy, L. L. Littlejohn, and R. Nichols,
On self-adjoint boundary conditions for singularSturm–Liouville operators bounded from below , J. Diff. Eq. , 6448–6491 (2020).[25] F. Gesztesy, L. L. Littlejohn, M. Piorkowski, and J. Stanfill,
The Jacobi operator and itsWeyl–Titchmarsh–Kodaira m -functions , preprint 2020.[26] F. Gesztesy, C. Macedo, and L. Streit, An exactly solvable periodic Schr¨odinger operator , J.Phys.
A18 , L503–L507 (1985).[27] F. Gesztesy, M. M. H. Pang, and J. Stanfill,
On domain properties of Bessel-type operators ,in preparation.[28] F. Gesztesy and L. Pittner,
On the Friedrichs extension of ordinary differential operatorswith strongly singular potentials , Acta Phys. Austriaca , 259–268 (1979).[29] F. Gesztesy and M. ¨Unal, Perturbative oscillation criteria and Hardy-type inequalities , Math.Nachr. , 121–144 (1998).[30] F. Gesztesy and M. Zinchenko,
Sturm–Liouville Operators, Their Spectral Theory, and SomeApplications , in preparation.[31] G. R. Goldstein, J. A. Goldstein, R. M. Mininni, and S. Romanelli,
Scaling and variants ofHardy’s inequality , Proc. Amer. Math. Soc. , 1165–1172 (2019).[32] G. H. Hardy,
Notes on some points in the integral calculus, LX. An inequality betweenintegrals , Messenger Math. , 150–156 (1925).[33] G. H. Hardy, J. E. Littlewood, and G. P´olya, Inequalities , Cambridge University Press,Cambridge, UK, reprinted, 1988.[34] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and A. Laptev,
A geometrical version ofHardy’s inequality , J. Funct. Anal. , 539–548 (2002).[35] L. Infeld and T. E. Hull,
The factorization method , Rev. Mod. Phys. , 21–68 (1951). REFINEMENT OF HARDY’S INEQUALITY 21 [36] K. J¨orgens and F. Rellich,
Eigenwerttheorie Gew¨ohnlicher Differentialgleichungen , Springer-Verlag, Berlin, 1976.[37] H. Kalf,
On the characterization of the Friedrichs extension of ordinary or elliptic differentialoperators with a strongly singular potential , J. Funct. Anal. , 230–250 (1972).[38] H. Kalf, A characterization of the Friedrichs extension of Sturm–Liouville operators , J. Lon-don Math. Soc. (2) , 511–521 (1978).[39] H. Kalf and J. Walter, Strongly singular potentials and essential self-adjointness of singularelliptic operators in C ∞ ( R n \{ } ), J. Funct. Anal. , 114–130 (1972).[40] A. Kostenko and G. Teschl, On the singular Weyl–Titchmarsh function of perturbed sphericalSchr¨odinger operators , J. Diff. Eq. , 3701–3739 (2011).[41] A. Kufner, L. Maligranda, and L.-E. Persson,
The Hardy Inequality. About its History andSome Related Results , Vydavatelsk´y Servis, Pilsen, 2007.[42] A. Kufner, L.-E. Persson, and N. Samko,
Weighted Inequalities of Hardy Type , 2nd ed., WorldScientific, Singapore, 2017.[43] A. Kufner,
Weighted Sobolev Spaces , A Wiley-Interscience Publication, John Wiley & Sons,1985.[44] E. Landau,
A note on a theorem concerning series of positive terms: extract from a letter ofProf. E. Landau to Prof. I. Schur , J. London Math. Soc. , 38–39 (1926).[45] W. Lotmar, Zur Darstellung des Potentialverlaufs bei zweiatomigen Molek¨ulen , Z. Physik , 528–533 (1935).[46] M. A. Naimark, Linear Differential Operators. Part II: Linear Differential Operators inHilbert Space , Transl. by E. R. Dawson, Engl. translation edited by W. N. Everitt, UngarPublishing, New York, 1968.[47] R. G. Nasibullin and R. V. Makarov,
Hardy’s inequalities with remainders and Lamb-typeequations , Siberian Math. J. , 1102–1119 (2020).[48] H.-D. Niessen and A. Zettl, Singular Sturm–Liouville problems: the Friedrichs extension andcomparison of eigenvalues , Proc. London Math. Soc. (3) , 545–578 (1992).[49] B. Opic and A. Kufner, Hardy-Type Inequalities , Pitman Research Notes in MathematicsSeries, Vol. 219. Longman Scientific & Technical, Harlow, 1990.[50] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (eds.),
NIST Handbook ofMathematical Functions , National Institute of Standards and Technology (NIST), U.S. Dept.of Commerce, and Cambridge Univ. Press, 2010.[51] D. B. Pearson,
Quantum Scattering and Spectral Theory , Academic Press, London, 1988.[52] L.-E. Persson and S. G. Samko,
A note on the best constants in some Hardy inequalities , J.Math. Inequalities , 437–447 (2015).[53] G. P¨oschl and E. Teller, Bemerkungen zur Quantenmechanik des anharmonischen Oszilla-tors , Zeitschr. Physik , 143–151 (1933).[54] F. Rellich, Halbbeschr¨ankte gew¨ohnliche Differentialoperatoren zweiter Ordnung . Math. Ann. , 343–368 (1951). (German.)[55] N. Rosen and P. M. Morse,
On the Vibrations of Polyatomic Molecules , Phys. Rev. ,210–217 (1932).[56] R. Rosenberger, A new characterization of the Friedrichs extension of semibounded Sturm–Liouville operators , J. London Math. Soc. (2) , 501–510 (1985).[57] F. L. Scarf, Discrete states for singular potential problems , Phys. Rev. , 2170–2176 (1958).[58] F. L. Scarf,
New soluble energy band problem , Phys. Rev. , 1137–1140 (1958).[59] G. Teschl,
Mathematical Methods in Quantum Mechanics. With Applications to Schr¨odingerOperators , 2nd ed., Graduate Studies in Math., Vol. 157, Amer. Math. Soc., RI, 2014.[60] J. Weidmann,
Linear Operators in Hilbert Spaces , Graduate Texts in Mathematics, Vol. 68,Springer, New York, 1980.[61] J. Weidmann,
Lineare Operatoren in Hilbertr¨aumen. Teil II: Anwendungen , Teubner,Stuttgart, 2003.[62] A. Zettl,
Sturm–Liouville Theory , Mathematical Surveys and Monographs, Vol. 121, Amer.Math. Soc., Providence, RI, 2005.
Department of Mathematics, Baylor University, Sid Richardson Bldg., 1410 S. 4thStreet, Waco, TX 76706, USA
Email address : [email protected]
URL : Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
Email address : [email protected] URL : Department of Mathematics, Baylor University, Sid Richardson Bldg., 1410 S. 4thStreet, Waco, TX 76706, USA
Email address : [email protected]
URL ::