CUBO, A Mathematical Journal Vol. 24, no. 01, pp. 115–165, April 2022 DOI: 10.4067/S0719-06462022000100115 Optimality of constants in power-weighted Birman–Hardy–Rellich-Type inequalities with logarithmic refinements Fritz Gesztesy 1 Isaac Michael 2 Michael M. H. Pang 3 1Department of Mathematics, Baylor University, Sid Richardson Bldg., 1410 S., 4th Street, Waco, TX 76706, USA. Fritz Gesztesy@baylor.edu 2Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA. imichael@lsu.edu 3Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. pangm@missouri.edu ABSTRACT The principal aim of this paper is to establish the optimality (i.e., sharpness) of the constants A(m, α) and B(m, α), m ∈ N, α ∈ R, of the form A(m, α) = 4 −m m ∏ j=1 (2j − 1 − α) 2 , B(m, α) = 4 −m m ∑ k=1 m ∏ j=1 j 6=k (2j − 1 − α) 2 , in the power-weighted Birman–Hardy–Rellich-type inte- gral inequalities with logarithmic refinement terms recently proved in [41], namely, ˆ ρ 0 dx x α ∣ ∣f (m) (x) ∣ ∣ 2 > A(m, α) ˆ ρ 0 dx x α−2m ∣ ∣f(x) ∣ ∣ 2 + B(m, α) N ∑ k=1 ˆ ρ 0 dx x α−2m k ∏ p=1 [lnp(γ/x)] −2 ∣ ∣f(x) ∣ ∣ 2 , f ∈ C ∞ 0 ((0, ρ)), m, N ∈ N, α ∈ R, ρ, γ ∈ (0, ∞), γ > eN ρ. Here the iterated logarithms are given by ln1( · ) = ln( · ), lnj+1( · ) = ln(lnj( · )), j ∈ N, and the iterated exponentials are defined via e0 = 0, ej+1 = e ej , j ∈ N0 = N ∪ {0}. Moreover, we prove the analogous sequence of inequalities on the exterior interval (r, ∞) for f ∈ C∞0 ((r, ∞)), r ∈ (0, ∞), and once again prove optimality of the constants involved. Accepted: 23 February, 2022 Received: 28 June, 2021 c©2022 F. Gesztesy et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462022000100115 https://orcid.org/0000-0001-8554-9745 https://orcid.org/0000-0003-1697-2018 https://orcid.org/0000-0002-5262-8882 mailto:Fritz_Gesztesy@baylor.edu mailto:imichael@lsu.edu mailto:pangm@missouri.edu CUBO, A Mathematical Journal Vol. 24, no. 01, pp. 115–167, April 2022 DOI: 10.4067/S0719-06462022000100115 RESUMEN El objetivo principal de este art́ıculo es establecer la optima- lidad (i.e. la precisión) de las constantes A(m, α) y B(m, α), m ∈ N, α ∈ R, de la forma A(m, α) = 4 −m m ∏ j=1 (2j − 1 − α) 2 , B(m, α) = 4 −m m ∑ k=1 m ∏ j=1 j 6=k (2j − 1 − α) 2 , en las desigualdades integrales de tipo Birman–Hardy– Rellich pesadas por potencias con términos de refinamiento logaŕıtmicos recientemente demostradas en [41], es decir, ˆ ρ 0 dx x α ∣ ∣f (m) (x) ∣ ∣ 2 > A(m, α) ˆ ρ 0 dx x α−2m ∣ ∣f(x) ∣ ∣ 2 + B(m, α) N ∑ k=1 ˆ ρ 0 dx x α−2m k ∏ p=1 [lnp(γ/x)] −2 ∣ ∣f(x) ∣ ∣ 2 , f ∈ C ∞ 0 ((0, ρ)), m, N ∈ N, α ∈ R, ρ, γ ∈ (0, ∞), γ > eN ρ. Acá los logaritmos iterados están dados por ln1( · ) = ln( · ), lnj+1( · ) = ln(lnj( · )), j ∈ N, y las exponenciales iteradas están definidas por e0 = 0, ej+1 = e ej , j ∈ N0 = N ∪ {0}. Más aún, probamos la secuencia análoga de desigualdades en el intervalo exterior (r, ∞) para f ∈ C∞0 ((r, ∞)), r ∈ (0, ∞), y una vez más probamos la optimalidad de las constantes involucradas. Keywords and Phrases: Birman-Hardy-Rellich inequalities, logarithmic refinements. 2020 AMS Mathematics Subject Classification: 26D10, 34A40, 35A23, 34L10. Accepted: 23 February, 2022 Received: 28 June, 2021 c©2022 F. Gesztesy et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462022000100115 CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 117 1 Introduction and notations employed Given the notation introduced in (1.4)–(1.8) we will prove in this paper that the constants A(m,α) and the N constants B(m,α) appearing in the power-weighted Birman–Hardy–Rellich-type integral inequalities with logarithmic refinement terms, ˆ ρ 0 dxxα ∣∣f(m)(x) ∣∣2 > A(m,α) ˆ ρ 0 dxxα−2m ∣∣f(x) ∣∣2 + B(m,α) N∑ k=1 ˆ ρ 0 dxxα−2m k∏ p=1 [lnp(γ/x)] −2 ∣∣f(x) ∣∣2, (1.1) f ∈ C∞0 ((0,ρ)), m,N ∈ N, α ∈ R, ρ,γ ∈ (0,∞), γ > eNρ, recently proved in [41], are optimal (i.e., sharp). Moreover, we prove optimality of A(m,α) and the N constants B(m,α) for the analogous sequence of inequalities on the exterior interval (r,∞), that is, ˆ ∞ r dxxα ∣∣f(m)(x) ∣∣2 > A(m,α) ˆ ∞ r dxxα−2m ∣∣f(x) ∣∣2 + B(m,α) N∑ k=1 ˆ ∞ r dxxα−2m k∏ p=1 [lnp(x/Γ)] −2 ∣∣f(x) ∣∣2, (1.2) f ∈ C∞0 ((r,∞)), m,N ∈ N, α ∈ R, r,Γ ∈ (0,∞), r > eNΓ. Of course, (1.1) (resp., (1.2)) extends to N = 0, ρ = ∞ (resp., to N = 0, r = 0) upon disregarding all logarithmic terms (i.e., upon putting B(m,α) = 0). In their simplest (i.e., unweighted) form, the Birman–Hardy–Rellich inequalities, as recorded by Birman in 1961, and in English translation in 1966 [19] (see also [45, pp. 83–84]), are given by ˆ ρ 0 dx ∣∣f(m)(x) ∣∣2 > [(2m − 1)!!]2 22m ˆ ρ 0 dxx−2m|f(x)|2, (1.3) f ∈ Cm0 ((0,ρ)), m ∈ N, 0 < ρ 6 ∞. The case m = 1 in (1.3) represents Hardy’s celebrated inequality [51], [52, Sect. 9.8] (see also [61, Chs. 1, 3, App.]), the case m = 2 is due to Rellich [81, Sect. II.7]. The power-weighted extension of (1.3) is then represented by the first line of (1.1) (i.e., by deleting the second line in (1.1) which contains additional logarithmic refinements). Even though a detailed history of the power-weighted Birman–Hardy–Rellich inequalities was provided in the companion paper [41], we will now repeat the highlights of this history for matters of completeness. We start with the observation that the inequalities (1.3) and their power weighted generalizations, that is, the first line in (1.1), are known to be strict, that is, equality holds in (1.3), resp., in the first line in (1.1) (in fact, for the entire inequality (1.1)) if and only if f = 0 on (0,ρ). Moreover, 118 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) these inequalities are optimal, meaning, the constants [(2m − 1)!!]2/22m in (1.3), respectively, the constants A(m,α) in (1.1) are sharp, although, this must be qualified and will be revisited below as different authors frequently prove sharpness for different function spaces. In the present one-dimensional context at hand, sharpness of (1.3) (and typically, its power weighted version, the first line in (1.1)), are often proved in an integral form (rather than the currently presented differential form) where f(m) on the left-hand side is replaced by F and f on the right-hand side by m repeated integrals over F . For pertinent one-dimensional sources, we refer, for instance, to [14, pp. 3–5], [22], [24, pp. 104–105], [42, 49, 51], [52, pp. 240–243], [61, Ch. 3], [62, pp. 5–11], [64, 72, 80]. We also note that higher-order Hardy inequalities, including various weight functions, are discussed in [60, Sect. 5], [61, Chs. 2–5], [62, Chs. 1–4], [63], and [79, Sect. 10] (however, Birman’s sequence of inequalities (1.3) is not mentioned in these sources). In addition, there are numerous sources which treat multi-dimensional versions of these inequalities on various domains Ω ⊆ Rn, which, when specialized to radially symmetric functions (e.g., when Ω represents a ball), imply one-dimensional Birman–Hardy–Rellich-type inequalities with power weights under various restrictions on these weights. However, none of the results obtained in this manner imply (1.1), under optimal hypotheses on α and γ. We also mention that a large number of these references treat the Lp-setting, and in some references x ∈ (a,b) is replaced by d(x), the distance of x to the boundary of (a,b), respectively, Ω, but this represents quite a different situation (especially in the multi-dimensional context) and hence is not further discussed in this paper. To put the logarithmic refinements in (1.1) (i.e., the second line in (1.1)) into some perspective and to compare with existing results in the literature, we offer the following comments: originally, logarithmic refinements of Hardy’s inequality started with oscillation theoretic considerations going back to Hartman [53] (see also [54, pp. 324–325]) and have been used in connection with Hardy’s inequality in [38, 43], and more recently, in [39, 40]. Since then there has been enormous activity in this context and we mention, for instance, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, Chs. 3, 5], [16, 17, 18, 21, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 39, 44, 46, 47], [48, Chs. 2, 6, 7], [56, 57, 65, 66, 67, 68, 70, 71, 74, 76, 77], [81, Sect. 2.7], [82, 83, 84, 88, 89, 90, 91]. The vast majority of these