A survey of some norm inequalities
aa r X i v : . [ m a t h . F A ] J a n A SURVEY OF SOME NORM INEQUALITIES
FRITZ GESZTESY, ROGER NICHOLS, AND JONATHAN STANFILL
Dedicated with great pleasure to Henk de Snoo on the happy occasion of his 75th birthday
Abstract.
We survey some classical norm inequalities of Hardy, Kallman,Kato, Kolmogorov, Landau, Littlewood, and Rota of the type k Af k X ≤ C k f k X (cid:13)(cid:13) A f (cid:13)(cid:13) X , f ∈ dom (cid:0) A (cid:1) , and recall that under exceedingly stronger hypotheses on the operator A and/or the Banach space X , the optimal constant C in these inequalities di-minishes from 4 (e.g., when A is the generator of a C contraction semigroupon a Banach space X ) all the way down to 1 (e.g., when A is a symmetricoperator on a Hilbert space H ).We also survey some results in connection with an extension of the Hardy–Littlewood inequality involving quadratic forms as initiated by Everitt. Introduction
We dedicate this note with great pleasure to Henk de Snoo, whose exemplaryscholarship over the years deserves our undivided respect and admiration. HappyBirthday, Henk, we hope our modest contribution to norm inequalities will givesome joy.
This is a survey of a number of classical norm inequalities due to Hardy, Kallman,Kato, Kolmogorov, Landau, Littlewood, and Rota of the type k Af k X ≤ C k f k X (cid:13)(cid:13) A f (cid:13)(cid:13) X , f ∈ dom (cid:0) A (cid:1) , (1.1)and some of their higher-order extensions. In particular, we recall that underexceedingly stronger hypotheses on the operator A and/or the Banach space X ,such as,( i ) A is the generator of a C contraction semigroup on a Banach space X ,( ii ) A is the generator of a C contraction semigroup on a Hilbert space H ,( iii ) A is the generator of a C group of isometries on a Banach space X ,( iv ) A = cS ∗ , where c ∈ C and S is maximally symmetric in a Hilbert space butnot self-adjoint,( v ) A is a symmetric operator on a Hilbert space H ,the optimal constant C in these inequalities diminishes from 4 to 1 in the process. Date : February 2, 2021.2020
Mathematics Subject Classification.
Primary: 47A30, 34L40; Secondary: 47B25, 47B44.
Key words and phrases.
Hardy–Littlewood, Kallman–Rota, and Landau–Kolmogorovinequalities.Appeared in
Complex Analysis and Operator Theory , No. 23 (2021). Historically, this type of investigations started with Landau [75] in 1913 whoproved k f ′ k L ∞ ((0 , ∞ ); dx ) ≤ k f k L ∞ ((0 , ∞ ); dx ) k f ′′ k L ∞ ((0 , ∞ ); dx ) , f ∈ W , ∞ ((0 , ∞ )) (1.2)(the constant 4 being optimal), followed by Hardy and Littlewood [49] who derivedthe L -analog of (1.2) in 1932, k f ′ k L ((0 , ∞ ); dx ) ≤ k f k L ((0 , ∞ ); dx ) k f ′′ k L ((0 , ∞ ); dx ) , f ∈ W , ((0 , ∞ )) (1.3)(again, with best possible constant 2). These authors also proved the analogs ofinequalities (1.2) and (1.3) on the whole line R , with (optimal) constants 2 (see alsoHadamard [47] in this context) and 1, respectively, followed by fundamental workof Kolmogorov [61] in 1939. These early investigations led to extensive subsequentwork in this area as will be shown in the bulk of this survey.Inequalities of the type (1.2), (1.3) were abstracted in the form (1.1) with C = 4by Kallman and Rota [56] in the context where A is the generator of a C contractionsemigroup on a Banach space X . That the constant can be diminished from 4 to 2in the Hilbert space context was shown by Kato [58] in 1971. Again, this markedthe beginning of numerous subsequent works, especially in connection with higher-order analogs of the estimate (1.1).In Section 2 we survey the case of norm inequalities for generators of C semi-groups in Banach and Hilbert spaces. The case where A generates a C groupin Banach and Hilbert spaces is recalled in Section 3. Some inequalities for frac-tional powers of generators of contraction semigroups are surveyed in Section 4.Extensions of the Hardy–Littlewood inequality in Hilbert spaces are recalled inSection 5. In particular, we discuss an extension initiated by Everitt [29] in 1971involving quadratic forms of general Sturm–Liouville operators and then add someconsiderations naturally involving the Friedrichs extension A F of a symmetric op-erator A bounded from below. The explicitly solvable example associated with thedifferential expression τ α,β,γ = x − α (cid:20) − ddx x β ddx + (2 + α − β ) γ − (1 − β ) x β − (cid:21) ,α > − , β < , γ ∈ (0 , , x ∈ (0 , ∞ ) , (1.4)is analyzed in some detail in our final Section 6.For other surveys of many aspects of integral inequalities we refer, for instance,to [5], [23], [26], [30], [68], [69], [72], and [73].Finally, some comments regarding our notation: All Hilbert spaces H are as-sumed to be complex in this survey and a symmetric operator A in H is alwaysassumed to be densely defined (such that A ⊆ A ∗ ).2. Norm Inequalities for Generators of C Semigroups
We begin by considering inequalities concerning (infinitesimal) generators G of C semigroups of bounded operators in the form T ( t ) = e tG , t ∈ [0 , ∞ ), on aBanach space X , in particular, { T ( t ) } t ∈ [0 , ∞ ) ⊂ B ( X ) satisfies( i ) T (0) = I X ;( ii ) T ( s ) T ( t ) = T ( s + t ), s, t ∈ [0 , ∞ );( iii ) [0 , ∞ ) ∋ t T ( t ) f ∈ X is continuous for each f ∈ X (w.r.t. the topology on X , i.e., T ( · ) f ∈ C ([0 , ∞ ) , X )). SURVEY OF SOME NORM INEQUALITIES 3
We will especially be interested in the case of C contraction semigroups, thatis, those satisfying k T ( t ) k B ( X ) ≤ t ∈ [0 , ∞ ).Given a C semigroup T ( t ), t ∈ [0 , ∞ ), its generator G is then defined as usualvia Gf = lim t ↓ t − [ T ( t ) f − f ] , (2.1)with dom( G ) ⊆ X consisting precisely of those f ∈ X for which the limit in (2.1)exists in the norm k · k X of X . Theorem 2.1. ( The Kallman–Rota inequality [56] , see also [44, Theorem 9.8]) .Let c ∈ C \{ } and suppose that cA generates a C contraction semigroup T ( t ) , t ∈ [0 , ∞ ) , on a Banach space X . Then k Af k X ≤ k f k X (cid:13)(cid:13) A f (cid:13)(cid:13) X , f ∈ dom (cid:0) A (cid:1) . (2.2) Sketch of proof.
There are at least two well-known proofs of Theorem 2.1. Withoutloss of generality we assume that c = 1 in the following.( i ) The standard semigroup proof, presented, for instance, in [44, Theorem 9.8],repeatedly uses the fact ( d/ds ) T ( s ) f = AT ( s ) f = T ( s ) Af , f ∈ dom( A ), and thenderives the relation tAf = T ( t ) f − f − ˆ t ds ( t − s ) T ( s ) A f, f ∈ dom (cid:0) A (cid:1) . (2.3)The triangle inequality and the contraction property of T ( · ) imply k Af k X ≤ t − [ k T ( t ) f k X + k f k X ] + t − ˆ t ds ( t − s ) (cid:13)(cid:13) T ( s ) A f (cid:13)(cid:13) X ≤ t − [ k f k X + k f k X ] + t − ˆ t ds ( t − s ) (cid:13)(cid:13) A f (cid:13)(cid:13) X = 2 t k f k X + t (cid:13)(cid:13) A f (cid:13)(cid:13) X , f ∈ dom (cid:0) A (cid:1) . (2.4)If A f = 0, letting t ↑ ∞ in (2.4) shows that Af = 0. If A f = 0, one minimizesthe right-hand side of (2.4) over t > t = 2 k f k / X (cid:14)(cid:13)(cid:13) A f (cid:13)(cid:13) / X , proving (2.2).( ii ) A functional analytic proof of (2.2) was presented by Certain and Kurtz [13].It uses Landau’s inequality [75] k f ′ k L ∞ ((0 , ∞ ); dx ) ≤ k f k L ∞ ((0 , ∞ ); dx ) k f ′′ k L ∞ ((0 , ∞ ); dx ) ,f ∈ (cid:8) g ∈ L ∞ ((0 , ∞ ); dx ) (cid:12)(cid:12) g, g ′ ∈ AC ([0 , R ]) for all R > , (2.5) g ′′ ∈ L ∞ ((0 , ∞ ); dx ) (cid:9) = W , ∞ ((0 , ∞ )) , as input and then derives the following inequality from it: Let X be a Banach spacewith norm k · k X , and functions F : [0 , ∞ ) → X with ||| · ||| abbreviating ||| F ||| = sup t ∈ [0 , ∞ ) k F ( t ) k X . (2.6)Next, assume that F : [0 , ∞ ) → X has two continuous derivatives with ||| F ||| < ∞ and ||| F ′′ ||| < ∞ . If the functional ℓ ∈ X ∗ is normalized, k ℓ k X ∗ = 1, then F. GESZTESY, R. NICHOLS, AND J. STANFILL the function g ( t ) = ℓ ( F ( t )), t ∈ [0 , ∞ ), is twice continuously differentiable with g ( k ) ( t ) = ℓ (cid:0) F ( k ) ( t ) (cid:1) , 1 ≤ k ≤