references deals with analogous multi-dimensional settings (relevant to our setting in particular in the case of radially symmetric functions), several also with the Lp-context. For m > 2 the inequalities (1.1) and (1.2) proven in [41] were new in the following sense: the weight parameter α ∈ R is unrestricted (as opposed to prior results) and at the same time the conditions on the logarithmic parameters γ and Γ are sharp. The issue of sharpness of the constants A(m,α) and B(m,α) appearing in (1.1) is a rather delicate one and hence we offer the following remarks, the gist of which can be found in [41, Appendix A]. We start by noting that the smaller the underlying function space, the larger the efforts needed to prove optimality. Many of the results cited in the remainder of this remark, under particular restrictions on the weight parameter α, establish sharpness for larger classes of functions f which CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 119 do not automatically continue to hold in the C∞0 ((0,ρ))-context. It is this simple observation that adds considerable complexity to sharpness proofs for the space C∞0 ((0,ρ)). (The issue of dependence of optimal constants on the underlying function space is nicely illustrated in [30].) By the same token, optimality proofs obtained for C∞0 function spaces automatically hold for larger function spaces as long as the inequalities have already been established for the larger function spaces with the same constants A(m,α),B(m,α). This comment applies, in particular, to many papers that prove sharpness results in multi-dimensional situations for larger function spaces such as1 C∞0 (B(0;ρ)) or (homogeneous, weighted) Sobolev spaces rather than C ∞ 0 (B(0;ρ)\{0}). Unless C∞0 (B(0;ρ)\{0}) is dense in the appropriate norm, one cannot a priori assume that the optimal constants A(m,α̃) and B(m,α̃) (with α̃ appropriately depending on n, e.g., α̃ = α+n− 1) remain the same for C∞0 (B(0;ρ)) and C ∞ 0 (B(0;ρ)\{0}), say. At least in principle, they could actually increase for the space C∞0 (B(0;ρ)\{0}). Turning to a review of the existing literature, sharpness of the constant A(m,0), m ∈ N (i.e., in the unweighted case, α = 0), corresponding to the space C∞0 ((0,∞)) has been shown by Yafaev [91]. In fact, he also established this result for fractional m (in this context we also refer to appropriate norm bounds in Lp(Rn;dnx) of operators of the form |x| −β |−i∇| −β , 1 < p < n/β, see [13, Sect. 1.7],[14, 55, 58, 59, 78, 86], [87, Sects. 1.7, 4.2]). Sharpness of A(2,0) (i.e., in the unweighted Rellich case) was shown by Rellich [81, pp. 91–101] in connection with the space C∞0 ((0,∞)); his multi-dimensional results also yield sharpness of A(2,n−1) for n ∈ N, n > 3, again for C∞0 ((0,∞)); in this context see also [14, Corollary 6.3.5]. An exhaustive study of optimality of A(2, α̃) (i.e., Rellich inequalities with power weights) for the space C∞0 (Ω\{0}) for cones Ω ⊆ R n, n > 2, appeared in Caldiroli and Musina [21]. The authors, in particular, describe situations where A(2, α̃) has to be replaced by other constants and also treat the special case of radially symmetric functions in detail. Additional results for power weighted Rellich inequalities appeared in [74, 75]; further extensions of power weighted Rellich inequalities with sharp constants on C∞0 (R n\{0}) were obtained in [69]; for optimal power weighted Hardy, Rellich, and higher-order inequalities on homogeneous groups, see [82, 83]. Many of these references also discuss sharp (power weighted) Hardy inequalities, implying optimality for A(1, α̃). Moreover, replacing f(x) by F(x) = ´ x 0 dtf(t) ( or F(x) = ´∞ x dtf(t) ) , optimality of the Hardy constant A(1,0) for larger, Lp-based function spaces, can already be found in [52, Sect. 9.8] (see also [14, Theorem 1.2.1], [61, Ch. 3], [62, pp. 5– 11], [64, 72, 80], in connection with A(1,α)). We mention that Theorems 4.1 and 4.7, which assert optimality of A(m,α) in (1.1) and (1.2), were already proved in [41, Theorem A.1] using a different method. Sharpness results for A(m,α) and B(m,α) together are much less frequently discussed in the literature, even under suitable restrictions on m and α. The results we found primarily follow upon specializing multi-dimensional results for function spaces such as C∞0 (Ω\{0}), or C ∞ 0 (Ω), Ω ⊆ R n 1Here B(0; ρ) ⊆ Rn denotes the open ball in Rn, n > 2, with center at the origin x = 0 and radius ρ > 0. 120 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) open, and appropriate restrictions on m, α, and n > 2, for radially symmetric functions to the one- dimensional case at hand (cf. the previous paragraph). In this context we mention that the Hardy case m = 1, without a weight function, is studied in [1, 2, 5, 9, 20, 23, 26, 36, 50, 57, 65, 85, 89] (all for N = 1), and in [10, 28, 46] (all for N ∈ N); the case with power weight functions is discussed in [17, 47], [48, Ch. 6] (for N ∈ N); see also [66]. The Rellich case m = 2 with a general power weight on C∞0 (Ω\{0}) is discussed in [21] (for N = 1); the Rellich case m = 2, without weight function on C∞0 (Ω), is studied in [26, 27, 29] (all for N = 1), the case N ∈ N is studied in [4]; the case of additional power weights is treated in [47], [48, Ch. 6], [71]. The general case m ∈ N is discussed in [6] (for N = 1) and in [15, 47], [48, Ch. 6], [90] (all for N ∈ N and including power weights, but with additional restrictions). Employing oscillation theory, sharpness of the unweighted Hardy case A(1,0) = B(1,0) = 1/4, with N ∈ N, was proved in [43]. As will become clear in the course of this paper, the special results available on sharpness of the N constants B(m,α) are all saddled with considerable complexity, especially, for larger values of N ∈ N. For this reason only sharpness of the constants A(m,α) was derived in [41, Appendix A] and sharpness of A(m,α) and B(m,α) was postponed to this paper which therefore should be viewed as a companion of [41]. In Section 2 (a very massive one) we establish all the preliminary results, culminating in Lemmas 2.13 and 2.14, required in the remainder of this paper. The methods used in this section are adaptations of those in [15, Sect. 3]. The basic approximation procedure is introduced in Section 3, with Corollaries 3.12 and 3.13 summarizing the principal results. Our final Section 4 then proves optimality of the N + 1 constants A(m,α) and B(m,α) for the interval (0,ρ) in Theorems 4.1 and 4.2 and for the interval (r,∞) in Theorems 4.7 and 4.8 based on Lemmas 2.13 and 2.14 and Corollaries 3.12 and 3.13. We also mention that Theorems 4.2 and 4.8 still hold if the repeated log-terms lnp( · ) (see (1.5) below) are replaced by the type of repeated log-terms used, for example, in [15, 16, 17, 90].2 We conclude this introduction by establishing the principal notation used in this paper: for j ∈ N0 (with N0 = N ∪ {0}) we define ej by e0 = 0, e1 = 1, ej+1 = e ej, j ∈ N. (1.4) For N ∈ N, γ,ρ ∈ (0,∞), with γ > ρeN, and 1 6 j 6 N, we define lnj(γ/x), for 0 < x < ρ, by ln1(γ/x) = ln(γ/x), lnj+1(γ/x) = ln(lnj(γ/x)), 1 6 j 6 N − 1. (1.5) For the rest of this paper we shall assume that N ∈ N ∪ {0}, m ∈ N, α ∈ R, γ,ρ ∈ (0,∞), with 2Detailed proofs of Theorems 4.2 and 4.8 for the type of log-terms used in [15, 16, 17, 90] are available from the authors upon request. CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 121 γ > ρeN+1. We shall write A(m,α) = 4−m m∏ j=1 (2j − 1 − α)2, (1.6) B(m,α) = 4−m m∑ k=1 m∏ j=1,j 6=k (2j − 1 − α)2. (1.7) Note that if α ∈ R\{2j − 1}16j6m, one has B(m,α) = A(m,α) m∑ j=1 (2j − 1 − α)−2. (1.8) We assume ψ ∈ C∞(R) satisfies the following properties: (i) ψ is non-increasing, (1.9) (ii) ψ(x) =    1, x 6 8ρ/10, 0, x > 9ρ/10. (1.10) For g ∈ C∞((0,ρ)) we shall write JN[g] = ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 − A(m,α) ˆ ρ 0 dxxα−2m|g(x)|2 − B(m,α) N∑ k=1 ˆ ρ 0 dxxα−2m|g(x)|2 k∏ j=1 [lnj(γ/x)] −2, (1.11) provided that ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 < ∞, ˆ ρ 0 dxxα−2m|g(x)|2 < ∞. (1.12) For j = 0,1, . . . ,N and β ∈ R we introduce σ0(β) = (2m − 1 − α + β)/2, σj(β) = −(1 − β)/2, j = 1, . . . ,N. (1.13) For 0 6 j 6 k 6 N and ε = (ε0,ε1, . . . ,εN), where ε0,ε1, . . . ,εN > 0, we shall write Γj,k(ε) = Γj,k(ε0,ε1, . . . ,εN), = ˆ ρ 0 dxx−1+ε0 [ln1(γ/x)] −1−ε1 · · · [lnj(γ/x)] −1−εj × [lnj+1(γ/x)] −εj+1 · · · [lnk(γ/x)] −εk × [lnk+1(γ/x)] 1−εk+1 · · · [lnN(γ/x)] 1−εN [ψ(x)]2. (1.14) 122 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) In particular, if N ∈ N, Γ0,0(ε) = ˆ ρ 0 dxx−1+ε0 N∏ k=1 [lnk(γ/x)] 1−εk [ψ(x)]2, Γ0,k(ε) = ˆ ρ 0 dxx−1+ε0 k∏ ℓ=1 [lnℓ(γ/x)] −εℓ N∏ p=k+1 [lnp(γ/x)] 1−εp[ψ(x)]2, k = 1, . . . ,N, Γk,k(ε) = ˆ ρ 0 dxx−1+ε0 k∏ ℓ=1 [lnℓ(γ/x)] −1−εℓ N∏ p=k+1 [lnp(γ/x)] 1−εp[ψ(x)]2, k = 1, . . . ,N, ΓN,N(ε) = ˆ ρ 0 dxx−1+ε0 N∏ ℓ=1 [lnℓ(γ/x)] −1−εℓ[ψ(x)]2. (1.15) For k ∈ N we shall write Pk for the polynomial Pk(σ) = σ(σ − 1) · · · (σ − k + 1), σ ∈ R. (1.16) For β = (β0,β1, . . . ,βN), where β0,β1, . . . ,βN ∈ R, we introduce vβ(x) = vβ0,β1,...,βN (x) =    xσ0(β0), 0 < x < ρ, N = 0, xσ0(β0) ∏N ℓ=1[lnℓ(γ/x)] −σℓ(βℓ), 0 < x < ρ, N ∈ N, (1.17) and fβ(x) = fβ0,β1,...,βN (x) = vβ(x)ψ(x), 0 < x < ρ. (1.18) If N ∈ N and ε1 = (ε1, . . . ,εN), where ε1, . . . ,εN > 0, we define hℓ,ε1 : (0,ρ) → R, ℓ ∈ N, iteratively by h1,ε 1 (x) = h1,ε1,...,εN (x) = N∑ k=1 σk(εk) k∏ j=1 [lnj(γ/x)] −1, hℓ+1,ε 1 (x) = xh′ℓ,ε1 (x), ℓ ∈ N. (1.19) Note that, since γ/x > γ/ρ > eN+1, one infers that [lnj(γ/x)] −1 6 1, 0 < x < ρ, j = 1, . . . ,N. (1.20) For 0 6 j 6 k 6 N and β0,β1, . . . ,βN ∈ R, we define aj,k(β) = aj,k(β0,β1, . . . ,βN) by a0,0(β) = [ Pm(σ0(β0)) ]2 − A(m,α), aN,N(β) = σN(βN) { Pm(σ0(β0))P ′′ m(σ0(β0))[σN (βN) + 1] + [ P ′m(σ0(β0)) ]2 σN(βN ) } , aj,j(β) = σj(βj) { Pm(σ0(β0))P ′′ m(σ0(β0))[σj(βj) + 1] + [ P ′m(σ0(β0)) ]2 σj(βj) } − B(m,α), 1 6 j 6 N − 1, a0,j(β) = 2σj(βj)Pm(σ0(β0))P ′ m(σ0(β0)), 1 6 j 6 N, aj,k(β) = σk(βk) { Pm(σ0(β0))P ′′ m(σ0(β0))[2σj(βj) + 1] + 2 [ P ′m(σ0(β0)) ]2 σj(βj) } , 1 6 j < k 6 N. (1.21) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 123 If N ∈ N, β0,β1, . . . ,βN ∈ R, and 1 6 j 6 k 6 N, then we define bj,k(β) = bj,k(β0,β1, . . . ,βN) by bj,j(β) = 1 4 [ Pm(σ0(β0))P ′′ m(σ0(β0)) + [ P ′m(σ0(β0)) ]2] (βj − β 2 j ) + aj,j(β), 1 6 j 6 N, bj,k(β) = aj,k(β) − 1 4 [ Pm(σ0(β0))P ′′ m(σ0(β0)) + [ P ′m(σ0(β0)) ]2] (1 − 2βj)(1 − βk), 1 6 j < k 6 N. (1.22) For the rest of this paper we shall assume that M ∈ (0,∞) is fixed and that ε0,ε1, . . . ,εN ∈ (0,M), constants denoted by cj,j ∈ N, will depend on N ∈ N∪{0}, γ,ρ ∈ (0,∞) with γ > ρeN+1, m ∈ N, α ∈ R, M ∈ (0,∞), and ψ ∈ C∞(R), but will be independent of ε0,ε1, . . . ,εN ∈ (0,M). 2 Preliminary results We mention again that the methods used in this section are adapted from [15, Sect. 3]. Lemma 2.1. Let j ∈ {1, . . . ,N + 1} and β ∈ R. Then, for all 0 < x < ρ, d dx [lnj(γ/x)] −β = βx−1[ln1(γ/x)] −1 · · · [lnj−1(γ/x)] −1[lnj(γ/x)] −1−β. (2.1) Proof. For j = 1, (2.1) clearly holds. Suppose that (2.1) holds for j ∈ {1, . . . ,N}. Then d dx [lnj+1(γ/x)] −β = d dx [ln(lnj(γ/x))] −β = −β[lnj+1(γ/x)] −1−β[lnj(γ/x)] −1 d dx [lnj(γ/x)] −(−1) = −β[lnj+1(γ/x)] −1−β[lnj(γ/x)] −1(−1)x−1 j−1∏ k=1 [lnk(γ/x)] −1 = βx−1 j∏ k=1 [lnk(γ/x)] −1[lnj+1(γ/x)] −1−β. (2.2) The result now follows by induction. Lemma 2.2. (i) [ Pm(σ0(0)) ]2 = A(m,α). (ii) 1 4 {[ P ′m(σ0(0)) ]2 − Pm(σ0(0))P ′′ m(σ0(0)) } = B(m,α). Proof. Since (i) is clear, we only need to prove (ii). Since both sides of (ii) are continuous in α, we may assume that α ∈ R\{1,3, . . . ,2m − 1}. For σ ∈ R\{0,1, . . . ,m − 1} one gets P ′m(σ) = (σ − 1)(σ − 2) · · · (σ − m + 1) + σ(σ − 2) · · · (σ − m + 1) + · · · + σ(σ − 1) · · · (σ − m + 2) = σ−1Pm(σ) + (σ − 1) −1Pm(σ) + · · · + (σ − m + 1) −1Pm(σ), (2.3) 124 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) hence P ′m(σ) [ Pm(σ) ]−1 = m−1∑ j=0 (σ − j)−1, (2.4) thus, differentiating both sides, Pm(σ)P ′′ m(σ) − [P ′ m(σ)] 2 = −[Pm(σ)] 2 m−1∑ j=0 (σ − j)−2. (2.5) Put σ = (2m − 1 − α)/2. Then σ ∈ R\{0,1, . . .,m − 1} if and only if α ∈ R\{1,3, . . . ,2m − 1}. So, by (2.5), part (i), and (1.8), for α ∈ R\{1,3, . . .,2m − 1}, one obtains [P ′m((2m − 1 − α)/2)] 2 − Pm((2m − 1 − α)/2)P ′′ m((2m − 1 − α)/2) = [Pm((2m − 1 − α)/2)] 2 m−1∑ j=0 ( 2m − 1 − α 2 − j )−2 , (2.6) that is, [P ′m(σ0(0))] 2 − Pm(σ0(0))P ′′ m(σ0(0)) = 4 [ Pm(σ0(0)) ]2 m−1∑ j=0 (2(m − j) − 1 − α)−2 = 4A(m,α) m∑ j=1 (2j − 1 − α)−2 = 4B(m,α). (2.7) Remark 2.3. Let hℓ,ε 1 : (0,ρ) → R, ℓ ∈ N, be as in (1.19). For all ℓ ∈ N with ℓ > 3, there exists c1(ℓ) > 0 such that for all ε1, . . . ,εN ∈ (0,M) one has |hℓ,ε 1 (x)| 6 c1(ℓ)[ln(γ/x)] −3, 0 < x < ρ. (2.8) Lemma 2.4. Suppose N ∈ N. Let vε = vε0,ε1,...εN : (0,ρ) → (0,∞) be defined as in (1.17). Then, for τ ∈ N, v(τ)ε (x) = x σ0(ε0)−τ N∏ j=1 [lnj(γ/x)] −σj (εj) { Pτ(σ0(ε0)) + P ′τ(σ0(ε0))h1,ε1(x) + (1/2)P ′′ τ (σ0(ε0))[h1,ε1(x)] 2 + (1/2)P ′′τ (σ0(ε0))h2,ε1(x) (2.9) + Eτ,ε(x) } , 0 < x < ρ, where Eτ,ε(x) is of the form Eτ,ε(x) = Eτ,ε0,ε1,...,εN (x) = Q(τ)∑ j=1 pτ,j[h1,ε 1 (x)]wτ,j,1 · · · [hτ,ε 1 (x)]wτ,j,τ , 0 < x < ρ, (2.10) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 125 for some Q(τ) ∈ N,wτ,j,k ∈ N ∪ {0} for all j ∈ {1, . . . ,Q(τ)} and k ∈ {1, . . . ,τ}, pτ,j ∈ R for all j ∈ {1, . . . ,Q(τ)}. Moreover, there exists c2 = c2(τ) > 0, independent of ε0,ε1, . . . ,εN, such that ∣∣pτ,j[h1,ε 1 (x)]wτ,j,1 · · · [hτ,ε 1 (x)]wτ,j,τ ∣∣ 6 c2[ln(γ/x)]−3, 0 < x < ρ, (2.11) for all j ∈ {1, . . . ,Q(τ)}. Hence ∣∣Eτ,ε(x) ∣∣ 6 c2Q(τ)[ln(γ/x)]−3, 0 < x < ρ. (2.12) Proof. We prove this result by induction on τ ∈ N. For brevity we shall write σj = σj(εj),j = 0,1, . . . ,N, in this proof. For τ = 1 we have, by Lemma 2.1, v′ε(x) = x σ0−1 N∏ j=1 [lnj(γ/x)] −σj ( σ0 + h1,ε1(x) ) , 0 < x < ρ. (2.13) For τ = 2 we have v′′ε (x) = x σ0−2 N∏ j=1 [lnj(γ/x)] −σj ( σ0 − 1 + h1,ε 1 (x) )( σ0 + h1,ε 1 (x) ) + xσ0−1 N∏ j=1 [lnj(γ/x)] −σj ( x−1h2,ε 1 (x) ) = xσ0−2 N∏ j=1 [lnj(γ/x)] −σj { σ0(σ0 − 1) + (2σ0 − 1)h1,ε 1 (x) + [h1,ε 1 (x)]2 + h2,ε 1 (x) } . (2.14) For τ = 3 we have v′′′ε (x) = x σ0−3 N∏ j=1 [lnj(γ/x)] −σj ( σ0 − 2 + h1,ε 1 (x) ){ σ0(σ0 − 1) + (2σ0 − 1)h1,ε 1 (x) + [h1,ε 1 (x)]2 + h2,ε 1 (x) } + xσ0−3 N∏ j=1 [lnj(γ/x)] −σj { (2σ0 − 1)h2,ε 1 (x) + 2h1,ε 1 (x)h2,ε 1 (x) + h3,ε 1 (x) } = xσ0−3 N∏ j=1 [lnj(γ/x)] −σj { P3(σ0) + P ′ 3(σ0)h1,ε1(x) + (1/2)P ′′3 (σ0)[h1,ε1(x)] 2 + (1/2)P ′′3 (σ0)h2,ε1(x) + E3,ε(x) } , (2.15) where E3,ε(x) = [h1,ε1(x)] 3 + 3h1,ε1(x)h2,ε1(x) + h3,ε1(x), (2.16) hence the result holds for τ = 3 by Remark 2.3 and (1.20). Next, we assume that the lemma holds 126 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) for τ ∈ N. Differentiating (2.9) yields v(τ+1)ε (x) = x σ0−τ−1 N∏ j=1 [lnj(γ/x)] −σj ( σ0 − τ + h1,ε 1 (x) ){ Pτ (σ0) + P ′τ (σ0)h1,ε1(x) + (1/2)P ′′ τ (σ0)[h1,ε1(x)] 2 + (1/2)P ′′τ (σ0)h2,ε1(x) + Eτ,ε(x) } + xσ0−τ−1 N∏ j=1 [lnj(γ/x)] −σj { P ′τ (σ0)h2,ε1(x) + P ′′τ (σ0)h1,ε1(x)h2,ε1(x) + (1/2)P ′′ τ (σ0)h3,ε1(x) + xE ′ τ,ε(x) } = xσ0−(τ+1) N∏ j=1 [lnj(γ/x)] −σj { Pτ (σ0)(σ0 − τ) + [ Pτ (σ0) + P ′τ (σ0)(σ0 − τ) ] h1,ε 1 (x) + [ (1/2)P ′′τ (σ0)(σ0 − τ) + P ′ τ(σ0) ] [h1,ε 1 (x)]2 + [ (1/2)P ′′τ (σ0)(σ0 − τ) + P ′ τ (σ0) ] h2,ε1(x) + Eτ+1,ε(x) } = xσ0−(τ+1) N∏ j=1 [lnj(γ/x)] −σj { Pτ+1(σ0) + P ′ τ+1(σ0)h1,ε1(x) + (1/2)P ′′τ+1(σ0)[h1,ε1(x)] 2 + (1/2)P ′′τ+1(σ0)h2,ε1(x) + Eτ+1,ε(x) } , (2.17) where Eτ+1,ε(x) = (1/2)P ′′ τ (σ0)[h1,ε1(x)] 3 + (3/2)P ′′τ (σ0)h1,ε1(x)h2,ε1(x) + (σ0 − τ)Eτ,ε(x) + h1,ε 1 (x)Eτ,ε(x) + (1/2)P ′′ τ (σ0)h3,ε1(x) + xE ′ τ,ε(x). (2.18) Thus, by (1.19), Eτ+1,ε(x) can be written in the form Eτ+1,ε(x) = Q(τ+1)∑ j=1 pτ+1,j[h1,ε 1 (x)]wτ+1,j,1 · · · [hτ+1,ε 1 (x)]wτ+1,j,τ+1 (2.19) for some Q(τ + 1) ∈ N,wτ+1,j,k ∈ N ∪ {0} for j ∈ {1, . . . ,Q(τ + 1)} and k ∈ {1, . . . ,τ + 1}, pτ+1,j ∈ R for j ∈ {1, . . . ,Q(τ +1)}. By (2.18), (1.19), (1.20), and Remark 2.3, there exists c̃2 > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that, for all 0 < x < ρ, ∣∣pτ+1,j[h1,ε 1 (x)]wτ+1,j,1 · · · [hτ+1,ε 1 (x)]wτ+1,j,τ+1 ∣∣ 6 c̃2[ln(γ/x)]−3. (2.20) Hence the lemma holds for τ + 1. Lemma 2.5. Suppose N ∈ N. Let vε = vε0,ε1,...,εN : (0,ρ) → (0,∞) be defined as in (1.17), fε = fε0,ε1,...,εN : (0,ρ) → [0,∞) be defined as in (1.18), and, for 0 6 j 6 k 6 N, aj,k(ε) = aj,k(ε0,ε1, . . . ,εN) be defined as in (1.21). Let G1,ε = G1(ε0,ε1, . . . ,εN) ∈ R be defined by 3 ˆ ρ 0 dxxα ∣∣f(m)ε (x) ∣∣2 = ˆ ρ 0 dxxα ∣∣v(m)ε (x) ∣∣2[ψ(x)]2 + G1,ε. (2.21) 3One notes that, since ε0 > 0, (1.10) and Lemma 2.4 imply that the integrals in (2.21) are finite and hence G1,ε is