2. Hence Landau’s inequality (2.5) for g yields (cid:0) sup t ∈ [0 , ∞ ) | ℓ ( F ′ ( t )) | (cid:1) ≤ t ∈ [0 , ∞ ) | ℓ ( F ( t )) | sup t ∈ [0 , ∞ ) | ℓ ( F ′′ ( t )) |≤ ||| F ||| ||| F ′′ ||| (2.7)since k ℓ k X ∗ = 1. Taking the supremum over all ℓ ∈ X ∗ with k ℓ k X ∗ = 1, employingthe fact that k F ( t ) k X = sup ℓ ∈X ∗ , k ℓ k X∗ =1 | ℓ ( F ( t )) | , t ∈ [0 , ∞ ), proves ||| F ′ ||| ≤ ||| F ||| ||| F ′′ ||| . (2.8)If A is the generator of a C contraction semigroup T ( t ), t ∈ [0 , ∞ ), then for f ∈ dom (cid:0) A (cid:1) , F ( t ) = T ( t ) f is twice continuously differentiable and combining(2.8) with (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup t ∈ [0 , ∞ ) (cid:13)(cid:13) T ( t ) A k f (cid:13)(cid:13) X = (cid:13)(cid:13) A k f (cid:13)(cid:13) X , k = 0 , , , (2.9)yields the estimate (2.2). (cid:3) Remark . ( i ) For additional literature in the context of Hardy, Kallman, Landau,Littlewood, Rota inequalities we refer to [17], [33], [35], [42], [45], [53], [65], [66],[68], [69], [70], [71], [73, Ch. 2].( ii ) The identical strategy of proof in [13] extends the higher-order L ∞ -inequalities, (cid:13)(cid:13) f ( k ) (cid:13)(cid:13) nL ∞ ((0 , ∞ ); dx ) ≤ C n,k ( ∞ , R + ) k f k n − kL ∞ ((0 , ∞ ); dx ) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) kL ∞ ((0 , ∞ ); dx ) ,k ∈ { , . . . , n − } , n ∈ N , n ≥ ,f ∈ (cid:8) g ∈ L ∞ ((0 , ∞ ); dx ) (cid:12)(cid:12) g, . . . , g ( n − ∈ AC ([0 , R ]) for all R > , (2.10) g ( n ) ∈ L ∞ ((0 , ∞ ); dx ) (cid:9) = W n, ∞ ((0 , ∞ )) , to higher-order analogs of the Kallman–Rota estimate (2.2) of the form, (cid:13)(cid:13) A k f (cid:13)(cid:13) n X ≤ C n,k ( ∞ , R + ) k f k n − k X (cid:13)(cid:13) A n f (cid:13)(cid:13) k X , f ∈ dom (cid:0) A n (cid:1) ,k ∈ { , . . . , n − } , n ∈ N , n ≥ . (2.11)For additional results in the higher-order cases, see, for instance, [3], [9], [14], [42],[46], [64], [66], [72], [73, Ch. 1], [77], [81], [85], [86], [90], [91], [95].( iii ) For norm inequalities in connection with generators of cosine operator functions(cf. [2, Sects. 3.14–3.16], [36, Ch. II], [44, Sect. II.8]) we refer, for instance, to [76],[90]. The case of analytic semigroups (cf. [2, Sect. 3.7], [19, Sect. 2.5], [44, Sect. 1.5])is treated in [90]. ⋄ Theorem 2.1 can be rewritten replacing the contraction semigroup by a uniformlybounded semigroup, that is, a semigroup such that for some M ≥ k T ( t ) k B ( X ) ≤ M, t ∈ [0 , ∞ ) . (2.12) Corollary 2.3. ( Pazy [83, Lemma 2.8]) . Let c ∈ C \{ } and suppose that cA generates a C semigroup T ( t ) satisfying k T ( t ) k B ( X ) ≤ M , t ∈ [0 , ∞ ) , on a Banachspace X . Then k Af k X ≤ M k f k X (cid:13)(cid:13) A f (cid:13)(cid:13) X , f ∈ dom (cid:0) A (cid:1) . (2.13) SURVEY OF SOME NORM INEQUALITIES 5
Proof.
One obtains this result by introducing the constant M when bounding T ( t )in equation (2.4), noting that M ≥ T (0) = I X . (cid:3) Example 2.4. ( See, [44, Example 9.10]) . Consider A = d/dx in L p ((0 , ∞ ); dx ) , p ∈ [1 , ∞ ) ∪ {∞} , and abbreviate R + = (0 , ∞ ) . Then k f ′ k L p ((0 , ∞ ); dx ) ≤ C , ( p, R + ) k f k L p ((0 , ∞ ); dx ) k f ′′ k L p ((0 , ∞ ); dx ) ,f ∈ dom (cid:18) d dx (cid:19) = (cid:8) g ∈ L p ((0 , ∞ ); dx ) (cid:12)(cid:12) g, g ′ ∈ AC ([0 , R ]) for all R > , (2.14) g ′′ ∈ L p ((0 , ∞ ); dx ) (cid:9) = W ,p ((0 , ∞ )) , where C , (1 , R + ) = 5 / , C , (2 , R + ) = 2 ,C , ( p, R + ) ≤ C , ( ∞ , R + ) , p ∈ [1 , ∞ ) ∪ {∞} . (2.15) Landau [75] proved the case p = ∞ in 1913; Hardy and Littlewood [49] proved thecase p = 2 in 1932 ( cf. also [17] , [73, Theorem 2.2]) ; Berdyshev [7] proved thecase p = 1 in 1971. The best possible constant C , ( p, R + ) in (2.14) is not knownotherwise, however, it is known to be a continuous function of p ( cf. [73, Theorem2.19]) ; for more details see [38] , [73, Ch. 2] . Theorem 2.1 can be refined in the case of Hilbert spaces as seen in the nextresult, the original proof of which is due to Kato [58].
Theorem 2.5. ( Kato [58] , see also [44, Theorem 9.9] , [74]) . Let c ∈ C \{ } andsuppose that cA generates a C contraction semigroup T ( t ) , t ∈ [0 , ∞ ) , on a Hilbertspace H . Then k Af k H ≤ k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) H , f ∈ dom (cid:0) A (cid:1) , (2.16) and equality holds if and only if for some D ∈ [0 , ∞ ) , c A f + DcAf + D f = 0 and Re (cid:0)(cid:0) f, c A f (cid:1) H (cid:1) = 0 . (2.17) In the case of equality one necessarily has | c | (cid:13)(cid:13) A f (cid:13)(cid:13) H = D k f k H , f ∈ dom (cid:0) A (cid:1) . (2.18) In fact, the hypothesis that cA generates a C contraction semigroup is unneces-sary, it suffices to assume that Re(( f, cAf ) H ) ≤ , f ∈ dom( A ) . (2.19) Sketch of proof.
We briefly follow the proof provided in [74], assuming without lossof generality that c = 1. In the following, let d > f ∈ dom (cid:0) A (cid:1) , thenRe(([ dA + I H ] f, dA [ dA + I H ] f ) H ) ≤ dAf, dAf ) H + (cid:0)(cid:2) d A + dA + I H (cid:3) f, (cid:2) d A + dA + I H (cid:3) f (cid:1) H ≤ (cid:0) d A f, d A f (cid:1) H + k f k H , (2.21)and hence d k Af k H ≤ d (cid:13)(cid:13) A f (cid:13)(cid:13) H + k f k H . (2.22) F. GESZTESY, R. NICHOLS, AND J. STANFILL
Dividing (2.22) by d and taking d → ∞ yields that if (cid:13)(cid:13) A f (cid:13)(cid:13) H = 0, then k Af k H = 0and hence (2.16) holds. Otherwise, let d = k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) − H and then (2.22) oncemore yields (2.16).Next, assume that for some 0 = f ∈ dom (cid:0) A (cid:1) , k Af k H = 2 k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) H . (2.23)If A f = 0 then k Af k H = 0 and (2.17) holds with D = 0. If A f = 0, let d = k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) − H and inserting this into (2.21) implies d A f + dAf + f = 0and hence the first relation in (2.17) with D = d − . Working back from (2.21) to(2.20) yields0 = Re(([ dA + I H ] f, dA [ dA + I H ] f ) H ) = Re (cid:0)(cid:0) [ dA + I H ] f, (cid:2) d A + dA (cid:3) f (cid:1) H (cid:1) = Re (cid:0)(cid:0) d A f, [ − f ] (cid:1) H (cid:1) = − d Re (cid:0)(cid:0) f, A f (cid:1) H (cid:1) , (2.24)implying the last relation in (2.17).Conversely, suppose that for some D ∈ (0 , ∞ ), A f + D Af + D f = 0 and Re (cid:0)(cid:0) f, A f (cid:1) H (cid:1) = 0 . (2.25)From (2.25) one obtains k Af k H = D − (cid:13)(cid:13) A f + D f (cid:13)(cid:13) H = D − h(cid:13)(cid:13) A f (cid:13)(cid:13) H + D k f k H + 2 D Re (cid:0)(cid:0) f, A f (cid:1) H (cid:1)i = D − (cid:13)(cid:13) A f (cid:13)(cid:13) H + D k f k H . (2.26)Since k Af k H ≤ k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) H , it follows that D − (cid:13)(cid:13) A f (cid:13)(cid:13) H − k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) H + D k f k H ≤ , (2.27)hence (cid:2) D − (cid:13)(cid:13) A f (cid:13)(cid:13) H − D k f k H (cid:3) ≤