well-defined. CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 127 Then there exists c3 > 0, independent of ε0,ε1, . . . ,εN, such that ∣∣G1,ε ∣∣ 6 c3, (2.22) and JN−1[fε] = G1,ε + ∑ 06j6k6N aj,k(ε)Γj,k(ε) + ˆ ρ 0 dxx2(σ0(ε0)−m)+α N∏ j=1 [lnj(γ/x)] −2σj (εj)G2,ε(x)[ψ(x)] 2, (2.23) where G2,ε = G2,ε0,ε1,...,εN : (0,ρ) → R satisfies ∣∣G2,ε(x) ∣∣ 6 c3[ln(γ/x)]−3, 0 < x < ρ. (2.24) Proof. We shall write σj = σj(εj), j = 0,1, . . . ,N, in this proof. By Lemma 2.4 we have ∣∣v(m)ε (x) ∣∣2[ψ(x)]2 = x2(σ0−m) N∏ j=1 [lnj(γ/x)] −2σj [ Pm(σ0) + P ′m(σ0)h1,ε1(x) + 1 2 P ′′m(σ0)[h1,ε1(x)] 2 + 1 2 P ′′m(σ0)h2,ε1(x) + Em,ε(x) ]2 [ψ(x)]2 = x2(σ0−m) N∏ j=1 [lnj(γ/x)] −2σj {[ Pm(σ0) ]2 + 2Pm(σ0)P ′ m(σ0)h1,ε1(x) + [ Pm(σ0)P ′′ m(σ0) + [ P ′m(σ0) ]2] [h1,ε 1 (x)]2 + Pm(σ0)P ′′ m(σ0)h2,ε1(x) + G2,ε(x) } [ψ(x)]2, (2.25) where, by Lemma 2.4, G2,ε = G2,ε0,ε1,...,εN : (0,ρ) → R satisfies ∣∣G2,ε(x) ∣∣ 6 c4[ln(γ/x)]−3, 0 < x < ρ, (2.26) for some c4 > 0 independent of ε0,ε1, . . . ,εN ∈ (0,M). Direct computation shows ˆ ρ 0 dxx2(σ0−m)+α N∏ j=1 [lnj(γ/x)] −2σj h1,ε 1 (x)[ψ(x)]2 = N∑ j=1 σjΓ0,j(ε), (2.27) ˆ ρ 0 dxx2(σ0−m)+α N∏ j=1 [lnj(γ/x)] −2σj [h1,ε1(x)] 2[ψ(x)]2 = N∑ j=1 σ2j Γj,j(ε) + 2 ∑ 16j 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that |G1,ε| 6 c5. Thus Lemma 2.5 is proved upon putting c3 = max{c4,c5}. Lemma 2.6. Let k ∈ {0,1, . . . ,N} and β0,β1, . . . ,βk > 0. Then ˆ ρ 0 dxx−1+β0[ln1(γ/x)] −1−β1 · · · [lnk(γ/x)] −1−βk < ∞ (2.33) if and only if    β0 > 0, or β0 = 0 and β1 > 0, or β0 = β1 = 0 and β2 > 0, ... or β0 = β1 = · · · = βk−1 = 0 and βk > 0. (2.34) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 129 Proof. This follows from Lemma 2.1 and (1.20). Lemma 2.7. Let β ∈ (−∞,1). Then there exists c6 = c6(β) > 0, independent of ε0 ∈ (0,M), such that ˆ ρ 0 dx−1+ε0[ln1(γ/x)] −β[ψ(x)]2 6 c6ε −1+β 0 . (2.35) Proof. Writing τ = ε−10 [ln(γ/ρ)] −1 > 0, and using the change of variables s = ε−10 [ln(γ/x)] −1 ( i.e., x = γe −1 ε0s ) , ds = ε−10 x −1[ln(γ/x)]−2dx ( i.e., dx = γε−10 s −2e −1 ε0s ds ) , (2.36) one obtains ˆ ρ 0 dxx−1+ε0 [ln(γ/x)]−β[ψ(x)]2 6 ˆ ρ 0 dxx−1+ε0 [ln(γ/x)]−β = γε0ε −1+β 0 ˆ τ 0 dss−2+βe −1 s 6 ( γε0 ˆ ∞ 0 dss−2+βe −1 s ) ε −1+β 0 . (2.37) Lemma 2.8. Suppose N > 2. Let β ∈ (−∞,1) and 1 6 j 6 N − 1. Then there exists c7 = c7(β) > 0, independent of εj ∈ (0,M), such that ˆ ρ 0 dxx−1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj [lnj+1(γ/x)] −β[ψ(x)]2 6 c7ε −1+β j . (2.38) Proof. Writing τ = ε−1j [lnj+1(γ/ρ)] −1 > 0, and using the change of variables s = ε−1j [lnj+1(γ/x)] −1, (2.39) so that, by Lemma 2.1, ds = ε−1j x −1[ln1(γ/x)] −1 · · · [lnj(γ/x)] −1[lnj+1(γ/x)] −2dx, (2.40) one gets ˆ ρ 0 dxx−1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj [lnj+1(γ/x)] −β[ψ(x)]2 6 εj ˆ τ 0 ds [lnj(γ/x)] −εj [lnj+1(γ/x)] 2−β. (2.41) By (2.39) one has (εjs) −1 = ln(lnj(γ/x)) ( i.e., lnj(γ/x) = e 1 εjs ) . (2.42) Hence ˆ ρ 0 dxx−1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj [lnj+1(γ/x)] −β[ψ(x)]2 6 ˆ τ 0 dsεje −1 s (εjs) −2+β 6 ( ˆ ∞ 0 dse −1 s s−2+β ) ε −1+β j . (2.43) 130 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) Next, we need to introduce some more notation: For τ ∈ {0,1, . . . ,N − 1} and τ < j 6 k 6 N we write ( Γτ (ε) ) j,k = ˆ ρ 0 dx { x−1 τ∏ ℓ=1 [lnℓ(γ/x)] −1 j∏ ℓ=τ+1 [lnℓ(γ/x)] −1−εℓ k∏ ℓ=j+1 [lnℓ(γ/x)] −εℓ × N∏ ℓ=k+1 [lnℓ(γ/x)] 1−εℓ[ψ(x)]2 } . (2.44) By Lemma 2.6, ( Γτ (ε) ) j,k is well-defined for τ ∈ {0,1, . . . ,N − 1} and τ < j 6 k 6 N as the integral on the right-hand side of (2.44) is finite. Lemma 2.9. (i) There exists c8 > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that ε0Γ0,0(ε) = N∑ j=1 (1 − εj)Γ0,j(ε) + G3,ε, (2.45) and for j = 1, . . . ,N, ε0Γ0,j(ε) = − j∑ k=1 εkΓk,j(ε) + N∑ k=j+1 (1 − εk)Γj,k(ε) + G4,j,ε, (2.46) where ∣∣G3,ε ∣∣ 6 c8, ∣∣G4,j,ε ∣∣ 6 c8. (2.47) (ii) Suppose N > 2. Let 1 6 j 6 N − 1. Then there exists c9 = c9(j) > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that εj ( Γj−1(ε) ) j,j = N∑ k=j+1 (1 − εk) ( Γj−1(ε) ) j,k + G5,j,ε j , (2.48) where εj = (εj, . . . ,εN), and, for j + 1 6 k 6 N, εj ( Γj−1(ε) ) j,k = − k∑ ℓ=j+1 εℓ ( Γj−1(ε) ) ℓ,k + N∑ ℓ=k+1 (1 − εℓ) ( Γj−1(ε) ) k,ℓ + G6,j,k,ε j , (2.49) and where ∣∣G5,j,ε j ∣∣ 6 c9, ∣∣G6,j,k,ε j ∣∣ 6 c9. (2.50) (iii) There exists c10 > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that ε20Γ0,0(ε) − 2ε0 N∑ j=1 (1 − εj)Γ0,j(ε) = N∑ j=1 (εj − ε 2 j)Γj,j(ε) − ∑ 16j 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that ∣∣G3,ε ∣∣ 6 c8, ∣∣G4,j,ε ∣∣ 6 c8. (2.57) (ii) One has d dx ( [lnj(γ/x)] −εj N∏ k=j+1 [lnk(γ/x)] 1−εk [ψ(x)]2 ) − 2[lnj(γ/x)] −εj N∏ k=j+1 [lnk(γ/x)] 1−εkψ(x)ψ′(x) = εjx −1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj N∏ k=j+1 [lnk(γ/x)] 1−εk [ψ(x)]2 − (1 − εj+1)x −1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj [lnj+1(γ/x)] −εj+1 N∏ k=j+2 [lnk(γ/x)] 1−εk [ψ(x)]2 ... − (1 − εN)x −1 j−1∏ k=1 [lnk(γ/x)] −1[lnj(γ/x)] −1−εj N∏ k=j+1 [lnk(γ/x)] −εk [ψ(x)]2, (2.58) integrating both sides in (2.58) yields G5,j,ε j = εj ( Γj−1(ε) ) j,j − N∑ k=j+1 (1 − εk) ( Γj−1(ε) ) j,k . (2.59) Similarly one obtains, for j + 1 6 k 6 N, d dx ( [lnj(γ/x)] −εj · · · [lnk(γ/x)] −εk [lnk+1(γ/x)] 1−εk+1 · · · [lnN(γ/x)] 1−εN [ψ(x)]2 ) − 2[lnj(γ/x)] −εj · · · [lnk(γ/x)] −εk [lnk+1(γ/x)] 1−εk+1 · · · [lnN(γ/x)] 1−εN ψ(x)ψ′(x) = εjx −1 j−1∏ ℓ=1 [lnℓ(γ/x)] −1[lnj(γ/x)] −1−εj k∏ ℓ=j+1 [lnℓ(γ/x)] −εℓ N∏ ℓ=k+1 [lnℓ(γ/x)] 1−εℓ[ψ(x)]2+ ... + εkx −1 j−1∏ ℓ=1 [lnℓ(γ/x)] −1 k∏ ℓ=j [lnℓ(γ/x)] −1−εℓ N∏ ℓ=k+1 [lnℓ(γ/x)] 1−εℓ[ψ(x)]2 − (1 − εk+1)x −1 × j−1∏ ℓ=1 [lnℓ(γ/x)] −1 k∏ ℓ=j [lnℓ(γ/x)] −1−εℓ[lnk+1(γ/x)] −εk+1 N∏ ℓ=k+2 [lnℓ(γ/x)] 1−εℓ[ψ(x)]2− ... − (1 − εN)x −1 j−1∏ ℓ=1 [lnℓ(γ/x)] −1 k∏ ℓ=j [lnℓ(γ/x)] −1−εℓ N∏ ℓ=k+1 [lnℓ(γ/x)] −εℓ[ψ(x)]2, (2.60) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 133 integrating both sides in (2.60) yields G6,j,k,ε j = k∑ ℓ=j εℓ ( Γj−1(ε) ) ℓ,k − N∑ ℓ=k+1 (1 − εℓ) ( Γj−1(ε) ) k,ℓ . (2.61) By (1.10), there exists c9 > 0, independent of εj, . . . ,εN ∈ (0,M), such that ∣∣G5,j,ε j ∣∣ 6 c9, ∣∣G6,j,k,ε j ∣∣ 6 c9, (2.62) for 1 6 j 6 N − 1 and j + 1 6 k 6 N. (iii) By (i) we have ε20Γ0,0(ε) − 2ε0 N∑ j=1 (1 − εj)Γ0,j(ε) = −ε0 N∑ j=1 (1 − εj)Γ0,j(ε) + ε0G3,ε = − N∑ j=1 (1 − εj) { − j∑ k=1 εkΓk,j(ε) + N∑ k=j+1 (1 − εk)Γj,k(ε) + G4,j,ε } + ε0G3,ε = N∑ j=1 j∑ k=1 (1 − εj)εkΓk,j(ε) − N∑ j=1 N∑ k=j+1 (1 − εj)(1 − εk)Γj,k(ε) + G7,ε, (2.63) where there exists c10 > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that ∣∣G7,ε ∣∣ 6 c10. (2.64) Thus ε20Γ0,0(ε) − 2ε0 N∑ j=1 (1 − εj)Γ0,j(ε) = N∑ j=1 (εj − ε 2 j)Γj,j(ε) + ∑ 16j 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), with the following property: Given any fixed ε1, . . . ,εN ∈ (0,M), there exists a decreasing sequence {ε0,ℓ} ∞ ℓ=1 ⊆ (0,M) and L0 ∈ R such that ε0,ℓ ↓ 0 as ℓ ↑ ∞, |L0| 6 c11, and, writing fε = fε0,ℓ,ε1,...,εN as defined in (1.18), lim ℓ↑∞ JN−1[fε] = ∑ 16j6k6N bj,k(0,ε1, . . . ,εN) ( Γ0(ε) ) j,k + L0. (2.66) 134 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) Proof. We first note that by Lemma 2.7, there exists c12 > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that for all ε0,ε1, . . . ,εN ∈ (0,M) we have Γ0,0(ε) = ˆ ρ 0 dxx−1+ε0 [ln1(γ/x)] 1−ε1 · · · [lnN(γ/x)] 1−εN [ψ(x)]2 6 ˆ ρ 0 dxx−1+ε0 [ln1(γ/x)] 3/2 { [ln1(γ/x)] −1/2 N∏ k=2 [lnk(γ/x)] } [ψ(x)]2 6 c12ε −5/2 0 . (2.67) For j = 1, . . . ,N, by Lemma 2.7, there exists c13 = c13(j) > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that for all ε0,ε1, . . . ,εN ∈ (0,M) we have Γ0,j(ε) = ˆ ρ 0 dxx−1+ε0 j∏ k=1 [lnk(γ/x)] −εk N∏ k=j+1 [lnk(γ/x)] 1−εk [ψ(x)]2 6 ˆ ρ 0 dxx−1+ε0 [ln1(γ/x)] 1/2 { [ln1(γ/x)] −1/2 N∏ k=j+1 [lnk(γ/x)] } [ψ(x)]2 6 c13ε −3/2 0 . (2.68) Since we are fixing ε1, . . . ,εn ∈ (0,M), for 0 6 j 6 k 6 N, we shall consider aj,k(ε) = aj,k(ε0,ε1, . . . ,εN) as functions of ε0 ∈ (0,M) only. Then a0,0(ε0) = [ Pm(σ0(ε0)) ]2 − A(m,α), a′0,0(ε0) = Pm(σ0(ε0))P ′ m(σ0(ε0)), a′′0,0(ε0) = 1 2 { Pm(σ0(ε0))P ′′ m(σ0(ε0)) + [ P ′m(σ0(ε0)) ]2 } , a (k) 0,0(ε0) = 2 −k { dk dσk ([ Pm(σ) ]2) ∣∣∣∣ σ=σ0(ε0) } , k = 3, . . . ,2m. (2.69) Similarly one has, for j = 1, . . . ,N, and k = 2, . . . ,2m − 1, a0,j(ε0) = 2σj(εj)Pm(σ0(ε0))P ′ m(σ0(ε0)), a′0,j(ε0) = σj(εj) {[ P ′m(σ0(ε0)) ]2 + Pm(σ0(ε0))P ′′ m(σ0(ε0)) } , a (k) 0,j (ε0) = 2 −(k−1)σj(εj) { dk dσk ( Pm(σ)P ′ m(σ) )∣∣∣∣ σ=σ0(ε0) } . (2.70) Thus, by Lemma 2.2, a0,0(ε0) = a0,0(0) + a ′ 0,0(0)ε0 + 1 2 a′′0,0(0)ε 2 0 + ε 3 0 ( 2m∑ k=3 (k!)