0, or D − (cid:13)(cid:13) A f (cid:13)(cid:13) H = D k f k H , equivalently, (cid:13)(cid:13) A f (cid:13)(cid:13) H = D k f k H . (2.28)Substituting (2.28) into (2.26) yields equality in (2.16). (cid:3) The following result is a natural extension of Example 2.4 in the case p = 2 andit illustrates a particular case of Theorem 2.5. While this observation is not new,we record its proof due to its simplicity: Lemma 2.6. ( Ljubiˇc [77, Theorem 5] , see also Chernoff [14] and Kwong and Zettl [67]) . Suppose S is a maximally symmetric, non-self-adjoint operator in a Hilbertspace H . Let A = c S ∗ for some c ∈ C \{ } . Then k Af k H ≤ k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) H , f ∈ dom (cid:0) A (cid:1) . (2.29) The constant is optimal and the case of equality in (2.29) is determined as in (2.17) applied to ± iS ∗ .Proof. Since S is maximally symmetric yet non-self-adjoint, one of its deficiencyindices equals zero, that is,dim(ker( S ∗ − iI H )) = 0 , or dim(ker( S ∗ + iI H )) = 0 , (2.30) SURVEY OF SOME NORM INEQUALITIES 7 but not both are zero simultaneously. Consequently (cf. [59, Sect. V.3.4]), thespectrum of S is either the closed complex upper half-plane or the closed complexlower half-plane, respectively, σ ( S ) = C + or σ ( S ) = C − . (2.31)Thus, σ ( iS ∗ ) = { z ∈ C | Re( z ) ≤ } , or σ ( iS ∗ ) = { z ∈ C | Re( z ) ≥ } , (2.32)and hence one of ± iS ∗ is m-accretive and thus generates a contraction semigroupon H . (Actually, one of them satisfies (2.19) which suffices for our argument.) Since A = c S ∗ , the estimate (2.29) (and the case of equality therein) now follows from(2.16) (and (2.17)). That the constant 2 is optimal is a consequence of Example2.4 with p = 2. (cid:3) Remark . ( i ) We emphasize that the constants to be found in Theorems 2.1 and2.5 are optimal as can be seen from Example 2.4. The equality condition given inTheorem 2.5, in the case p = 2 in Example 2.4 then becomes f ′′ + Df ′ + D f = 0 and Re(( f, f ′′ ) L ((0 , ∞ ); dx ) ) = Re (cid:18) ˆ ∞ dx f ( x ) f ′′ ( x ) (cid:19) = 0 . (2.33)Hardy and Littlewood [49] (see also [50, p. 187–188]) showed these conditions de-termine f to be f C,D ( x ) = C e − Dx/ sin (cid:0) / D ( x/ − ( π/ (cid:1) , C ∈ C , D > , x ≥ . (2.34)( ii ) Kato’s original proof of Theorem 2.5 relied on properties of the Cayley trans-form of A . For additional results in this direction see, [14], [29], [43], [67], [84].( iii ) For Theorem 2.5 under weaker hypotheses on A see [51].( iv ) Higher-order inequalities of the type (2.11) with explicit constants in theHilbert space context were derived by Protter [85], thus extending Kato’s result(where k = 1, n = 2) to the higher-order case. Additional results appeared in [14],[67], [77, Theorem 7], [84], in particular, these authors prove the following result:If C n,k (2 , R + ) denote the constants in the L -inequalities, (cid:13)(cid:13) f ( k ) (cid:13)(cid:13) nL ((0 , ∞ ); dx ) ≤ C n,k (2 , R + ) k f k n − kL ((0 , ∞ ); dx ) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) kL ((0 , ∞ ); dx ) ,k ∈ { , . . . , n − } , n ∈ N , n ≥ ,f ∈ (cid:8) g ∈ L ((0 , ∞ ); dx ) (cid:12)(cid:12) g, . . . , g ( n − ∈ AC ([0 , R ]) for all R > , (2.35) g ( n ) ∈ L ((0 , ∞ ); dx ) (cid:9) = H n ((0 , ∞ )) , then the same constants feature in the corresponding higher-order inequalities in-volving the operator A as follows: Suppose that A is a nonzero constant multipleof a generator of a contraction semigroup T ( t ) , t ∈ [0 , ∞ ) ( equivalently, a nonzeroconstant multiple of a densely defined dissipative operator ) , on a Hilbert space H ,then (cid:13)(cid:13) A k f (cid:13)(cid:13) n H ≤ C n,k (2 , R + ) k f k n − k H (cid:13)(cid:13) A n f (cid:13)(cid:13) k H , f ∈ dom (cid:0) A n (cid:1) ,k ∈ { , . . . , n − } , n ∈ N , n ≥ . (2.36) F. GESZTESY, R. NICHOLS, AND J. STANFILL
We also recall the symmetry property [14], [67], [77, Corollary 1, p. 71], C n,k (2 , R + ) = C n,n − k (2 , R + ) , k ∈ { , . . . , n − } , n ∈ N , n ≥ C n,k ( ∞ , R + )).In particular, the extension of Lemma 2.6 to the case of the higher-order inequal-ities (2.36) (under the same hypothesis that A = c S ∗ , for S maximally symmetricand non-self-adjoint, c ∈ C \{ } ) has been observed by Ljubiˇc [77, Theorem 5] in1964, and was subsequently rederived by rather different means in Chernoff [14]and Kwong and Zettl [67]. ⋄ Norm Inequalities for Generators of C Groups
As all these results so far have concerned semigroups, it is natural for one to askwhat the corresponding results are when considering groups rather than semigroups.
Theorem 3.1. ( See, [20] , [46] , [56]) . Let c ∈ C \{ } and suppose that cA generatesa C group T ( t ) , t ∈ R , of isometries on a Banach space X . Then k Af k X ≤ k f k X (cid:13)(cid:13) A f (cid:13)(cid:13) X , f ∈ dom (cid:0) A (cid:1) . (3.1) Sketch of proof.
The two proofs mentioned in Remark 2.2 extend to the presentcase. For instance (choosing again c = 1 for brevity), the standard semigroupproof in [44, 2nd ed., Theorem 1.1, p. 237] proceeds in establishing the Taylor-typeformula T ( − t ) f = f − tAf + ˆ − t ds ( t + s ) T ( s ) A f, t ∈ [0 , ∞ ) , f ∈ dom (cid:0) A (cid:1) , (3.2)and from this one obtains k Af k X ≤ t k f k X + t (cid:13)(cid:13) A f (cid:13)(cid:13) X , t > . (3.3)If A f = 0, letting t ↑ ∞ in (3.3) shows that Af = 0. If A f = 0, one minimizesthe right-hand side of (3.3) over t > t = 2 / k f k / X (cid:14)(cid:13)(cid:13) A f (cid:13)(cid:13) / X . Substituting this minimum into (3.3) yields (3.1).Similarly, the functional analytic proof of Certain and Kurtz [13] extends to thepresent estimate (3.1) upon systematically replacing the half-line [0 , ∞ ) by R . (cid:3) Remark . ( i ) For additional references in this context see [42], [45], [65], [66],[68], [69], [70], [71], [73, Ch. 2].( ii ) The case of higher-order inequalities on R was studied by Kolmogorov [61], andagain the strategy of proof in Certain–Kurtz [13] extends the analog of (2.10) on R with smaller constants C n,k ( ∞ , R ) to the analog of the higher-order Kallman–Rotainequalities (2.11) with the same optimal constants C n,k ( ∞ , R ). For additionalresults in the higher-order case, see, [3], [42], [60], [72], [73, Ch. 1], [77], [85], [90],[91], [92].( iii ) In the special case where X is a Hilbert space H in Theorem 3.1, the constant2 on the right-hand side of (3.1) can be replaced by 1 (cf., [14]). Indeed, in this casethe group of isometries, T ( t ), t ∈ R , becomes a unitary group, hence its generator A is a skew-adjoint operator by Stone’s theorem and then the assertion k Af k H ≤ k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) H , f ∈ dom (cid:0) A (cid:1) , (3.4)is a special case of Lemma 4.3. ⋄ SURVEY OF SOME NORM INEQUALITIES 9
Once more the constant in Theorem 3.1 is optimal as can be seen from the nextexample.
Example 3.3.
Consider A = d/dx in L p ( R ; dx ) , p ∈ [1 , ∞ ) ∪ {∞} . Then k f ′ k L p ( R ; dx ) ≤ C , ( p, R ) k f k L p ( R ; dx ) k f ′′ k L p ( R ; dx ) ,f ∈ dom (cid:18) d dx (cid:19) = (cid:8) f ∈ L p ( R ; dx ) (cid:12)(cid:12) f, f ′ ∈ AC ([0 , R ]) for all R > , (3.5) f ′′ ∈ L p ( R ; dx ) (cid:9) = W ,p ( R ) , where C , (1 , R ) = 2 , C , (2 , R ) = 1 ,C , ( p, R ) ≤ C , ( ∞ , R ) , C , ( p, R ) ≤ C , ( p, R + ) , p ∈ [1 , ∞ ) ∪ {∞} . (3.6) Landau [75] in 1913, and Hadamard [47] in 1914 proved the case p = ∞ ; Hardy andLittlewood [49] proved the case p = 2 in 1932 ( cf. also [73, Theorem 2.2]) ; Berdyshev [7] proved the case p = 1 in 1971. The best possible constant C , ( p, R ) in (3.5) isnot known otherwise, however, it is again known to be a continuous function of p ( cf. [73, Theorem 2.19]) ; for more details see [37] , [73, Ch. 2] . In particular, Hardyand Littlewood [49] showed that equality holds in (3.5) when p = 2 only when f ≡ and the best constant can be proven by taking f of the form, f ( x ) = ( sin( x ) , | x | < nπ, , | x | > nπ, n ∈ N , (3.7) and smoothing near x = ± nπ , n ∈ N , so as to ensure f ′′ is continuous ( see also [50,p. 193]) . For p = ∞ , inequality (3.5) was generalized to higher-order derivativeson the real line by Kolmogorov in [61] , who showed that the sharp constants couldbe expressed through relations with the so-called Favard constants ( cf. [38]) . We refer to Kwong and Zettl [73] for an extensive treatment of the topic of L p -norm inequalities for derivatives.4. Inequalities for Fractional Powers of Generators ofContraction Semigroups
We now transition to the related topic of fractional powers of generators ofcontraction semigroups on Banach spaces.We start with the simple case of nonnegative, self-adjoint (generally, unbounded)operators in a Hilbert space. The following elementary inequality is derived in thebook by Krasnosel’skii, Pustylnik, Sobolevskii, and Zabreiko [62, p. 223]:
Theorem 4.1. ( See, [62, Theorem 12.1, p. 223]) . Let S be a nonnegative self-adjoint operator on a Hilbert space H . Then for each τ ∈ (0 , , (cid:13)(cid:13) S τ f (cid:13)(cid:13) H ≤ k f k − τ H k Sf k τ H , f ∈ dom( S ) . (4.1) Proof.
Using the spectral representation of the operator S (and observing σ ( S ) ⊆ [0 , ∞ )), S = ˆ σ ( S ) λ dE S ( λ ) , (4.2) implies k S τ f k H = ˆ σ ( S ) λ τ d k E S ( λ ) f k H , f ∈ dom( S τ ) . (4.3)It follows from H¨older’s inequality that (cid:13)(cid:13) S τ f (cid:13)(cid:13) H = ˆ σ ( S ) λ τ d k E S ( λ ) f k H ≤ (cid:18) ˆ σ ( S ) λ d k E S ( λ ) f k H (cid:19) τ (cid:18) ˆ σ ( S ) d k E S ( λ ) f k H (cid:19) − τ (4.4)= k Sf k τ H k f k − τ ) H , f ∈ dom( S ) . (cid:3) Remark . ( i ) As the elementary example S = I H shows, the constant 1 on theright-hand side of (4.1) is optimal.( ii ) One compares Theorem 4.1 to Theorem 2.5 as follows. Choose a self-adjointoperator S such that S = A , A ≥
0, then also S ≥ H . Theorem 4.1 thenimplies, (cid:13)(cid:13) A τ f (cid:13)(cid:13) H ≤ k f k − τ H (cid:13)(cid:13) A f (cid:13)(cid:13) τ H , f ∈ dom (cid:0) A (cid:1) , (4.5)and choosing τ = 1 / k Af k H ≤ k f k / H (cid:13)(cid:13) A f (cid:13)(cid:13) / H , f ∈ dom (cid:0) A (cid:1) . (4.6)Equivalently, this yields for any nonnegative (resp., nonpositive) self-adjoint oper-ator A a in a Hilbert space H that k Af k H ≤ k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) H , f ∈ dom (cid:0) A (cid:1) . (4.7)Comparing (4.7) to (2.16), one notes that the optimal constant has now been re-duced from 2 to 1 in inequality (2.16). ⋄ Actually, as pointed out in [77] (see also [74]), self-adjointness and nonnegativityof A in (4.7) are not needed at all. Indeed, it suffices to assume that A is a constantmultiple of a symmetric operator: Lemma 4.3.
Let c ∈ C \{ } and suppose that cA is a symmetric operator in aHilbert space H . Then k Af k H ≤ k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) H , f ∈ dom (cid:0) A (cid:1) , (4.8) with equality in (4.8) if and only if A f = Cf for some C ∈ C .Proof. Let f ∈ dom (cid:0) A (cid:1) , then Cauchy’s inequality implies k cAf k H = ( cAf, cAf ) H = (cid:0) f, c A f (cid:1) H ≤ | c | k f k H (cid:13)(cid:13) A f (cid:13)(cid:13) H . (4.9) (cid:3) Remark . Despite the most elementary nature of the proof of Lemma 4.3, itshould be noted that Naimark [80] proved the existence of a symmetric operator A in H such that dom (cid:0) A (cid:1) = { } , rendering (4.8) vacuous in this case. (Of course,this pathology cannot occur in the context of (4.7).) In this context we also referto Schm¨udgen [89]. ⋄ SURVEY OF SOME NORM INEQUALITIES 11
More generally, fractional powers of the 1st-order differentiation operator D = d/dx on C b ([0 , ∞ )) (the space of bounded, uniformly continuous functions on[0 , ∞ )) and associated norm inequalities were already proved in 1935 by Hardy,Landau, and Littlewood [48] employing the Riemann–Liouville formula for frac-tional derivatives (cid:0) ( − D ) α f (cid:1) ( x ) = − Γ( − α ) − ˆ ∞ dt t − α − [ f ( t ) − f ( x + t )] , < α < , f ∈ C b ([0 , ∞ )) . (4.10)Assuming that A generates a contraction semigroup T ( t ), t ∈ [0 , ∞ ), on theBanach space X , one can define fractional powers of − A via the formula (see, [8],[78, Theorem 6.1.6])( − A ) γ f = C − γ,k ˆ ∞ dt t − γ − [ I H − T ( t )] k f, f ∈ X , (4.11) C γ,k = ˆ ∞ dt t − γ − (cid:2) − e − t (cid:3) k , k − ≤ γ < k, k ∈ N . (4.12)Chernoff [14] proves the following two results: Theorem 4.5. ( Chernoff [14]) . Suppose that A generates a C contraction semi-group T ( t ) , t ∈ [0 , ∞ ) , on a Banach space X and let < α < β . Then (cid:13)(cid:13) ( − A ) α f (cid:13)(cid:13) β X ≤ C β,α ( ∞ , R + ) k f k β − α X (cid:13)(cid:13) ( − A ) β f (cid:13)(cid:13) α X , f ∈ dom (cid:0) A β (cid:1) . (4.13)In the Hilbert space context this result turns into the following: Theorem 4.6. ( Chernoff [14]) . Suppose that A generates a C contraction semi-group T ( t ) , t ∈ [0 , ∞ ) , on a Hilbert space H and let < α < β . Then (cid:13)(cid:13) ( − A ) α f (cid:13)(cid:13) β H ≤ C β,α (2 , R + ) k f k β − α H (cid:13)(cid:13) ( − A ) β f (cid:13)(cid:13) α H , f ∈ dom (cid:0) A β (cid:1) . (4.14)Here the constants C β,α ( ∞ , R + ) , C β,α (2 , R + ) are the same as in the context offractional powers of the maximally defined 1st-order differentiation operator in L ∞ ((0 , ∞ ); dx ) and L ((0 , ∞ ); dx ), respectively.The analogous theorems hold in case A generates a group of isometries T ( t ), t ∈ R , employing the constants C β,α ( ∞ , R ) , C β,α (2 , R ), instead.5. On Some Extensions of the Hardy–Littlewood Inequality
We finally discuss some extensions of the Hardy–Littlewood inequality in thecontext of L ((0 , ∞ ); dx ) and L ( R ; dx ).The following extensions of the Hardy–Littlewood inequality appeared in [10](see also [18], [30]), (cid:12)(cid:12) k f ′ k L ((0 , ∞ ); dx ) − µ k f k L ((0 , ∞ ); dx ) (cid:12)(cid:12) ≤ k f k L ((0 , ∞ ); dx ) k [ f ′′ + µf ] k L ((0 , ∞ ); dx ) ,µ ∈ [0 , ∞ ) , f ∈ H ((0 , ∞ )) , (5.1) (cid:12)(cid:12) k f ′ k L ( R ; dx ) − µ k f k L ( R ; dx ) (cid:12)(cid:12) ≤ k f k L ( R ; dx ) k [ f ′′ + µf ] k L ( R ; dx ) ,µ ∈ R , f ∈ H ( R ) . (5.2)Both estimates are strict, that is, equality holds in (5.1) or (5.2) if and only if f ≡ Theorem 5.1. ( Chernoff [14]) . Let c ∈ C \{ } and suppose that cA generates a C contraction semigroup T ( t ) , t ∈ [0 , ∞ ) , on a Hilbert space H . Then (cid:12)(cid:12) k Af k H − µ k f k H (cid:12)(cid:12) ≤ k f k H (cid:13)(cid:13)(cid:2) A + µI H (cid:3) f (cid:13)(cid:13) H , µ ∈ [0 , ∞ ) , f ∈ dom (cid:0) A (cid:1) . (5.3)A further extension of the Hardy–Littlewood inequality, naturally involving qua-dratic forms, was initiated by Everitt [29] in 1971 originally in the context of regu-lar Sturm–Liouville problems on the half-line (0 , ∞ ). To describe this problem oneneeds a few preparations.Suppose that −∞ < a < b ≤ ∞ , and assume that for all c ∈ ( a, b ), p − ∈ L (( a, c ); dx ) , p > a, b ) ,q ∈ L (( a, c ); dx ) , q real-valued a.e. on ( a, b ) , (5.4) r ∈ L (( a, c ); dx ) , r > a, b ) . Introducing the Sturm–Liouville differential expression (regular at the point a ) τ = r ( x ) − (cid:20) − ddx p ( x ) ddx + q ( x ) (cid:21) , x ∈ ( a, b ) , (5.5)assuming τ to be in the limit point case at the point b (i.e., for all c ∈ ( a, b ) andfor all z ∈ C , τ ψ = zψ , has at least one solution ψ ( z, · ) not in L (( c, b ); r ( x ) dx )),one is interested in L (( a, b ); r ( x ) dx )-realizations associated with τ and introducesminimal and maximal operators corresponding to τ via( T min f )( x ) = ( τ f )( x ) , x ∈ ( a, b ) ,f ∈ dom( T min ) = (cid:8) g ∈ L (( a, b ); r ( x ) dx ) (cid:12)(cid:12) g, g [1] ∈ AC ([ a, c ]) for all c ∈ ( a, b ); g ( a ) = 0 = ( pg ′ )( a ); τ g ∈ L (( a, b ); r ( x ) dx ) (cid:9) , (5.6)( T max f )( x ) = ( τ f )( x ) , x ∈ ( a, b ) ,f ∈ dom( T max ) = (cid:8) g ∈ L (( a, b ); r ( x ) dx ) (cid:12)(cid:12) g, g [1] ∈ AC ([ a, c ]) for all c ∈ ( a, b ); τ g ∈ L (( a, b ); r ( x ) dx ) (cid:9) . (5.7)Here we employed the notion of a quasi-derivative g [1] ( x ) = p ( x ) g ′ ( x ), x ∈ ( a, b ).One then infers that T min is closed, densely defined, and symmetric and T ∗ min = T max , T ∗ max = T min . (5.8)The problem posed by Everitt (in a slightly extended form) then reads as follows:Is there a constant K D = K D (( a, b ); p, q, r ) ∈ (0 , ∞ ) such that (cid:12)(cid:12)(cid:12)(cid:12) ˆ ba dx (cid:2) p ( x ) | f ′ ( x ) | + q ( x ) | f ( x ) | (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ K D (cid:18) ˆ ba r ( x ) dx | f ( x ) | (cid:19) / (cid:18) ˆ ba r ( x ) dx | ( τ f )( x ) | (cid:19) / , f ∈ D , (5.9)equivalently, (cid:12)(cid:12)(cid:12)(cid:12) ˆ ba dx h p ( x ) − (cid:12)(cid:12) f [1] ( x ) (cid:12)(cid:12) + q ( x ) | f ( x ) | i(cid:12)(cid:12)(cid:12)(cid:12) ≤ K D k f k L (( a,b ); r ( x ) dx ) k τ f k L (( a,b ); r ( x ) dx ) , f ∈ D , (5.10)for an appropriate linear subspace D of L (( a, b ); r ( x ) dx ) satisfyingdom( T min ) ⊆ D ⊆ dom( T max ) . (5.11) SURVEY OF SOME NORM INEQUALITIES 13
It is implicitly assumed that both terms are finite on the left-hand side of (5.9),(5.10). Introducing the sesquilinear form q ( f, g ) = ˆ ba dx h p ( x ) − f [1] ( x ) g [1] ( x ) + q ( x ) f ( x ) g ( x ) i , f ∈ D , (5.12)(still implicitly assuming that both terms on the right-hand side of (5.12) arefinite), one recognizes that Everitt’s problem is equivalent to the existence of K D = K D (( a, b ); p, q, r ) ∈ (0 , ∞ ) such that the integral inequality | q ( f, f ) | ≤ K D k f k L (( a,b ); r ( x ) dx ) k τ f k L (( a,b ); r ( x ) dx ) , f ∈ D , (5.13)holds. In particular, the special case a = 0 , b = ∞ , p ( x ) = r ( x ) = 1 , q ( x ) = 0 , τ = − d dx , x ∈ (0 , ∞ ) , (5.14) D = H ((0 , ∞ )) , K H ((0 , ∞ )) ( R + ; 1 , ,