−1a (k) 0,0(0)ε k−3 0 ) = Pm(σ0(0))P ′ m(σ0(0))ε0 + 1 4 { Pm(σ0(0))P ′′ m(σ0(0)) + [ P ′m(σ0(0)) ]2} ε20 + ( 2m∑ k=3 (k!)−12−k { dk dσk ([ Pm(σ) ]2) ∣∣∣∣ σ=σ0(0) } εk−30 ) ε30. (2.71) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 135 Put G8(ε0) = 2m∑ k=3 (k!)−12−k { dk dσk ([ Pm(σ) ]2) ∣∣∣∣ σ=σ0(0) } εk−30 , (2.72) then there exists c14 > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that ∣∣G8(ε0) ∣∣ 6 c14, ε0 ∈ (0,M). (2.73) Similarly, for j = 1, . . . ,N, a0,j(ε0) = a0,j(0) + a ′ 0,j(0)ε0 + 2m−1∑ k=2 (k!)−1a (k) 0,j (0)ε k 0 = 2σj(εj)Pm(σ0(0))P ′ m(σ0(0)) + σj(εj) {[ P ′m(σ0(0)) ]2 + Pm(σ0(0))P ′′ m(σ0(0)) } ε0 + ( 2m−1∑ k=2 (k!)−12−(k−1)σj(εj) { dk dσk ( Pm(σ)P ′ m(σ) )∣∣∣∣ σ=σ0(0) } εk−20 ) ε20. (2.74) For j = 1, . . . ,N, put G9,j(ε0,εj) = 2m−1∑ k=2 (k!)−12−(k−1)σj(εj) { dk dσk ( Pm(σ)P ′ m(σ) )∣∣∣∣ σ=σ0(0) } εk−20 , (2.75) then there exists c15 = c15(j) > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that ∣∣G9,j(ε0,εj) ∣∣ 6 c15, j = 1, . . . ,N, ε0,εj ∈ (0,M). (2.76) Hence, applying Lemma 2.9, a0,0(ε)Γ0,0(ε) + N∑ j=1 a0,j(ε)Γ0,j(ε) = Pm(σ0(0))P ′ m(σ0(0))ε0Γ0,0(ε) + 1 4 { Pm(σ0(0))P ′′ m(σ0(0)) + [ P ′m(σ0(0)) ]2} ε20Γ0,0(ε) + G8(ε0)ε 3 0Γ0,0(ε) + N∑ j=1 { 2σj(εj)Pm(σ0(0))P ′ m(σ0(0))Γ0,j(ε) + σj(εj) ([ P ′m(σ0(0)) ]2 + Pm(σ0(0))P ′′ m(σ0(0)) ) ε0Γ0,j(ε) + G9,j(ε0,εj)ε 2 0Γ0,j(ε) } = Pm(σ0(0))P ′ m(σ0(0)) { ε0Γ0,0(ε) − N∑ j=1 (1 − εj)Γ0,j(ε) } + 1 4 { Pm(σ0(0))P ′′ m(σ0(0)) + [ P ′m(σ0(0)) ]2} { ε20Γ0,0(ε) − 2ε0 N∑ j=1 (1 − εj)Γ0,j(ε) } + G8(ε0)ε 3 0Γ0,0(ε) + N∑ j=1 G9,j(ε0,εj)ε 2 0Γ0,j(ε) 136 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) = Pm(σ0(0))P ′ m(σ0(0))G3,ε + 1 4 { Pm(σ0(0))P ′′ m(σ0(0)) + [ P ′m(σ0(0)) ]2} × { N∑ j=1 (εj − ε 2 j)Γj,j(ε) − ∑ 16j 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that ∣∣G10,ε ∣∣ 6 c16, ε0,ε1, . . . ,εN ∈ (0,M). (2.79) Let {ε0,ℓ} ∞ ℓ=1 be any decreasing sequence in (0,M) with limℓ↑∞ ε0,ℓ = 0. Applying Lemma 2.5, (2.77), and (2.78), we have, with ε0 = ε0,ℓ, JN−1[fε] = G1,ε + ˆ ρ 0 dxx−1+ε0,ℓ N∏ j=1 [lnj(γ/x)] 1−εjG2,ε(x)[ψ(x)] 2 + a0,0(ε)Γ0,0(ε) + N∑ j=1 a0,j(ε)Γ0,j(ε) + ∑ 16j6k6N aj,k(ε)Γj,k(ε) = G1,ε + ˆ ρ 0 dxx−1+ε0,ℓ N∏ j=1 [lnj(γ/x)] 1−εjG2,ε(x)[ψ(x)] 2 + G10,ε + 1 4 { Pm(σ0(0))P ′′ m(σ0(0)) + [ P ′m(σ0(0)) ]2 }{ N∑ j=1 (εj − ε 2 j)Γj,j(ε) − ∑ 16j 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that ∣∣∣∣ ˆ ρ 0 dxx−1+ε0 N∏ j=1 [lnj(γ/x)] 1−εjG2,ε(x)[ψ(x)] 2 ∣∣∣∣ 6 ˆ ρ 0 dxc3x −1[ln1(γ/x)] −3/2 { [ln1(γ/x)] −1/2 N∏ j=2 [lnj(γ/x)] } [ψ(x)]2 6 c17 ˆ ρ 0 dxx−1[ln1(γ/x)] −3/2[ψ(x)]2 = c18 < ∞. (2.82) This, together with (2.22) and (2.79), implies that there exists c11 > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), such that ∣∣G11,ε ∣∣ 6 c11, ε0,ε1, . . . ,εN ∈ (0,M). (2.83) By compactness of [−c11,c11], there exist a subsequence {ε0,ℓp} ∞ p=1 and L0 ∈ [−c11,c11], such that lim p↑∞ G11(ε0,ℓp,ε1, . . . ,εN) = L0. (2.84) We shall regard this subsequence as {ε0,ℓ} ∞ ℓ=1. For 1 6 j 6 k 6 N we have, by monotone convergence, lim ℓ↑∞ Γj,k(ε0,ℓ,ε1, . . . ,εN) = ( Γ0(ε) ) j,k (ε1, . . . ,εN). (2.85) The lemma now follows from taking the limit ℓ ↑ ∞ in (2.80) and using (2.81) and (2.83)–(2.85). Lemma 2.11. Suppose N > 2. Then there exists a constant c19 > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), with the following property: Let p ∈ {1, . . . ,N − 1} and let εp+1, . . . ,εN ∈ (0,M) be fixed. Then there exist Lp ∈ R, with |Lp| 6 c19, and a decreasing sequence {εp,ℓ} ∞ ℓ=1 ⊆ (0,M) with εp,ℓ ↓ 0 as ℓ ↑ ∞, such that lim ℓ↑∞ ∑ p6j6k6N bj,k(0, . . . ,0,εp,ℓ,εp+1, . . . ,εN) ( Γp−1(ε) ) j,k = ∑ p+16j6k6N bj,k(0, . . . ,0,εp+1, . . . ,εN) ( Γp(ε) ) j,k + Lp. (2.86) Proof. By Lemma 2.2 one obtains bp,p(0, . . . ,0,εp,εp+1, . . . ,εN) = 1 4 { Pm(σ0(0))P ′′ m(σ0(0)) + [ P ′m(σ0(0)) ]2}( εp − ε 2 p ) − 1 2 (1 − εp) { Pm(σ0(0))P ′′ m(σ0(0)) 1 2 (1 + εp) − [ P ′m(σ0(0)) ]2 1 2 (1 − εp) } − B(m,α) = 1 4 { Pm(σ0(0))P ′′ m(σ0(0)) − [ P ′m(σ0(0)) ]2} εp = −B(m,α)εp, (2.87) 138 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) and, for j = p + 1, . . . ,N, one gets bp,j(0, . . . ,0,εp,εp+1, . . . ,εN) = σj(εj) { Pm(σ0(0))P ′′ m(σ0(0))εp − [ P ′m(σ0(0)) ]2 (1 − εp) } + 1 2 σj(εj) { Pm(σ0(0))P ′′ m(σ0(0)) + [ P ′m(σ0(0)) ]2} (1 − 2εp) = 1 2 σj(εj) { Pm(σ0(0))P ′′ m(σ0(0)) − [ P ′m(σ0(0)) ]2} = B(m,α)(1 − εj). (2.88) Thus, by Lemma 2.9, ∣∣∣∣bp,p(0, . . . ,0,εp,εp+1, . . . ,εN) ( Γp−1(ε) ) p,p + N∑ j=p+1 bp,j(0, . . . ,0,εp,εp+1, . . . ,εN) ( Γp−1(ε) ) p,j ∣∣∣∣ = ∣∣∣∣ − B(m,α) { εp ( Γp−1(ε) ) p,p − N∑ j=p+1 (1 − εj) ( Γp−1(ε) ) p,j }∣∣∣∣ 6 c19, (2.89) where c19 = B(m,α) max{c9(1), . . . ,c9(N − 1)} > 0 is once again independent of ε0,ε1, . . . ,εN ∈ (0,M). Hence by compactness of [−c19,c19] there exist a decreasing subsequence {εp,ℓ} ∞ ℓ=1 of {1 ℓ }∞ℓ=1 and Lp ∈ [−c19,c19] such that Lp = lim ℓ↑∞ bp,p(0, . . . ,0,εp,ℓ,εp+1, . . . ,εN) ( Γp−1(ε) ) p,p + N∑ j=p+1 bp,j(0, . . . ,0,εp,ℓ,εp+1, . . . ,εN) ( Γp−1(ε) ) p,j . (2.90) By monotone convergence lim ℓ↑∞ ∑ p+16j6k6N bj,k(0, . . . ,0,εp,ℓ,εp+1, . . . ,εN) ( Γp−1(ε) ) j,k = ∑ p+16j6k6N bj,k(0, . . . ,0,εp+1, . . . ,εN) ( Γp(ε) ) j,k . (2.91) The lemma now follows from (2.90), (2.91). Lemma 2.12. We have lim εN ↓0 bN,N(0, . . . ,0,εN) = B(m,α). (2.92) Proof. We have, by Lemma 2.2 lim εN ↓0 bN,N(0, . . . ,0,εN) = lim εN ↓0 aN,N(0, . . . ,0,εN) = lim εN ↓0 − 1 4 (1 − εN) { Pm(σ0(0))P ′′ m(σ0(0))(1 + εN) − [ P ′m(σ0(0)) ]2 (1 − εN) } = 1 4 {[ P ′m(σ0(0)) ]2 − Pm(σ0(0))P ′′ m(σ0(0)) } = B(m,α). (2.93) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 139 Lemma 2.13. Suppose N ∈ N. Then given any η > 0, there exist ε0,ε1, . . . ,εN ∈ (0,M) such that if fε = fε0,ε1,...,εN is as defined in (1.18), one has ∣∣∣∣JN−1[fε] [ ˆ ρ 0 dxxα−2m N∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2 ]−1 − B(m,α) ∣∣∣∣ 6 η. (2.94) Proof. Let c20 = max{c11,c19} > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M), where c11 and c19 are as in Lemmas 2.10 and 2.11. By Lemma 2.6 and monotone convergence one infers lim εN ↓0 ˆ ρ 0 dxx−1 N−1∏ j=1 [lnj(γ/x)] −1[lnN(γ/x)] −1−εN [ψ(x)]2 = ∞. (2.95) Thus, we can choose εN ∈ (0,M) sufficiently small such that ˆ ρ 0 dxx−1 N−1∏ j=1 [lnj(γ/x)] −1[lnN(γ/x)] −1−εN [ψ(x)]2 > 1, (2.96) and c20 [ ˆ ρ 0 dxx−1 N−1∏ j=1 [lnj(γ/x)] −1[lnN(γ/x)] −1−εN [ψ(x)]2 ]−1 < η, (2.97) and, by Lemma 2.12, ∣∣bN,N(0, . . . ,0,εN) − B(m,α) ∣∣ < η. (2.98) Thus, for any RN−1 ∈ [−c20,c20], one has ∣∣∣∣ { bN,N(0, . . . ,0,εN) ( ΓN−1(ε) ) N,N + RN−1 }[ˆ ρ 0 dxx−1 N−1∏ j=1 [lnj(γ/x)] −1 × [lnN(γ/x)] −1−εN [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ 6 ∣∣bN,N(0, . . . ,0,εN) − B(m,α) ∣∣ + c20 ∣∣∣∣ [ ˆ ρ 0 dxx−1 N−1∏ j=1 [lnj(γ/x)] −1[lnN(γ/x)] −1−εN [ψ(x)]2 ]−1∣∣∣∣ < 2η. (2.99) Suppose first that N > 2. Then, by Lemma 2.11, there exist LN−1 ∈ [−c19,c19] and a decreasing sequence {εN−1,ℓ} ∞ ℓ=1 ⊆ (0,M), with limℓ↑∞ εN−1,ℓ = 0, such that lim ℓ↑∞ ∑ N−16j6k6N bj,k(0, . . . ,0,εN−1,ℓ,εN) ( ΓN−2(ε) ) j,k = bN,N(0, . . . ,0,εN) ( ΓN−1(ε) ) N,N + LN−1. (2.100) 140 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) By (2.96) and monotone convergence, and replacing {εN−1,ℓ} ∞ ℓ=1 by a subsequence if necessary, one can assume that ˆ ρ 0 dx { x−1 N−2∏ j=1 [lnj(γ/x)] −1[lnN−1(γ/x)] −1−εN−1,ℓ[lnN(γ/x)] −1−εN [ψ(x)[2 } > 1, ℓ ∈ N. (2.101) Combining (2.97), (2.100), (2.101), and (2.99) with RN−1 = LN−1, and using monotone conver- gence, there exists εN−1 ∈ (0,M) satisfying ∣∣∣∣ { ∑ N−16j6k6N bj,k(0, . . . ,0,εN−1,εN) ( ΓN−2(ε) ) j,k } × ( ˆ ρ 0 dxx−1 N−2∏ j=1 [lnj(γ/x)] −1[lnN−1(γ/x)] −1−εN−1[lnN(γ/x)] −1−εN [ψ(x)]2 )−1 − B(m,α) ∣∣∣∣ 6 ∣∣∣∣ { ∑ N−16j6k6N bj,k(0, . . . ,0,εN−1,εN) ( ΓN−2(ε) ) j,k − [ bN,N(0, . . . ,0,εN) ( ΓN−1(ε) ) N,N + LN−1 ]}[ˆ ρ 0 dxx−1 N−2∏ j=1 [lnj(γ/x)] −1 × [lnN−1(γ/x)] −1−εN−1[lnN(γ/x)] −1−εN [ψ(x)]2 ]−1∣∣∣∣ + ∣∣∣∣ [ bN,N(0, . . . ,0,εN) ( ΓN−1(ε) ) N,N + LN−1 ][ˆ ρ 0 dxx−1 N−2∏ j=1 [lnj(γ/x)] −1 × [lnN−1(γ/x)] −1−εN−1[lnN(γ/x)] −1−εN [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ < η + 2η = 3η, (2.102) and c20 [ ˆ ρ 0 dxx−1 N−2∏ j=1 [lnj(γ/x)] −1 N∏ j=N−1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < η, (2.103) as well as ˆ ρ 0 dxx−1 N−2∏ j=1 [lnj(γ/x)] −1 N∏ j=N−1 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1. (2.104) One notes that by (2.102), (2.103), for all