1) = 2 , (5.15)or D = H ((0 , ∞ )) , K H ((0 , ∞ )) ( R + ; 1 , ,
1) = 1 , (5.16)yields the classical Hardy–Littlewood inequality [49], k f ′ k L ((0 , ∞ ); dx ) ≤ k f k L ((0 , ∞ ); dx ) k f ′′ k L ((0 , ∞ ); dx ) , f ∈ H ((0 , ∞ )) , (5.17)(cf. (2.14), (2.15) for p = 2) and k f ′ k L ((0 , ∞ ); dx ) ≤ k f k L ((0 , ∞ ); dx ) k f ′′ k L ((0 , ∞ ); dx ) , f ∈ H ((0 , ∞ )) , (5.18)as a special case of Lemma 4.3 since( A f )( x ) = f ′ ( x ) , x ∈ (0 , ∞ ) , f ∈ dom( A ) = H ((0 , ∞ )) , (5.19)in L ((0 , ∞ ); dx ) is skew-symmetric (i.e., iA is symmetric) and (cid:0) A f (cid:1) ( x ) = f ′′ ( x ) , x ∈ (0 , ∞ ) , f ∈ dom (cid:0) A (cid:1) = H ((0 , ∞ )) . (5.20)The precise analysis of the problem stated in connection with (5.13), originallydue to Everitt [29], is rather complex, utilizing results from the calculus of variationsand from Weyl–Titchmarsh theory. This was continued in Bennewitz [6], Evans andEveritt [23], Evans and Zettl [28], and Ph´ong [84] under more general hypotheseson p, q, r employing, in addition, von Neumann’s first formula for deficiency spaces(so adding some operator theory to this circle of ideas). To describe its solution wenow follow the discussion in [23].Let C ± = { ζ ∈ C | ± Im( z ) > } , and suppose that θ π/ ( z, · ) , φ π/ ( z, · ) aresolutions of τ u = zu (entire w.r.t. z ) satisfying the initial conditions θ π/ ( z, a ) = 0 , θ [1] π/ ( z, a ) = 1 ,φ π/ ( z, a ) = − , φ [1] π/ ( z, a ) = 0 , (5.21)and ψ π/ , ± ( z, · ) is a (Neumann-type) Weyl–Titchmarsh solution ψ π/ , ± ( z, · ) = θ π/ ( z, · ) + m π/ , ± ( z ) φ π/ ( z, · ) ∈ L (( a, b ); r ( x ) dx ) , z ∈ C ± , (5.22)with m π/ , ± ( · ) the (Neumann-type) Weyl–Titchmarsh m -function analytic in C ± ,such that m π/ , ± ( z ) = m π/ , ∓ ( z ) , z ∈ C ± . (5.23) Moreover, denoting z = ρe iϑ , ρ ∈ (0 , ∞ ), ϑ ∈ [0 , π ), one introduces L + ( ϑ ) = (cid:8) ρe iϑ (cid:12)(cid:12) ρ ∈ (0 , ∞ ) (cid:9) , L − ( ϑ ) = (cid:8) ρe i ( ϑ + π ) (cid:12)(cid:12) ρ ∈ (0 , ∞ ) (cid:9) , ϑ ∈ (0 , π/ ,ϑ ± = inf (cid:8) ϑ ∈ (0 , π/ (cid:12)(cid:12) for all ϕ ∈ [ ϑ, π/ , ∓ Im (cid:0) z m π/ , ± ( z ) (cid:1) ≥ , z ∈ L ± ( ϕ ) (cid:9) ,ϑ = max { ϑ + , ϑ − } ∈ [0 , π/ , (5.24) E ± = (cid:8) ρ ∈ (0 , ∞ ) (cid:12)(cid:12) z ∈ L ± ( ϑ ) , Im (cid:0) z m π/ , ± ( z ) (cid:1) = 0 (cid:9) ,Y ± ( ρ, x ) = Im( zψ π/ , ± ( z, x )) , z ∈ L ± ( ϑ ) , x ∈ [ a, b ) . Finally, τ is said to be in the strong limit point case at b iflim x ↑ b f ( x ) g [1] ( x ) = 0 , f, g ∈ dom( T max ) (5.25)(see, e.g., [52] for sufficient conditions on p, q, r guaranteeing the strong limit pointendpoint). We recall that the strong limit point property at b implies the limitpoint property at b (cf. [23]).Given these preparations, the principal result in [23] (see also [25]) reads asfollows: Theorem 5.2. ( Evans and Everitt [23]) . Assume the conditions (5.4) and supposethat τ is in the strong limit point case at b . Given the preparations in (5.4) – (5.25) ,the following assertions ( i ) – ( iv ) hold: ( i ) ϑ ∈ (0 , π/
2] ( in particular, ϑ > . ( ii ) The inequality (cid:12)(cid:12)(cid:12)(cid:12) lim d ↑ b ˆ da dx (cid:2) p ( x ) | f ′ ( x ) | + q ( x ) | f ( x ) | (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ K k f k L (( a,b ); r ( x ) dx ) k τ f k L (( a,b ); r ( x ) dx ) , f ∈ dom( T max ) , (5.26) is satisfied for some K ∈ (0 , ∞ ) if and only if < ϑ < π/ holds. ( iii ) If (5.27) holds, then the best constant K in inequality (5.26) , denoted by K dom( T max ) (( a, b ); p, q, r ) , is given by K dom( T max ) (( a, b ); p, q, r ) = [cos( ϑ )] − . (5.28) In particular, K dom( T max ) (( a, b ); p, q, r ) > . ( iv ) If (5.27) is satisfied and K is given by [cos( ϑ )] − in (5.28) , then the elements f ∈ dom( T max ) yielding equality in (5.26) are determined according to the followingthree mutually exclusive situations: ( α ) f = 0 . ( β ) There exists = f ∈ dom( T max ) such that τ f = 0 and f ( a ) = 0 or f [1] ( a ) = 0( but not both ) , in which case either side of (5.26) vanishes. ( γ ) E + ∪ E − = ∅ , f ( x ) = CY ± ( r, x ) , r ∈ E ± , C ∈ C \{ } , x ∈ [ a, b ) .Remark . ( i ) Returning to the classical Hardy–Littlewood case (see, (2.14),(2.15) for p = 2 or (5.14), (5.15), (5.17)) one obtains m π/ , ± ( z ) = iz − / , ψ π/ , ± ( z, x ) = − iz − / e iz / x , z ∈ C ± , x ∈ (0 , ∞ ) ,ϑ + = π/ , ϑ − = 0 , ϑ = π/ , K H ((0 , ∞ )) ( R + ; 1 , ,
1) = 2 , (5.29) SURVEY OF SOME NORM INEQUALITIES 15 E + = (0 , ∞ ) , E − = ∅ ,Y + ( ρ, x ) = e − ρx/ sin (cid:0)(cid:0) / ρx/ (cid:1) − ( π/ (cid:1) , ρ ∈ (0 , ∞ ) , x ∈ [0 , ∞ ) , confirming (5.17) once again.Reference [28] treats the analog of Theorem 5.2 and the Hardy–Littlewood exam-ple (5.29) with the Neumann-type m -function m π/ , ± ( · ) replaced by the Dirichlet-type m -function m , ± ( · ) = − /m π/ , ± ( · ).( ii ) Everitt and Zettl [35] showed that the example a = 0 , b = ∞ , p ( x ) = x β , q ( x ) = 0 , r ( x ) = x α , α > − , β < ,τ = − x − α ddx x β ddx , x ∈ (0 , ∞ ) , (5.30)leads to m π/ , ± ( z ) = (2 + α − β ) − β ) / (2+ α − β ) − β Γ((3 + α − β ) / (2 + α − β ))Γ((1 + α ) / (2 + α − β )) × e iπ (1 − β ) / (2+ α − β ) z − (1 − β ) / (2+ α − β ) , z ∈ C ± , (5.31)and hence to K dom( T max ) ( R + ; x β , , x α ) = [cos( π (1 + α ) / (3 + 2 α − β ))] − . (5.32)In particular, Everitt and Zettl [35] (see also [33]) showed that in the special case β = 0, the constant K dom( T max ) (cid:0) R + ; 1 , , x α (cid:1) takes on every value in (1 , ∞ ) as α ranges in ( − , ∞ ).For numerous additional explicit examples see, [5], [11], [23], [24], [25], [26], [27],[29], [30], [32], [33], [34]; for numerical investigations in this context we refer to [12].( iii ) Theorem 5.2 treats the case where a is a regular endpoint and the endpoint b is in the strong limit point case. The case where b is either a regular or a limitcircle nonoscillatory endpoint is treated in [25]. Moreover, the case where a and b are in the strong limit point case (with constant K = 1 in the analog of (5.26), andlim c ↓ a,d ↑ b ´ dc dx ... on the l.h.s. of (5.26)) is considered in [31]. ⋄ Next, we briefly return to quadratic forms and follow up on some ideas presentedby Ph´ong [84].Let A be a symmetric operator in the Hilbert space H bounded from below, thatis, A ⊆ A ∗ and for some c ∈ R , A > cI H . We denote by A the closure of A in H ,and introduce the associated forms in H , q A ( f, g ) = ( f, Ag ) H , f, g ∈ dom( q A ) = dom( A ) , (5.33) q A ( f, g ) = ( f, Ag ) H , f, g ∈ dom( q A ) = dom( A ) , (5.34)then the closures of q A and q A coincide in H (cf., e.g., [4, Lemma 5.1.12]) q A = q A (5.35)and the first representation theorem for forms (see, e.g., [21, Theorem 4.2.4], [59,Theorem VI.2.1, Sect. VI.2.3]) yields q A ( f, g ) = ( f, A F g ) H , f ∈ dom( q A ) , g ∈ dom( A F ) , (5.36)where A F ≥ cI H represents the self-adjoint Friedrichs extension of A . Due to thefact (5.35), one infers (cf., e.g., [4, Lemma 5.3.1]) A F = ( A ) F . (5.37) The second representation theorem for forms (see, e.g., [21, Theorem 4.2.8], [59,Theorem VI.2.123]) then yields the additional result q A ( f, g ) = (cid:0) ( A F − cI H ) / f, ( A F − cI H ) / g (cid:1) H + c k f k H ,f, g ∈ dom( q A ) = dom (cid:0) | A F | / (cid:1) . (5.38)Moreover, one has the fact (see, e.g., [4, Theorem 5.3.3], [21, Corollary 4.2.7])dom( A F ) = dom( q A ) ∩ dom( A ∗ ) = dom (cid:0) | A F | / (cid:1) ∩ dom( A ∗ ) . (5.39)Returning to (5.33), we now consider an extension Q of the form q A in H satis-fying Q ( f, g ) = ( f, Ag ) H , f ∈ dom( Q ) , g ∈ dom( A ) . (5.40)Then Cauchy’s inequality (cf. also Lemma 4.3) implies the elementary estimate | Q ( f, f ) | ≤ k f k H k Af k H , f ∈ dom( A ) , (5.41)and Ph´ong [84, p. 35–36] then points out via the following counterexample thatthe estimate (5.41), in general, permits no extension of the type: There exists aconstant K ∈ (0 , ∞ ) such that | Q ( f, f ) | ≤ K k f k H k A ∗ f k H , f ∈ dom( Q ) ∩ dom( A ∗ ) . (5.42) Example 5.4.
Consider τ = − d dx , x ∈ (0 , , ( T ,min f )( x ) = − f ′′ ( x ) , x ∈ (0 , , f ∈ dom( T ,min ) = H ((0 , , ( T ,max f )( x ) = − f ′′ ( x ) , x ∈ (0 , , f ∈ dom( T ,max ) = H ((0 , ,T ,max = T ∗ ,min , T ∗ ,max = T ,min , ker( T ,max ) = ker( T ∗ ,min ) = lin . span { u , u } ,u ( x ) = 1 , u ( x ) = x, x ∈ (0 , ,Q ( f, g ) = ( f ′ , g ′ ) L ((0 , dx ) , f, g ∈ dom( Q ) = H ((0 , ,q T ,min ( f, g ) = ( f, ( − g ′′ )) L ((0 , dx ) , f, g ∈ dom( q T ,min ) = H ((0 , ,q T ,min ( f, g ) = ( f ′ , g ′ ) L ((0 , dx ) , f, g ∈ dom( q T ,min ) = H ((0 , , ( T ,min,F f )( x ) = − f ′′ ( x ) , x ∈ (0 , ,f ∈ dom( T ,min,F ) = H ((0 , ∩ H ((0 , . (5.43) Then Q ( u , u ) = 1 ,T ∗ ,min u = T ,max u = 0 , (5.44) and hence there exists no K ∈ (0 , ∞ ) such that | Q ( f, f ) | ≤ K k f k L ((0 , dx ) k f ′′ k L ((0 , dx ) ,f ∈ dom( Q ) ∩ dom( T ∗ ,min ) = dom( T ∗ ,min ) = H ((0 , , (5.45) holds. ( Indeed, taking f = u yields on the l.h.s. of the inequality in (5.45) and on its r.h.s. ) SURVEY OF SOME NORM INEQUALITIES 17
Given the negative answer to the problem formulated in connection with (5.42),we now describe an elementary affirmative approach involving the Friedrichs exten-sion A F of A . Lemma 5.5.
Suppose A is symmetric and bounded from below in H . Then, inaddition to (5.41) one has | q A ( f, f ) | ≤ k f k H k A F f k H = k f k H k A ∗ f k H ,f ∈ dom( A F ) = dom( q A ) ∩ dom( A ∗ ) . (5.46) Proof.
It suffices to combine (5.36) and (5.39). (cid:3)
Due to (5.35), (5.37), one can systematically replace A by A in the above con-siderations. Remark . ( i ) Inequality (5.41) applied to T ,min in Example 5.4 yields k f ′ k L ((0 , dx ) ≤ k f k L ((0 , dx ) k f ′′ k L ((0 , dx ) , f ∈ H ((0 , , (5.47)and applying Lemma 5.5 to T ,min in Example 5.4 then implies the improvement k f ′ k L ((0 , dx ) ≤ k f k L ((0 , dx ) k f ′′ k L ((0 , dx ) , f ∈ H ((0 , ∩ H ((0 , . (5.48)( ii ) Similarly, Lemma 5.5 applied in the context of (5.14)–(5.20) yields the followingimprovement of (5.16) and (5.18) in the form k f ′ k L ((0 , ∞ ); dx ) ≤ k f k L ((0 , ∞ ); dx ) k f ′′ k L ((0 , ∞ ); dx ) , f ∈ H ((0 , ∩ H ((0 , ∞ )) , (5.49)that is, K H ((0 , ∞ )) ∩ H ((0 , ∞ )) = 1 . (5.50)Indeed, this follows from the fact (cf. (5.20)) (cid:0)(cid:0) − A (cid:1) ∗ f (cid:1) ( x ) = − f ′′ ( x ) , x ∈ (0 , ∞ ) , f ∈ dom (cid:0)(cid:0) − A (cid:1) ∗ (cid:1) = H ((0 , ∞ )) , (cid:0)(cid:0) − A (cid:1) F f (cid:1) ( x ) = − f ′′ ( x ) , x ∈ (0 , ∞ ) , (5.51) f ∈ dom (cid:0)(cid:0) − A (cid:1) F (cid:1) = H ((0 , ∞ )) ∩ H ((0 , ∞ )) . ⋄ We will provide one more explicitly solvable example illustrating Lemma 5.5,but due to the complexity of the example we will present it separately in the nextsection.6.
An Explicitly Solvable Example: A Generalized Bessel Equation
In this section we analyze the following explicitly solvable example detailed in(6.1), (6.2) below.Let a = 0, b = ∞ in (5.4), and consider the concrete example p ( x ) = x β , r ( x ) = x α , q ( x ) = (2 + α − β ) γ − (1 − β ) x β − ,α > − , β < , γ ≥ , x ∈ (0 , ∞ ) . (6.1)Then τ α,β,γ = x − α (cid:20) − ddx x β ddx + (2 + α − β ) γ − (1 − β ) x β − (cid:21) ,α > − , β < , γ ≥ , x ∈ (0 , ∞ ) , (6.2) is singular at the endpoint 0 (since the potential, q is not integrable near x = 0)and in the limit point case at ∞ . Furthermore, τ α,β,γ is in the limit circle case at x = 0 if 0 ≤ γ < x = 0 when γ ≥ y ,α,β,γ ( z, x ) = x (1 − β ) / J γ (cid:0) z / x (2+ α − β ) / / (2 + α − β ) (cid:1) , γ ≥ , (6.3) y ,α,β,γ ( z, x ) = ( x (1 − β ) / J − γ (cid:0) z / x (2+ α − β ) / / (2 + α − β ) (cid:1) , γ / ∈ N ,x (1 − β ) / Y γ (cid:0) z / x (2+ α − β ) / / (2 + α − β ) (cid:1) , γ ∈ N , γ ≥ , (6.4)where J ν ( · ) , Y ν ( · ) are the standard Bessel functions of order ν ∈ R (cf. [1, Ch. 9]).In the following we assume that γ ∈ [0 ,
1) (6.5)to ensure the limit circle case at x = 0. In this case it suffices to focus on thegeneralized boundary values at the singular endpoint x = 0 following [41]. Forthis purpose we introduce principal and nonprincipal solutions u ,α,β,γ (0 , · ) and b u ,α,β,γ (0 , · ) of τ α,β,γ u = 0 at x = 0 by u ,α,β,γ (0 , x ) = (1 − β ) − x [1 − β +(2+ α − β ) γ ] / , γ ∈ [0 , , b u ,α,β,γ (0 , x ) = ( (1 − β )[(2 + α − β ) γ ] − x [1 − β − (2+ α − β ) γ ] / , γ ∈ (0 , , (1 − β ) x (1 − β ) / ln(1 /x ) , γ = 0 ,α > − , β < , x ∈ (0 , . (6.6) Remark . Since the singularity of q at x = 0 renders τ α,β,γ singular at x = 0(unless, of course, γ = (1 − β ) / (2+ α − β ), in which case τ α,β, (1 − β ) / (2+ α − β ) is regularat x = 0), there is a certain freedom in the choice of the multiplicative constant inthe principal solution u ,α,β,γ of τ α,β,γ u = 0 at x = 0. Our choice of (1 − β ) − in(6.6) reflects continuity in the parameters when comparing to boundary conditionsin the regular case (cf. [41, Remark 3.12 ( ii )]), that is, in the case α > − β < γ = (1 − β ) / (2 + α − β ) treated in [35] (see Remark 5.3 ( ii )). ⋄ According to [41] the generalized boundary values for g ∈ dom( T α,β,γ,max ) arethen of the form e g (0) = − W ( u ,α,β,γ (0 , · ) , g )(0)= ( lim x ↓ g ( x ) (cid:14)(cid:2) (1 − β )[(2 + α − β ) γ ] − x [1 − β − (2+ α − β ) γ ] / (cid:3) , γ ∈ (0 , , lim x ↓ g ( x ) (cid:14)(cid:2) (1 − β ) x (1 − β ) / ln(1 /x ) (cid:3) , γ = 0 , (6.7) e g ′ (0) = W ( b u ,α,β,γ (0 , · ) , g )(0)= lim x ↓ (cid:2) g ( x ) − e g (0)(1 − β )[(2 + α − β ) γ ] − x [1 − β − (2+ α − β ) γ ] / (cid:3)(cid:14)(cid:2) (1 − β ) − x [1 − β +(2+ α − β ) γ ] / (cid:3) , γ ∈ (0 , , lim x ↓ (cid:2) g ( x ) − e g (0)(1 − β ) x (1 − β ) / ln(1 /x ) (cid:3)(cid:14)(cid:2) (1 − β ) − x (1 − β ) / (cid:3) , γ = 0 . (6.8) SURVEY OF SOME NORM INEQUALITIES 19
Next, introducing the standard normalized fundamental system of solutions φ α,β,γ, ( z, · ) , θ α,β,γ, ( z, · ) of τ α,β,γ u = zu , z ∈ C , that is real-valued for z ∈ R and entire with respect to z ∈ C by e φ α,β,γ, ( z,
0) = 0 , e φ ′ α,β,γ, ( z,
0) = 1 , e θ α,β,γ, ( z,
0) = 1 , e θ ′ α,β,γ, ( z,