RN−2 ∈ [−c20,c20], ∣∣∣∣ { ∑ N−16j6k6N bj,k(0, . . . ,0,εN−1,εN) ( ΓN−2(ε) ) j,k + RN−2 } × [ ˆ ρ 0 dxx−1 N−2∏ j=1 [lnj(γ/x)] −1 N∏ j=N−1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 141 6 ∣∣∣∣ { ∑ N−16j6k6N bj,k(0, . . . ,0,εN−1,εN) ( ΓN−2(ε) ) j,k } × [ ˆ ρ 0 dxx−1 N−2∏ j=1 [lnj(γ/x)] −1 N∏ j=N−1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ + c20 ( ˆ ρ 0 dxx−1 N−2∏ j=1 [lnj(γ/x)] −1 N∏ j=N−1 [lnj(γ/x)] −1−εj [ψ(x)]2 )−1 < 3η + η = 4η. (2.105) So we have chosen εN−1,εN ∈ (0,M). If N − 1 > 2, then, by Lemma 2.11, there exist LN−2 ∈ [−c19,c19] and a decreasing sequence {εN−2,ℓ} ∞ ℓ=1 ⊆ (0,M) with limℓ↑∞ εN−2,ℓ = 0 such that lim ℓ↑∞ ∑ N−26j6k6N bj,k(0, . . . ,0,εN−2,ℓ,εN−1,εN) ( ΓN−3(ε) ) j,k = ∑ N−16j6k6N bj,k(0, . . . ,0,εN−1,εN) ( ΓN−2(ε) ) j,k + LN−2. (2.106) By (2.104) and monotone convergence, and replacing {εN−2,ℓ} ∞ ℓ=1 by a subsequence, if necessary, one can assume that ˆ ρ 0 dxx−1 N−3∏ j=1 [lnj(γ/x)] −1[lnN−2(γ/x)] −1−εN−2,ℓ × N∏ j=N−1 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1, ℓ ∈ N. (2.107) Combining (2.103), (2.106), (2.107), and (2.105) with RN−2 = LN−2, and monotone convergence, there exists εN−2 ∈ (0,M) satisfying ∣∣∣∣ { ∑ N−26j6k6N bj,k(0, . . . ,0,εN−2,εN−1,εN) ( ΓN−3(ε) ) j,k } × [ ˆ ρ 0 dxx−1 N−3∏ j=1 [lnj(γ/x)] −1 N∏ j=N−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ 6 ∣∣∣∣ { ∑ N−26j6k6N bj,k(0, . . . ,0,εN−2,εN−1,εN) ( ΓN−3(ε) ) j,k − [ ∑ N−16j6k6N bj,k(0, . . . ,0,εN−1,εN) ( ΓN−2(ε) ) j,k + LN−2 ]} × [ ˆ ρ 0 dxx−1 N−3∏ j=1 [lnj(γ/x)] −1 N∏ j=N−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1∣∣∣∣ + ∣∣∣∣ [ ∑ N−16j6k6N bj,k(0, . . . ,0,εN−1,εN) ( ΓN−2(ε) ) j,k + LN−2 ] × [ ˆ ρ 0 dxx−1 N−3∏ j=1 [lnj(γ/x)] −1 N∏ j=N−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ < η + 4η = 5η, (2.108) 142 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) and c20 [ ˆ ρ 0 dxx−1 N−3∏ j=1 [lnj(γ/x)] −1 N∏ j=N−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < η, (2.109) as well as ˆ ρ 0 dxx−1 N−3∏ j=1 [lnj(γ/x)] −1 N∏ j=N−2 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1, (2.110) such that for all RN−3 ∈ [−c20,c20] one infers ∣∣∣∣ { ∑ N−26j6k6N bj,k(0, . . . ,0,εN−2,εN−1,εN) ( ΓN−3(ε) ) j,k + RN−3 } × [ ˆ ρ 0 dxx−1 N−3∏ j=1 [lnj(γ/x)] −1 N∏ j=N−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ 6 ∣∣∣∣ { ∑ N−26j6k6N bj,k(0, . . . ,0,εN−2,εN−1,εN) ( ΓN−3(ε) ) j,k } × [ ˆ ρ 0 dxx−1 N−3∏ j=1 [lnj(γ/x)] −1 N∏ j=N−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ + c20 [ ˆ ρ 0 dxx−1 N−3∏ j=1 [lnj(γ/x)] −1 N∏ j=N−2 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < 5η + η = 6η. (2.111) Repeating the argument above N − 1 times (or if N = 1) one arrives at the following fact: there exist ε1, . . . ,εN ∈ (0,M) such that ∣∣∣∣ { ∑ 16j6k6N bj,k(0,ε1, . . . ,εN) ( Γ0(ε) ) j,k }[ ˆ ρ 0 dxx−1 N∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ 6 (2N − 1)η, (2.112) and c20 [ ˆ ρ 0 dxx−1 N∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < η, (2.113) as well as ˆ ρ 0 dxx−1 N∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1, (2.114) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 143 so that for all R0 ∈ [−c20,c20] one obtains ∣∣∣∣ { ∑ 16j6k6N bj,k(0,ε1, . . . ,εN) ( Γ0(ε) ) j,k + R0 } × [ ˆ ρ 0 dxx−1 N∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ 6 ∣∣∣∣ { ∑ 16j6k6N bj,k(0,ε1, . . . ,εN) ( Γ0(ε) ) j,k }[ ˆ ρ 0 dxx−1 N∏ j=1 [lnj(γ/x)] −1−εj × [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ + c20 [ ˆ ρ 0 dxx−1 N∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 < (2N − 1)η + η = 2Nη. (2.115) Then, by Lemma 2.10, there exist L0 ∈ [−c20,c20] and a decreasing sequence {ε0,ℓ} ∞ ℓ=1 ⊆ (0,M) with limℓ↑∞ ε0,ℓ = 0 such that lim ℓ↑∞ JN−1[fε0,ℓ,ε1,...,εN ] = ∑ 16j6k6N bj,k(0,ε1, . . . ,εN) ( Γ0(ε) ) j,k + L0. (2.116) By (2.114) and monotone convergence, and replacing {ε0,ℓ} ∞ ℓ=1 by a subsequence if necessary, we can assume that ˆ ρ 0 dxx−1+ε0,ℓ N∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 > 1, ℓ ∈ N. (2.117) Combining (2.112), (2.113), (2.115) with R0 = L0, (2.116), (2.117), and monotone convergence, there exists ε0 ∈ (0,M) satisfying ∣∣∣∣JN−1[fε0,ε1,...,εN ] [ ˆ ρ 0 dxx−1+ε0 N∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ 6 ∣∣∣∣ [ JN−1[fε0,ε1,...,εN ] − { ∑ 16j6k6N bj,k(0,ε1, . . . ,εn) ( Γ0(ε) ) j,k + L0 }] × [ ˆ ρ 0 dxx−1+ε0 N∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1∣∣∣∣ + ∣∣∣∣ { ∑ 16j6k6N bj,k(0,ε1, . . . ,εn) ( Γ0(ε) ) j,k + L0 } × [ ˆ ρ 0 dxx−1+ε0 N∏ j=1 [lnj(γ/x)] −1−εj [ψ(x)]2 ]−1 − B(m,α) ∣∣∣∣ < η + 2Nη = (2N + 1)η. (2.118) Lemma 2.14. Suppose N = 0 and let fε0 be as defined on (1.18). Then lim ε0↓0 ˆ ρ 0 dxxα ∣∣f(m)ε0 (x) ∣∣2 [ ˆ ρ 0 dxxα−2m|fε0(x)| 2 ]−1 = A(m,α). (2.119) 144 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) Proof. By (1.10) we have lim ε0↓0 ˆ ρ 0 dxxα−2m|fε0(x)| 2 > lim ε0↓0 ˆ (0.8)ρ 0 dxx−1+ε0 = ∞. (2.120) In addition, one has f(m)ε0 (x) = m∑ j=0 ( m j ) Pj(σ0(ε0))x σ0(ε0)−jψ(m−j)(x), 0 < x < ρ. (2.121) Thus, for all 0 < x < ρ, xα ∣∣f(m)ε0 (x) ∣∣2 = m∑ j,k=0 ( m j )( m k ) Pj(σ0(ε0))Pk(σ0(ε0))x α+2σ0(ε0)−j−kψ(m−j)(x)ψ(m−k)(x) = [ Pm(σ0(ε0)) ]2 x−1+ε0[ψ(x)]2 + G12(ε0,x) = A(m,α − ε0)x −1+ε0[ψ(x)]2 + G12(ε0,x), (2.122) where, again by (1.10), ∣∣G12(ε0,x) ∣∣ 6 c21, ε0 ∈ (0,M), 0 < x < ρ, (2.123) for some c21 > 0, independent of ε0,ε1, . . . ,εN ∈ (0,M). Hence, ˆ ρ 0 dxxα ∣∣f(m)ε0 (x) ∣∣2 = A(m,α − ε0) ˆ ρ 0 dxx−1+ε0 [ψ(x)]2 + ˆ ρ 0 dxG12(ε0,x), (2.124) and the lemma follows by dividing both sides of (2.124) by ˆ ρ 0 dxxα−2m|fε0(x)| 2 = ˆ ρ 0 dxx−1+ε0 [ψ(x)]2 (2.125) and applying (2.120), (2.123). 3 The approximation procedure We start with some more notation. For the remainder of this paper we shall assume ε0,ε1, . . . ,εN ∈ (0,ρ/20), that is, we shall assume M = ρ/20. Let fε = fε0,ε1,...,εN be as defined in (1.18). Then for δ ∈ (0,ρ/20), we shall write, recalling ε = (ε0,ε1, . . . ,εN), f(δ),ε(x) =    0, x < δ or ρ 6 x, fε(x), δ 6 x < ρ. (3.1) We shall let h ∈ C∞(R) satisfy the following properties: (i) h is even on R, (3.2) (ii) h(x) > 0, x ∈ R, (3.3) (iii) supp(h) ⊆ (−1,1), (3.4) (iv) ˆ 1 −1 dxh(x) = 1, (3.5) (v) h is non-increasing on [0,∞). (3.6) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 145 For ε > 0 we write hε(x) = ε −1h(x/ε), x ∈ R. (3.7) For δ ∈ (0,ρ/20) and ε ∈ (0,δ/4], we write f(δ,ε),ε = f(δ),ε ∗ hε. (3.8) Remark 3.1. (i) Since h is even, we have f(δ,ε),ε(x) = ˆ ∞ −∞ dtε−1h(t/ε)f(δ),ε(x − t) = ˆ ∞ −∞ dtε−1h(−t/ε)f(δ),ε(x − t) = ˆ ∞ −∞ duε−1h(u/ε)f(δ),ε(x + u) = ˆ ∞ −∞ dr ε−1h((r − x)/ε)f(δ),ε(r) = ˆ x+ε x−ε drε−1h((r − x)/ε)f(δ),ε(r), x ∈ R. (3.9) (ii) Since ε ∈ (0,δ/4], supp(f(δ,ε),ε) ⊆ [3δ/4,73ρ/80]. Hence, f(δ,ε),ε ∈ C ∞ 0 ((0,ρ)). (3.10) (iii) Let g ∈ L∞(R),x ∈ R,τ ∈ R\{0}. For 0 < ε 6 δ/4 < ρ/80, let gε = hε ∗ g. By the sequence of change of variables in (3.9), we have τ−1[gε(x + τ) − gε(x)] = ˆ ∞ −∞ dr (τε)−1{h((r − x − τ)/ε) − h((r − x)/ε)}g(r) = − ˆ ∞ −∞ dr (τε)−1h′((r − x − λ(x,r,τ)τ)/ε)(τ/ε)g(r) = −ε−2 ˆ ∞ −∞ dr h′((r − x − λ(x,r,τ)τ)/ε)g(r), (3.11) where 0 6 λ(x,r,τ) 6 1, x,r ∈ R. (3.12) Since h′,g ∈ L∞(R) and, for −1 6 τ 6 1, supph′([ · − x − λ(x, · ,τ)τ]/ε) ⊆ [x − ε − 1,x + ε + 1], (3.13) applying the dominated convergence theorem we get g′ε(x) = lim τ→0 τ−1[gε(x + τ) − gε(x)] = − lim τ→0 ε−2 ˆ ∞ −∞ dr h′((r − x − λ(x,r,τ)τ)/ε)g(r) = −ε−2 lim τ→0 ˆ x+ε+1 x−ε−1 dr h′((r − x − λ(x,r,τ)τ)/ε)g(r) = −ε−2 ˆ x+ε+1 x−ε−1 dr h′((r − x)/ε)g(r) = −ε−2 ˆ x+ε x−ε dr h′((r − x)/ε)g(r). (3.14) 146 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) Let δ ∈ (0,ρ/20). For technical convenience, so that we can use the general theory of convo- lution, we shall write f̃(δ),ε for a function in C ∞ 0 (R) satisfying: (i) f̃(δ),ε(x) = f(δ),ε, x > δ, (ii) f̃(δ),ε(x) > 0, −∞ < x < ∞. (3.15) Constants denoted by νj,j ∈ N, will depend on N ∈ N ∪ {0}, γ,ρ ∈ (0,∞) with γ > ρeN+1, m ∈ N, α ∈ R, h,ψ ∈ C∞(R), and ε0,ε1, . . . ,εN ∈ (0,ρ/20), but are independent of δ ∈ (0,ρ/20) and ε ∈ (0,δ/4). ⋄ Lemma 3.2. For all k ∈ N ∪ {0} there exists ν1 = ν1(k) > 0 such that ∣∣f(k)ε (x) ∣∣ 6 ν1x[2(m−k)−1−α+(ε0/2)]/2, 0 < x < ρ. (3.16) Proof. This lemma follows from Lemma 2.4, the product rule f(k)ε (x) = k∑ j=0 ( k j ) v(k−j)ε (x)ψ (j)(x), 0 < x < ρ, (3.17) and that, for all β > 0, the function t 7→ t−βln(t) is bounded on (1,∞). Lemma 3.3. For j = 1, . . . ,m, and x ∈ [3δ/4,5δ/4], we have, writing θ = δ/4, f (j) (δ,θ),ε (x) = j∑ k=1 (−1)k+1θ−kh(k−1)((δ −x)/θ)f (j−k) (δ),ε (δ) +θ−1 ˆ x+θ δ drh((r −x)/θ)f (j) (δ),ε (r). (3.18) Proof. For 3δ/4 6 x 6 5δ/4 we have, by (3.14) f′(δ,θ),ε(x) = −θ −2 ˆ x+θ x−θ dr h′((r − x)/θ)f(δ),ε(r) = −θ−2 ˆ x+θ δ dr h′((r − x)/θ)f(δ),ε(r) = −θ−1 ˆ x+θ δ dr d dr [h((r − x)/θ)]f(δ),ε(r) = −θ−1 { h((r − x)/θ)f(δ),ε(r) ∣∣∣∣ x+θ δ − ˆ x+θ δ dr h((r − x)/θ)f′(δ),ε(r) } = −θ−1 { − h((δ − x)/θ)f(δ),ε(δ) − ˆ x+θ δ drh((r − x)/θ)f′(δ),ε(r) } = θ−1h((δ − x)/θ)f(δ),ε(δ) + θ −1 ˆ x+θ δ drh((r − x)/θ)f′(δ),ε(r). (3.19) Suppose j ∈ {1, . . . ,m − 1} and that for all x ∈ [3δ/4,5δ/4] one has f (j) (δ,θ),ε (x) = j∑ k=1 (−1)k+1θ−kh(k−1)((δ −x)/θ)f (j−k) (δ),ε (δ) +θ−1 ˆ x+θ δ drh((r −x)/θ)f (j) (δ),ε (r), (3.20) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 147 then, by (3.14), one concludes f (j+1) (δ,θ),ε (x) = j∑ k=1 (−1)k+1θ−k(−1/θ)h(k)((δ − x)/θ)f (j−k) (δ),ε (δ) + d dx ( θ−1 ˆ x+θ x−θ drh((r − x)/θ)f (j) (δ),ε (r) ) = j∑ k=1 (−1)kθ−(k+1)h(k)((δ − x)/θ)f (j−k) (δ),ε (δ) − 1 θ2 ˆ x+θ x−θ dr h′((r − x)/θ)f (j) (δ),ε (r) = j+1∑ k=2 (−1)k+1θ−kh(k−1)((δ − x)/θ)f (j+1−k) (δ),ε (δ) − 1 θ ˆ x+θ δ dr ( d dr [h((r − x)/θ)] ) f (j) (δ),ε (r) = j+1∑ k=2 (−1)k+1θ−kh(k−1)((δ − x)/θ)f (j+1−k) (δ),ε (δ) − 1 θ { h((r − x)/θ)f (j) (δ),ε (r) ∣∣∣∣ x+θ δ − ˆ x+θ δ drh((r − x)/θ)f (j+1) (δ),ε (r) } = j+1∑ k=2 (−1)k+1θ−kh(k−1)((δ − x)/θ)f (j+1−k) (δ),ε (δ) + 1 θ h((δ − x)/θ)f (j) (δ),ε (δ) + 1 θ ˆ x+θ δ dr h((r − x)/θ)f (j+1) (δ),ε (r) = j+1∑ k=1 (−1)k+1θ−kh(k−1)((δ − x)/θ)f (j+1−k) (δ),ε (δ) + 1 θ ˆ x+θ δ dr h((r − x)/θ)f (j+1) (δ),ε (r). (3.21) Hence, Lemma 3.3 follows by induction. Corollary 3.4. There exists ν2 > 0 such that for all δ ∈ (0,ρ/20), ∣∣f(m) (δ,δ/4),ε (x) ∣∣ 6 ν2x[−1−α+(ε0/2)]/2, 3δ/4 6 x 6 5δ/4. (3.22) Proof. Let Km = sup {∣∣h(k)(t) ∣∣ ∣∣ − 1 6 t 6 1, k = 0,1, . . . ,m } . (3.23) By Lemmas 3.2 and 3.3 we have for x ∈ [3δ/4,5δ/4], ∣∣f(m) (δ,δ/4),ε (x) ∣∣ 6 m∑ k=1 4kδ−kKmν1(m − k)δ [2k−1−α+(ε0/2)]/2 + 4δ−1(6δ 4 − δ)Km sup {∣∣f(m) (δ),ε (r) ∣∣ ∣∣δ 6 r 6 6δ/4 } = m∑ k=1 4kKmν1(m − k)δ [−1−α+(ε0/2)]/2 + 2Kmν1(m) sup { r[−1−α+(ε0/2)]/2 ∣∣δ 6 r 6 6δ/4 } 148 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) 6 Km ( m∑ k=1 4kν1(m − k) ){ (4/3)[−1−α+(ε0/2)]/2 + (4/5)[−1−α+(ε0/2)]/2 } × x[−1−α+(ε0/2)]/2 + 2Kmν1(m) { (4/5)[−1−α+(ε0/2)]/2 + 2[−1−α+(ε0/2)]/2 } x[−1−α+(ε0/2)]/2 = ν2x [−1−α+(ε0/2)]/2, (3.24) where ν2 = Km ( m∑ k=1 4kν1(m − k) ){ (4/3)[−1−α+(ε0/2)]/2 + (4/5)[−1−α+(ε0/2)]/2 } + 2Kmν1(m) { (4/5)[−1−α+(ε0/2)]/2 + 2[−1−α+(ε0/2)]/2 } . (3.25) Lemma 3.5. There exists ν3 > 0 such that for all δ ∈ (0,ρ/20) we have ∣∣f(m) (δ,δ/4),ε (x) ∣∣ 6 ν3x[−1−α+(ε0/2)]/2, 5δ/4 6 x 6 ρ. (3.26) Proof. We first note that, for 5δ/4 6 x 6 73ρ/80, f(δ,δ/4),ε(x) = ˆ x+δ/4 x−δ/4 dr (4/δ)h(4(r − x)/δ)f(δ),ε(r) = ˆ x+δ/4 x−δ/4 dr (4/δ)h(4(r − x)/δ)f̃(δ),ε(r) = ( hδ/4 ∗ f̃(δ),ε ) (x), (3.27) hence f (m) (δ,δ/4),ε (x) = ( hδ/4 ∗ f̃ (m) (δ),ε ) (x) = 4δ−1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f̃ (m) (δ),ε (r) = 4δ−1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f (m) (δ),ε (r), (3.28) therefore, by Lemma 3.2, ∣∣f(m) (δ,δ/4),ε (x) ∣∣ 6 sup {∣∣f(m) (δ),ε (r) ∣∣ ∣∣x − (δ/4) 6 r 6 x + (δ/4) } 6 ν1(m) sup { r[−1−α+(ε0/2)]/2 ∣∣x − (δ/4) 6 r 6 x + (δ/4) } 6 ν1(m) sup { r[−1−α+(ε0/2)]/2 ∣∣3x/4 6 r 6 5x/4 } 6 ν1(m) { (3/4)[−1−α+(ε0/2)]/2 + (5/4)[−1−α+(ε0/2)]/2 } x[−1−α+(ε0/2)]/2. (3.29) By Remark 3.1 (ii), supp(f(δ,δ/4),ε) ⊆ [3δ/4,73ρ/80]. So (3.26) holds for x ∈ [73ρ/80,ρ], completing the proof. CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 149 Lemma 3.6. On any compact interval [a,b] ⊆ (0,ρ], f (m) (δ,δ/4),ε converges to f (m) ε uniformly as δ ↓ 0. Proof. Choose δ0 ∈ (0,ρ/20) such that 0 < 5δ0/4 < a. Then for all 0 < δ < δ0 and x ∈ [a,b], f(δ,δ/4),ε(x) = 4δ −1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f(δ),ε(r) = 4δ−1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f(δ0),ε(r) = 4δ−1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f̃(δ0),ε(r) = ( hδ/4 ∗ f̃(δ0),ε ) (x). (3.30) Since f̃(δ0),ε ∈ C ∞ 0 (R), f (m) (δ,δ/4),ε (x) = ( hδ/4 ∗ f̃ (m) (δ0),ε ) (x), x ∈ [a,b], −→ δ↓0 f̃ (m) (δ0),ε (x) uniformly for x ∈ [a,b], = f(m)ε (x). (3.31) Corollary 3.7. We have lim δ↓0 ˆ ρ 0 dxxα ∣∣f(m) (δ,δ/4),ε (x) ∣∣2 = ˆ ρ 0 dxxα ∣∣f(m)ε (x) ∣∣2. (3.32) Proof. Let ν4 = max{ν2,ν3} > 0. Then by Corollary 3.4 and Lemma 3.5, we have, for all δ ∈ (0,ρ/20), xα ∣∣f(m) (δ,δ/4),ε (x) ∣∣2 6 ν24x −1+(ε0/2), 0 < x < ρ. (3.33) By Lemma 3.6 we have lim δ↓0 xα ∣∣f(m) (δ,δ/4),ε (x) ∣∣2 = xα ∣∣f(m)ε (x) ∣∣2, 0 < x < ρ. (3.34) Since x 7→ ν4x −1+(ε0/2) is integrable on (0,ρ), the corollary now follows by dominated convergence. Lemma 3.8. There exists ν5 > 0 such that for all δ ∈ (0,ρ/20) we have |f(δ,δ/4),ε(x)| 6 ν5x [2m−1−α+(ε0/2)]/2, 3δ/4 6 x 6 5δ/4. (3.35) 150 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) Proof. For 3δ/4 6 x 6 5δ/4 we have |f(δ,δ/4),ε(x)| = ∣∣∣∣4δ −1 ˆ x+δ/4 δ drh(4(r − x)/δ)f(δ),ε(r) ∣∣∣∣ 6 sup{|f(δ),ε(r)| |δ 6 r 6 6δ/4} = sup{|fε(r)| |δ 6 r 6 3δ/2} 6 ν1(0) sup { r[2m−1−α+(ε0/2)]/2 ∣∣δ 6 r 6 3δ/2 } 6 ν1(0) { (4/5)[2m−1−α+(ε0/2)]/2 + 2[2m−1−α+(ε0/2)]/2 } x[2m−1−α+(ε0/2)]/2. (3.36) Lemma 3.9. There exists ν6 > 0 such that for all δ ∈ (0,ρ/20) we have |f(δ,δ/4),ε(x)| 6 ν6x [2m−1−α+(ε0/2)]/2, 5δ/4 6 x < ρ. (3.37) Proof. For x ∈ [5δ/4,ρ) we have |f(δ,δ/4),ε(x)| = ∣∣∣∣4δ −1 ˆ x+δ/4 x−δ/4 dr h(4(r − x)/δ)f(δ),ε(r) ∣∣∣∣ 6 sup{|f(δ),ε(r)| |x − δ/4 6 r 6 x + δ/4} 6 ν1(0) sup { r[2m−1−α+(ε0/2)]/2 ∣∣3x/4 6 r 6 5x/4 } 6 ν1(0) { (3/4)[2m−1−α+(ε0/2)]/2 + (5/4)[2m−1−α+(ε0/2)]/2 } x[2m−1−α+(ε0/2)]/2. (3.38) Lemma 3.10. On any compact interval [a,b] ⊆ (0,ρ], f(δ,δ/4),ε converges to fε uniformly as δ ↓ 0. Proof. Choose δ0 ∈ (0,ρ/20) with 0 < 5δ0/4 < a. By (3.30), for all 0 < δ < δ0, we have f(δ,δ/4),ε(x) = ( hδ/4 ∗ f̃(δ0),ε ) (x), a 6 x 6 b. (3.39) Since f̃(δ0),ε ∈ C ∞ 0 (R), we have f(δ,δ/4),ε(x) = ( hδ/4 ∗ f̃(δ0),ε ) (x) −→ δ↓0 f̃(δ0),ε(x) uniformly for x ∈ [a,b] = fε(x). (3.40) Corollary 3.11. For k ∈ {0,1, . . . ,N} we have lim δ↓0 ˆ ρ 0 dxxα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣f(δ,δ/4),ε(x) ∣∣2 = ˆ ρ 0 dxxα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2. (3.41) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 151 Proof. Let ν7 = max{ν5,ν6} > 0. By Lemmas 3.8 and 3.9 we have, for all δ ∈ (0,ρ/20) and x ∈ (0,ρ), xα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣f(δ,δ/4),ε(x) ∣∣2 6 ν27x −1+(ε0/2) k∏ j=1 [lnj(γ/x)] −2. (3.42) By Lemma 3.10 we have for x ∈ (0,ρ), lim δ↓0 xα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣f(δ,δ/4),ε(x) ∣∣2 = xα−2m k∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2. (3.43) Since x 7→ x−1+(ε0/2) ( ∏k j=1[lnj(γ/x)] −2 ) is integrable on (0,ρ), the corollary now follows by dominated convergence. Corollary 3.12. Suppose N ∈ N. Then there exists a family {gδ,ε}δ∈(0,(0.05)ρ) ⊆ C ∞ 0 ((0,ρ)) such that lim δ↓0 JN−1[gδ,ε] ( ˆ ρ 0 dxxα−2m N∏ j=1 [lnj(γ/x)] −2 ∣∣gδ,ε(x) ∣∣2 )−1 = JN−1[fε] ( ˆ ρ 0 dxxα−2m N∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2 )−1 . (3.44) Proof. For δ ∈ (0,ρ/20) put gδ,ε = f(δ,δ/4),ε. Then gδ,ε ∈ C ∞ 0 ((0,ρ)) by Remark 3.1 (ii). The result now follows from Corollaries 3.7 and 3.11. Corollary 3.13. Suppose N = 0. Then there exists a family {gδ,ε}δ∈(0,(0.05)ρ) ⊆ C ∞ 0 ((0,ρ)) such that lim δ↓0 ˆ ρ 0 dxxα ∣∣g(m)δ,ε (x) ∣∣2 ( ˆ ρ 0 dxxα−2m ∣∣gδ,ε(x) ∣∣2 )−1 = ˆ ρ 0 dxxα ∣∣f(m)ε (x) ∣∣2 ( ˆ ρ 0 dxxα−2m ∣∣fε(x) ∣∣2 )−1 . (3.45) Proof. The proof of this corollary is the same as that of Corollary 3.12. 4 Principal results on optimal constants In our final section we now prove optimality of the constants A(m,α) and B(m,α). Starting with the interval (0,ρ), we first establish optimality of A(m,α) in (1.1). Theorem 4.1. Suppose that N = 0. Then, given any η > 0, there exists g ∈ C∞0 ((0,ρ)) such that ∣∣∣∣ ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 [ ˆ ρ 0 dxxα−2m ∣∣g(x) ∣∣2 ]−1 − A(m,α) ∣∣∣∣ 6 η. (4.1) In particular, the constant A(m,α) in (1.1) is sharp. 