0) = 0 , z ∈ C , (6.9)one obtains explicitly, φ α,β,γ, ( z, x ) = ( (1 − β ) − (2 + α − β ) γ Γ(1 + γ ) z − γ/ y ,α,β,γ ( z, x ) , γ ∈ (0 , , (1 − β ) − y ,α,β, ( z, x ) , γ = 0 ,z ∈ C , x ∈ (0 , ∞ ) , (6.10) θ α,β,γ, ( z, x ) = (1 − β )(2 + α − β ) − γ − γ − Γ(1 − γ ) z γ/ y ,α,β,γ ( z, x ) , γ ∈ (0 , , (1 − β )(2 + α − β ) − [ − πy ,α,β, ( z, x )+(ln( z ) − α − β ) + 2 γ E ) y ,α,β, ( z, x )] , γ = 0 ,z ∈ C , x ∈ (0 , ∞ ) , (6.11) W ( θ α,β,γ, ( z, · ) , φ α,β,γ, ( z, · )) = 1 , z ∈ C , (6.12)where Γ( · ) denotes the Gamma function, and γ E = 0 . . . . represents Euler’sconstant.Since τ α,β,γ is in the limit point case at ∞ (actually, it is in the strong limitpoint case at infinity since q is bounded on any interval of the form [ R, ∞ ), R > τ α,β,γ =(1 − β ) / (2+ α − β ) has been shown in [35]),in order to find the m -function corresponding to the Friedrichs (resp., Dirichlet)boundary condition at x = 0, one considers the requirement ψ α,β,γ, ( z, · ) = θ α,β,γ, ( z, · ) + m α,β,γ, ( z ) φ α,β,γ, ( z, · ) ∈ L ((0 , ∞ ); x α dx ) ,z ∈ C \ R . (6.13)This implies ψ α,β,γ, ( z, x ) = i (1 − β )(2 + α − β ) − γ − γ − Γ(1 − γ ) sin( πγ ) z γ/ × x (1 − β ) / H (1) γ (cid:0) z / x (2+ α − β ) / / (2 + α − β ) (cid:1) , γ ∈ (0 , ,iπ (1 − β ) / (2 + α − β ) x (1 − β ) / × H (1)0 (cid:0) z / x (2+ α − β ) / / (2 + α − β ) (cid:1) , γ = 0 ,z ∈ C \ [0 , ∞ ) , x ∈ (0 , ∞ ) , (6.14) m α,β,γ, ( z ) = − e − iπγ (1 − β ) (2 + α − β ) − γ − γ − × [Γ(1 − γ ) / Γ(1 + γ )] z γ , γ ∈ (0 , , (1 − β ) / (2 + α − β ) × [ iπ − ln( z ) + 2ln(2 + α − β ) − γ E ] , γ = 0 , (6.15) z ∈ C \ [0 , ∞ ) , where H (1) ν ( · ) is the Hankel function of the first kind and of order ν ∈ R (cf. [1,Ch. 9]). In particular, the results (6.14) and (6.15) coincide with the ones obtainedin [41] when α = β = 0 and [35] when γ = (1 − β ) / (2 + α − β ). L -realizations associated with the differential expression τ α,β,γ are then intro-duced as usual by( T α,β,γ,max f )( x ) = ( τ α,β,γ f )( x ) , x ∈ (0 , ∞ ) ,f ∈ dom( T α,β,γ,max ) = (cid:8) g ∈ L ((0 , ∞ ); x α dx ) (cid:12)(cid:12) g, g ′ ∈ AC loc ((0 , ∞ )); (6.16) τ α,β,γ g ∈ L ((0 , ∞ ); x α dx ) (cid:9) , ( T α,β,γ,min f )( x ) = ( τ α,β,γ f )( x ) , x ∈ (0 , ∞ ) ,f ∈ dom( T α,β,γ,min ) = (cid:8) g ∈ dom( T α,β,γ,max ) (cid:12)(cid:12) e g (0) = 0 , e g ′ (0) = 0 (cid:9) , (6.17)in particular, T ∗ α,β,γ,min = T α,β,γ,max , T ∗ α,β,γ,max = T α,β,γ,min . (6.18)Thus, following Kalf [54] (see also [82], [88]) one obtains for the Friedrichs extension T α,β,γ,F of T α,β,γ,min ,( T α,β,γ,F f )( x ) = ( τ α,β,γ f )( x ) , x ∈ (0 , ∞ ) ,f ∈ dom( T α,β,γ,F ) = (cid:8) g ∈ dom( T α,β,γ,max ) (cid:12)(cid:12) e g (0) = 0 (cid:9) . (6.19)Since u ,α,β,γ (0 , x ) = (1 − β ) − x [1 − β +(2+ α − β ) γ ] / > , γ ∈ [0 , , x ∈ (0 , , (6.20)standard oscillation theory implies that T α,β,γ,min ≥ T α,β,γ,F ≥ . (6.21)Incidentally, we note that T α,β,γ,F ≥ m α,β,γ, ( · ), proving that the spectral measure corresponding to T α,β,γ,F (i.e., the measure in the Nevanlinna–Herglotz representation of m α,β,γ, ( · ))has no support in ( −∞ , u ,α,β,γ ( x )[ x β u ′ ,α,β,γ ( x )] = x ↓ O (cid:0) x (2+ α − β ) γ (cid:1) , γ ∈ [0 , , (6.22) u ,α,β,γ ( x ) / b u ,α,β,γ ( x ) = x ↓ ( O (cid:0) x (2+ α − β ) γ (cid:1) , γ ∈ (0 , ,O (cid:0) [ln(1 /x )] − (cid:1) , γ = 0 , (6.23)in particular, u ,α,β,γ ( x ) (cid:2) x β u ′ ,α,β,γ ( x ) (cid:3) = x ↓ O (cid:0) u ,α,β,γ ( x ) / b u ,α,β,γ ( x ) (cid:1) for γ ∈ (0 , γ = 0), a condition isolated in [52], and that b u ,α,β,γ ( x ) (cid:2) x β b u ′ ,α,β,γ ( x ) (cid:3) = x ↓ ( O (cid:0) x − (2 − α + β ) γ (cid:1) , γ ∈ (0 , ,O (cid:0) [ln( x )] (cid:1) , γ = 0 , does not exist . (6.24)Turning our attention to x = ∞ , we now introduce (non-normalized) principaland nonprincipal solutions u ∞ ,α,β,γ (0 , · ) and b u ∞ ,α,β,γ (0 , · ) of τ α,β,γ u = 0 at x = ∞ by u ∞ ,α,β,γ (0 , x ) = x [1 − β − (2+ α − β ) γ ] / , γ ∈ [0 , , b u ∞ ,α,β,γ (0 , x ) = ( x [1 − β +(2+ α − β ) γ ] / , γ ∈ (0 , ,x (1 − β ) / ln( x ) , γ = 0 ,α > − , β < , x ∈ (1 , ∞ ) , (6.25) SURVEY OF SOME NORM INEQUALITIES 21 to obtain u ∞ ,α,β,γ ( x ) (cid:2) x β u ′∞ ,α,β,γ ( x ) (cid:3) = x ↑∞ O (cid:0) x − (2+ α − β ) γ (cid:1) , γ ∈ [0 , , (6.26) u ∞ ,α,β,γ ( x ) / b u ∞ ,α,β,γ ( x ) = x ↑∞ ( O (cid:0) x − (2+ α − β ) γ (cid:1) , γ ∈ (0 , ,O (cid:0) [ln( x )] − (cid:1) , γ = 0 , (6.27)hence, one infers once again that u ∞ ,α,β,γ ( x ) (cid:2) x β u ′∞ ,α,β,γ ( x ) (cid:3) = x ↑∞ O (cid:0) u ∞ ,α,β,γ ( x ) / b u ∞ ,α,β,γ ( x ) (cid:1) for γ ∈ (0 ,
1) (6.28)(but not when γ = 0), and that b u ∞ ,α,β,γ ( x ) (cid:2) x β b u ′∞ ,α,β,γ ( x ) (cid:3) = x ↑∞ ( O (cid:0) x (2 − α + β ) γ (cid:1) , γ ∈ (0 , ,O (cid:0) [ln( x )] (cid:1) , γ = 0 , does not exist . (6.29)Given (6.22)–(6.29), [88, Corollary 3] applies near x = 0 as well as near x = ∞ ,and thus one obtains in addition to (6.19) that( T α,β,γ,F f )( x ) = ( τ α,β,γ f )( x ) , x ∈ (0 , ∞ ) ,f ∈ dom( T α,β,γ,F ) = (cid:8) g ∈ dom( T α,β,γ,max ) (cid:12)(cid:12) g ′ ∈ L ((0 , ∞ ); x β dx ); (6.30) g ∈ L ((0 , ∞ ); x β − dx ) (cid:9) , α > − , β < , γ ∈ (0 , . Here we used the Monotone Convergence Theorem to conclude that ∞ > lim c ↓ , d ↑∞ ˆ dc dx x β − | g ( x ) | = lim c ↓ , d ↑∞ ˆ ∞ dx x β − χ [ c,d ] ( x ) | g ( x ) | = ˆ ∞ dx x β − | g ( x ) | , g ∈ dom( T α,β,γ,max ) . (6.31)One notes that the characterization (6.30) cannot hold for γ = 0 by simplychoosing g to equal u ,α,β, near x = 0 and u ∞ ,α,β, near x = ∞ (all integrandsthen are of the form O (1 /x ) near x = 0 , ∞ ). However, for γ > Lemma 6.2. ( Kalf and Walter [55, Lemma 1 (a)]) . Suppose that < p a.e. on (0 , ∞ ) , p − ∈ L ((0 , c ); dx ) for all c ∈ (0 , ∞ ) ,f ∈ AC loc ((0 , ∞ )) , f ′ ∈ L ((0 , ∞ ); p ( x ) dx ) , lim inf x ↓ | f ( x ) | = 0 . (6.32) Then lim x ↓ | u ( x ) | ´ x dt p ( t ) − = 0 , (6.33) and for all R ∈ (0 , ∞ ) ∪ {∞} , ˆ R dx p ( x ) | f ′ ( x ) | ≥ ˆ R dx | f ( x ) | p ( x ) h ´ x dt p ( t ) − i . (6.34)The generalized version of weighted Hardy inequalities, especially, its integralinequality version (replacing f ( x ) by ´ xa dt F ( t ) or ´ bx dt F ( t ), and hence f ′ ( x ) by F ( x ), etc.) in L (( a, b ); dx ) was established by Talenti [93] and Tomaselli [94] in1969 and independently rediscovered by Chisholm and Everitt [15] in 1971 (see also [16] for a more general result in the conjugate index case 1 /p + 1 /q = 1, and [39],[40], and the references therein, for recent developments). In addition, a 1972 paperby Muckenhoupt [79] has further generalizations. For additional information on thefascinating history of this type of inequalities we refer to the excellent account inthe [63, Ch. 4, pp. 33–37].An application of Lemma 6.2 to the example at hand with p ( x ) = x β , β < ˆ R dx x β | f ′ ( x ) | ≥ (1 − β ) ˆ R dx x β − | f ( x ) | , β < , R ∈ (0 , ∞ ) ∪ {∞} , (6.35)assuming f ∈ AC loc ((0 , ∞ )) , f ′ ∈ L (cid:0) (0 , ∞ ); x β dx (cid:1) , lim inf x ↓ | f ( x ) | = 0 . (6.36)Since q is of the form, q ( x ) = C α,β,γ x β − , a combination of (6.30) and (6.35), (6.36)yields( T α,β,γ,F f )( x ) = ( τ α,β,γ f )( x ) , x ∈ (0 , ∞ ) ,f ∈ dom( T α,β,γ,F ) = (cid:8) g ∈ dom( T α,β,γ,max ) (cid:12)(cid:12) g ′ ∈ L ((0 , ∞ ); x β dx ); (6.37)lim inf x ↓ | g ( x ) | = 0 (cid:9) , α > − , β < , γ ∈ (0 , . Finally, combining Lemma 5.5 and (6.37) yields ˆ ∞ dx h x β | f ′ ( x ) | + 4 − (cid:2) (2 + α − β ) γ − (1 − β ) (cid:3) x β − | f ( x ) | i ≤ (cid:18) ˆ ∞ x α dx | f ( x ) | (cid:19) / (cid:18) ˆ ∞ x α dx | ( τ α,β,γ f )( x ) | (cid:19) / , (6.38) f ∈ dom( T α,β,γ,F ) , α > − , β < , γ ∈ (0 , , extending the special case α = β = 0 treated at the end of [34].Judging from Theorem 5.2 in the regular context at x = a , one might naivelyguess at first that an inequality of the kind of (6.38) might extend to other self-adjoint extensions of T α,β,γ,min (perhaps, even to T α,β,γ,max ). In the remainderof this section we will prove that this fails and that (6.38) cannot extend to anyself-adjoint extension of T α,β,γ,min other than the Friedrichs extension T α,β,γ,F .We start by noting (cf. [41, Sects. 3, 4]) all self-adjoint extensions of T α,β,γ,min are of the form,( T α,β,γ,δ f )( x ) = ( τ α,β,γ f )( x ) , x ∈ (0 , ∞ ) , δ ∈ [0 , π ) ,f ∈ dom( T α,β,γ,δ ) = (cid:8) g ∈ dom( T α,β,γ,max ) (cid:12)(cid:12) sin( δ ) e g ′ (0) + cos( δ ) e g (0) = 0 (cid:9) , (6.39)and hence the Friedrichs extension T α,β,γ,F of T α,β,γ,min corresponds to the case δ = 0. In order to describe the resolvent of T α,β,γ,δ we need a few preparations.Introducing, φ α,β,γ,δ ( z, x ) = cos( δ ) φ α,β,γ, ( z, x ) − sin( δ ) θ α,β,γ, ( z, x ) ,θ α,β,γ,δ ( z, x ) = sin( δ ) φ α,β,γ, ( z, x ) + cos( δ ) θ α,β,γ, ( z, x ) , (6.40) z ∈ C , x ∈ (0 , ∞ ) , δ ∈ [0 , π ) , observing W ( θ α,β,γ,δ ( z, · ) , φ α,β,γ,δ ( z, · )) = 1 , z ∈ C , (6.41) SURVEY OF SOME NORM INEQUALITIES 23 one considers the Weyl–Titchmarsh solutions ψ α,β,γ,δ ( z, · ) = θ α,β,γ,δ ( z, · ) + m α,β,γ,δ ( z ) φ α,β,γ,δ ( z, · ) ∈ L ((0 , ∞ ); x α dx ) ,z ∈ C \ R , δ ∈ [0 , π ) , (6.42)where m α,β,γ,δ ( z ) = − sin( δ ) + cos( δ ) m α,β,γ, ( z )cos( δ ) + sin( δ ) m α,β,γ, ( z ) , z ∈ C \ R , δ ∈ [0 , π ) . (6.43)Since τ α,β,γ is in the limit point case at ∞ , one concludes that ψ α,β,γ,δ ( z, · ) = C α,β,γ,δ ( z ) ψ α,β,γ, ( z, · ) (6.44)for some constant C α,β,γ,δ ( z ) ∈ C \{ } .One then obtains for the Green’s function G α,β,γ,δ ( z, · , · ) of T α,β,γ,δ (i.e., theintegral kernel of the resolvent ( T α,β,γ,δ − zI L ((0 , ∞ ); x α dx ) ) − ), G α,β,γ,δ ( z, x, x ′ ) = ( φ α,β,γ,δ ( z, x ) ψ α,β,γ,δ ( z, x ′ ) , < x ≤ x ′ < ∞ ,φ α,β,γ,δ ( z, x ′ ) ψ α,β,γ,δ ( z, x ) , < x ′ ≤ x < ∞ , z ∈ C \ R . (6.45)Next, consider g ∈ L ((0 , ∞ ); x α dx ), with supp ( g ) ⊂ (0 , ∞ ) compact (i.e., thesupport of g is away from 0 and from ∞ ), then f ( z, δ, · ) = ( T α,β,γ,δ − zI L ((0 , ∞ ); x α dx ) ) − g ∈ dom( T α,β,γ,δ ) , z ∈ C \ R , (6.46)and the set D α,β,γ,δ,z of such f ( z, δ, · ) forms an operator core of T α,β,γ,δ , thatis, T α,β,γ,δ (cid:12)(cid:12) D α,β,γ,δ,z = T α,β,γ,δ (since T α,β,γ,min is symmetric and the set of g ∈ L ((0 , ∞ ); x α dx ) with supp( g ) ⊂ (0 , ∞ ) compact is dense in L ((0 , ∞ ); x α dx ), see[87, Corollary, p. 257]).One then obtains from (6.45), f ( z, δ, x ) = ψ α,β,γ,δ ( z, x ) ˆ x ( x ′ ) α dx ′ φ α,β,γ,δ ( z, x ′ ) g ( x ′ )+ φ α,β,γ,δ ( z, x ) ˆ ∞ x ( x ′ ) α dx ′ ψ α,β,γ,δ ( z, x ′ ) g ( x ′ )= φ α,β,γ,δ ( z, x ) ˆ ∞ ( x ′ ) α dx ′ ψ α,β,γ,δ ( z, x ′ ) g ( x ′ ) , z ∈ C \ R , (6.47)for 0 < x < inf(supp ( g )). Changing z ∈ C \ R a bit if necessary, we may assumewithout loss of generality that ˆ ∞ ( x ′ ) α dx ′ ψ α,β,γ,δ ( z, x ′ ) g ( x ′ ) = 0 . (6.48)Thus, for some constant c α,β,γ ( z ) ∈ C \{ } , φ α,β,γ,δ ( z, x ) = x ↓ sin( δ ) c α,β,γ ( z ) x [(1 − β ) − (2+ α − β ) γ ] / , δ ∈ (0 , π ) , (6.49)and hence for all α > − β < γ ∈ (0 , δ ∈ (0 , π ),lim ε ↓ ˆ ∞ ε dx x β | f ′ ( z, δ, x ) | = ∞ , lim ε ↓ ˆ ∞ ε dx x β − | f ( z, δ, x ) | = ∞ , (6.50) provided (2 + α − β ) γ − (1 − β ) = 0. Thus, as long as q
0, no cancellation in theanalog of the left-hand side of (6.38), namely, ˆ ∞ dx h x β | f ′ ( z, δ, x ) | + 4 − (cid:2) (2 + α − β ) γ − (1 − β ) (cid:3) x β − | f ( z, δ, x ) | i , (6.51)can possibly occur. Hence, for any δ ∈ (0 , π ) (i.e., for all self-adjoint extensions of T α,β,γ,min other than the Friedrichs extension T α,β,γ,F ), there exists a core D α,β,γ,δ,z for T α,β,γ,δ such that the analog of the left-hand side of (6.38) diverges, renderingan analog of Theorem 5.2 for T α,β,γ,min moot.Apart from [35] in the special case γ = (1 − β ) / (2 + α − β ), implying q ≡ τ α,β, (1 − β ) / (2+ α − β ) regular at x = 0, we did not find a treatment of the example inthis section in the literature. Acknowledgments.
We are indebted to Man Kam Kwong and Lance Littlejohnfor helpful discussions.
Data Availability Statement.
Data sharing is not applicable to this article asno datasets were generated or analyzed during the current study.
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