152 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) Proof. Given any η > 0 there exists ε0 ∈ (0,ρ/20) such that ∣∣∣∣ ˆ ρ 0 dxxα ∣∣f(m)ε0 (x) ∣∣2 [ ˆ ρ 0 dxxα−2m ∣∣fε0(x) ∣∣2 ]−1 − A(m,α) ∣∣∣∣ 6 η/2, (4.2) by Lemma 2.14. With this value of ε0 ∈ (0,ρ/20), Corollary 3.13 implies that there exists g ∈ C∞0 ((0,ρ)) such that ∣∣∣∣ ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 [ ˆ ρ 0 dxxα−2m ∣∣g(x) ∣∣2 ]−1 − ˆ ρ 0 dxxα ∣∣f(m)ε0 (x) ∣∣2 [ ˆ ρ 0 dxxα−2m ∣∣fε0(x) ∣∣2 ]−1∣∣∣∣ 6 η/2. (4.3) Theorem 4.1 now follows from (4.2), (4.3). Next, we prove optimality of the N constants B(m,α) in (1.1): Theorem 4.2. Suppose that N ∈ N. Then for any η > 0, there exists g ∈ C∞0 ((0,ρ)) such that ∣∣∣∣ [ ˆ ρ 0 dxxα ∣∣g(m)(x) ∣∣2 − A(m,α) ˆ ρ 0 dxxα−2m ∣∣g(x) ∣∣2 − B(m,α) N−1∑ k=1 ˆ ρ 0 dxxα−2m ∣∣g(x) ∣∣2 k∏ p=1 [lnp(γ/x)] −2 ] × [ ˆ ρ 0 dxxα−2m N∏ j=1 [lnj(γ/x)] −2 ∣∣g(x) ∣∣2 ]−1 − B(m,α) ∣∣∣∣ 6 η. (4.4) In particular, successively increasing N through 1,2,3, . . . , demonstrates that the N constants B(m,α) in (1.1) are sharp. Together with Theorem 4.1, this theorem asserts that the N + 1 constants, A(m,α) and the N constants B(m,α), in (1.1) are sharp. Proof. Given any η > 0 there exist ε0,ε1, . . . ,εN ∈ (0,ρ/20) such that, writing fε = fε0,ε1,...,εN , ∣∣∣∣JN−1[fε] [ ˆ ρ 0 dxxα−2m N∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2 ]−1 − B(m,α) ∣∣∣∣ 6 η/2, (4.5) by Lemma 2.13. With these values of ε0,ε1, . . . ,εN ∈ (0,ρ/20), Corollary 3.12 implies that there exists g ∈ C∞0 ((0,ρ)) such that ∣∣∣∣JN−1[g] [ ˆ ρ 0 dxxα−2m N∏ j=1 [lnj(γ/x)] −2 ∣∣g(x) ∣∣2 ]−1 − JN−1[fε] [ ˆ ρ 0 dxxα−2m N∏ j=1 [lnj(γ/x)] −2 ∣∣fε(x) ∣∣2 ]−1∣∣∣∣ 6 η/2. (4.6) Theorem 4.2 now follows from (4.5), (4.6). CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 153 Next we turn to analogous results for the half line (r,∞). We start with some preparations. Writing Qm,α(λ) = ( λ2 − (1 − α)2 4 )( λ2 − (3 − α)2 4 ) · · · ( λ2 − (2m − 1 − α)2 4 ) = m∏ j=1 ( λ2 − (2j − 1 − α)2 4 ) = 2m∑ ℓ=0 kℓ(m,α)λ ℓ, (4.7) one infers that (i) k2j−1(m,α) = 0, j = 1, . . . ,m, (4.8) (ii) k2j(m,α) = (−1) m−j|k2j(m,α)|, j = 0,1, . . . ,m, (4.9) and thus, Qm,α(λ) = m∑ j=0 (−1)m−j|k2j(m,α)|λ 2j. (4.10) Lemma 4.3 ([41, Sect. 2 and proof of Theorem 3.1 (i)]). Suppose ρ̂ > eN+1 and α ∈ R\{1, . . . ,2m− 1}. For g ∈ C∞0 ((ρ̂,∞)) let w = wg ∈ C ∞ 0 ((ln(ρ̂),∞)) be defined by g(et) = e[(2m−1−α)/2]tw(t), t ∈ (ln(ρ̂),∞). (4.11) Then for all g ∈ C∞0 ((ρ̂,∞)), ˆ ∞ ρ̂ dy yα ∣∣g(m)(y) ∣∣2 = ˆ ∞ ln(ρ̂) dt m∑ j=0 |k2j(m,α)| ∣∣w(j)(t) ∣∣2, ˆ ∞ ρ̂ dy yα−2m|g(y)|2 = ˆ ∞ ln(ρ̂) dt |w(t)|2, (4.12) and, if N ∈ N, one also has, for k = 1, . . . ,N, ( et )α−2m∣∣g(et) ∣∣2 k∏ p=1 [lnp(e t)]−2 = e−t|w(t)|2t−2 k−1∏ p=1 [lnp(t)] −2, t ∈ (ln(ρ̂),∞). (4.13) Hence, if N ∈ N, [ ˆ ∞ ρ̂ dy yα ∣∣g(m)(y) ∣∣2 − A(m,α) ˆ ∞ ρ̂ dy yα−2m|g(y)|2 − B(m,α) ˆ ∞ ρ̂ dy yα−2m|g(y)|2 N−1∑ k=1 k∏ p=1 [lnp(y)] −2 ] × [ ˆ ∞ ρ̂ dy yα−2m|g(y)|2 N∏ p=1 [lnp(y)] −2 ]−1 154 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) = [ ˆ ∞ ln(ρ̂) dt m∑ j=0 |k2j(m,α)| ∣∣w(j)(t) ∣∣2 − A(m,α) ˆ ∞ ln(ρ̂) dt |w(t)|2 − B(m,α) ˆ ∞ ln(ρ̂) dt |w(t)|2t−2 N−1∑ k=1 k−1∏ p=1 [lnp(t)] −2 ] × [ ˆ ∞ ln(ρ̂) dt |w(t)|2t−2 N−1∏ p=1 [lnp(t)] −2 ]−1 , g ∈ C∞0 ((ρ̂,∞)). (4.14) Corollary 4.4. Lemma 4.3 holds for all α ∈ R, that is, it holds without the restriction α ∈ R\{1, . . . ,2m − 1}. Proof. We first note that by (4.7), for ℓ = 0,1, . . . ,2m, kℓ(m,α) is a polynomial in α and so it is continuous in α. For g ∈ C∞0 ((ρ̂,∞)), to emphasize that the definition of w = wg ∈ C ∞ 0 ((ln(ρ̂),∞)) in (4.11) depends also on α, we shall write, for all α ∈ R, wα(t) = e −[(2m−1−α)/2]tg(et), t ∈ (ln(ρ̂),∞). (4.15) Then, for j = 0,1, . . . ,m, one gets w(j)α (t) = j∑ k=0 S(j,k,α,t)g(k)(et), t ∈ (ln(ρ̂),∞), (4.16) where, for j ∈ {0,1, . . . ,m}, k ∈ {0,1, . . . ,j}, and t ∈ (ln(ρ̂),∞), α 7→ S(j,k,α,t) is continuous in α. We also note that, for g ∈ C∞0 ((ρ̂,∞)), supp(wα) = { t ∈ (ln(ρ̂),∞) |et ∈ supp(g) } (4.17) is independent of α ∈ R. Now let α ∈ {1, . . . ,2m − 1}. Then, by dominated convergence, for g ∈ C∞0 ((ρ̂,∞)), lim β→α ˆ ∞ ρ̂ dy yβ ∣∣g(m)(y) ∣∣2 = ˆ ∞ ρ̂ dy yα ∣∣g(m)(y) ∣∣2, lim β→α ˆ ∞ ρ̂ dy yβ−2m|g(y)|2 = ˆ ∞ ρ̂ dy yα−2m|g(y)|2, (4.18) and, if N ∈ N, one obtains lim β→α ˆ ∞ ρ̂ dy yβ−2m|g(y)|2 N−1∑ k=1 k∏ p=1 [lnp(y)] −2 = ˆ ∞ ρ̂ dy yα−2m|g(y)|2 N−1∑ k=1 k∏ p=1 [lnp(y)] −2, lim β→α ˆ ∞ ρ̂ dy yβ−2m|g(y)|2 N∏ p=1 [lnp(y)] −2 = ˆ ∞ ρ̂ dy yα−2m|g(y)|2 N∏ p=1 [lnp(y)] −2. (4.19) Similarly, for g ∈ C∞0 ((ρ̂,∞)), lim β→α ˆ ∞ ln(ρ̂) dt m∑ j=0 |k2j(m,β)| ∣∣w(j)β (t) ∣∣2 = ˆ ∞ ln(ρ̂) dt m∑ j=0 |k2j(m,α)| ∣∣w(j)α (t) ∣∣2, lim β→α A(m,β) ˆ ∞ ln(ρ̂) dy |wβ(t)| 2 = A(m,α) ˆ ∞ ln(ρ̂) dy |wα(t)| 2, (4.20) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 155 and, if N ∈ N, one has lim β→α B(m,β) ˆ ∞ ln(ρ̂) dt |wβ(t)| 2t−2 N−1∑ k=1 k−1∏ p=1 [lnp(t)] −2 = B(m,α) ˆ ∞ ln(ρ̂) dt |wα(t)| 2t−2 N−1∑ k=1 k−1∏ p=1 [lnp(t)] −2, (4.21) lim β→α ˆ ∞ ln(ρ̂) dt |wβ(t)| 2t−2 N−1∏ p=1 [lnp(t)] −2 = ˆ ∞ ln(ρ̂) dt |wα(t)| 2t−2 N−1∏ p=1 [lnp(t)] −2. (4.22) The corollary now follows from (4.18)–(4.22) and Lemma 4.3. Lemma 4.5 ([41, Sect. 2 and proof of Theorem 3.1 (iii)]). Suppose 1/ρ̃ > eN+1 and α ∈ R\{1, . . . ,2m − 1}. For g ∈ C∞0 ((0, ρ̃)) let u = ug ∈ C ∞ 0 ((ln(1/ρ̃),∞)) be defined by g(e−t) = e−[(2m−1−α)/2]tu(t), t ∈ (ln(1/ρ̃),∞). (4.23) Then, for all g ∈ C∞0 ((0, ρ̃)), ˆ ρ̃ 0 dy yα ∣∣g(m)(y) ∣∣2 = ˆ ∞ ln(1/ρ̃) dt m∑ j=0 |k2j(m,α)| ∣∣u(j)(t) ∣∣2, ˆ ρ̃ 0 dy yα−2m|g(y)|2 = ˆ ∞ ln(1/ρ̃) dt |u(t)|2, (4.24) and, if N ∈ N, we also have, for k = 1, . . . ,N, ( e−t )α−2m∣∣g(e−t) ∣∣2 k∏ p=1 [lnp(e t)]−2 = et|u(t)|2t−2 k−1∏ p=1 [lnp(t)] −2, t ∈ (ln(1/ρ̃),∞). (4.25) Hence, if N ∈ N, [ ˆ ρ̃ 0 dy yα ∣∣g(m)(y) ∣∣2 − A(m,α) ˆ ρ̃ 0 dy yα−2m|g(y)|2 − B(m,α) ˆ ρ̃ 0 dy yα−2m|g(y)|2 N−1∑ k=1 k∏ p=1 [lnp(1/y)] −2 ] × [ ˆ ρ̃ 0 dy yα−2m|g(y)|2 N∏ p=1 [lnp(1/y)] −2 ]−1 = [ ˆ ∞ ln(1/ρ̃) dt m∑ j=0 |k2j(m,α)| ∣∣u(j)(t) ∣∣2 − A(m,α) ˆ ∞ ln(1/ρ̃) dt |u(t)|2 − B(m,α) ˆ ∞ ln(1/ρ̃) dt |u(t)|2t−2 N−1∑ k=1 k−1∏ p=1 [lnp(t)] −2 ] × [ ˆ ∞ ln(1/ρ̃) dt |u(t)|2t−2 N−1∏ p=1 [lnp(t)] −2 ]−1 , g ∈ C∞0 ((0, ρ̃)). (4.26) 156 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) Corollary 4.6. Lemma 4.5 holds for all α ∈ R, that is, it holds without the restriction α ∈ R\{1, . . . ,2m − 1}. As the proof of this corollary is very similar to that of Corollary 4.4 we shall omit it. At this point we are ready to establish optimality of A(m,α) on the interval (r,∞) in (1.2). Theorem 4.7. Suppose that N = 0. Let r ∈ (1,∞). Then, for any η > 0, there exists ϕ ∈ C∞0 ((r,∞)) such that ∣∣∣∣ ˆ ∞ r dxxα ∣∣ϕ(m)(x) ∣∣2 [ ˆ ∞ r dxxα−2m|ϕ(x)|2 ]−1 − A(m,α) ∣∣∣∣ 6 η. (4.27) In particular, the constant A(m,α) in (1.2) is sharp. Proof. Put ρ = 1/r so that 1 > ρ. Applying Theorem 4.1, there exists g ∈ C∞0 ((0,ρ)) such that ∣∣∣∣ ˆ ρ 0 dy yα ∣∣g(m)(y) ∣∣2 [ ˆ ρ 0 dy yα−2m|g(y)|2 ]−1 − A(m,α) ∣∣∣∣ 6 η. (4.28) By Corollary 4.6, writing u(t) = e[(2m−1−α)/2]tg(e−t), t ∈ (ln(1/ρ),∞), (4.29) one obtains ∣∣∣∣ ˆ ∞ ln(1/ρ) dt m∑ j=0 |k2j(m,α)| ∣∣u(j)(t) ∣∣2 [ ˆ ∞ ln(1/ρ) dt |u(t)|2 ]−1 − A(m,α) ∣∣∣∣ 6 η. (4.30) Introducing ϕ(x) = x(2m−1−α)/2u(ln(x)), x ∈ (1/ρ,∞) = (r,∞), (4.31) Corollary 4.4 implies ∣∣∣∣ ˆ ∞ r dxxα ∣∣ϕ(m)(x) ∣∣2 [ ˆ ∞ r dxxα−2m|ϕ(x)|2 ]−1 − A(m,α) ∣∣∣∣ 6 η, (4.32) concluding the proof since ϕ ∈ C∞0 ((r,∞)). Next, we prove optimality of the N constants B(m,α) in (1.2): Theorem 4.8. Suppose that N ∈ N. Let r,Γ ∈ (0,∞) satisfy r > ΓeN+1. Then, for any η > 0, there exists ϕ ∈ C∞0 ((r,∞)) such that ∣∣∣∣ [ ˆ ∞ r dxxα ∣∣ϕ(m)(x) ∣∣2 − A(m,α) ˆ ∞ r dxxα−2m|ϕ(x)|2 − B(m,α) N−1∑ k=1 ˆ ∞ r dxxα−2m|ϕ(x)|2 k∏ p=1 [lnp(x/Γ)] −2 ] × [ ˆ ∞ r dxxα−2m|ϕ(x)|2 N∏ p=1 [lnp(x/Γ)] −2 ]−1 − B(m,α) ∣∣∣∣ 6 η. (4.33) CUBO 24, 1 (2022) Optimality of constants in power-weighted Birman inequalities 157 In particular, successively increasing N through 1,2,3 . . . , demonstrates that the N constants B(m,α) in (1.2) are sharp. Together with Theorem 4.7, this theorem asserts that the N + 1 constants, A(m,α) and the N constants B(m,α), in (1.2) are sharp. Proof. Put ρ = Γ/r so that 1 > ρeN+1. Applying Theorem 4.2 with γ = 1, there exists g ∈ C∞0 ((0,ρ)) such that ∣∣∣∣ [ ˆ ρ 0 dy yα ∣∣g(m)(y) ∣∣2 − A(m,α) ˆ ρ 0 dy yα−2m|g(y)|2 − B(m,α) ˆ ρ 0 dy yα−2m|g(y)|2 N−1∑ k=1 k∏ p=1 [lnp(1/y)] −2 ] × [ ˆ ρ 0 dy yα−2m|g(y)|2 N∏ p=1 [lnp(1/y)] −2 ]−1 − B(m,α) ∣∣∣∣ 6 η. (4.34) By Corollary 4.6, writing u(t) = e[(2m−1−α)/2]tg(e−t), t ∈ (ln(1/ρ),∞), (4.35) one has ∣∣∣∣ [ ˆ ∞ ln(1/ρ) dt m∑ j=0 |k2j(m,α)| ∣∣u(j)(t) ∣∣2 − A(m,α) ˆ ∞ ln(1/ρ) dt |u(t)|2 − B(m,α) ˆ ∞ ln(1/ρ) dt |u(t)|2t−2 N−1∑ k=1 k−1∏ p=1 [lnp(t)] −2 ] × [ ˆ ∞ ln(1/ρ) dt |u(t)|2t−2 N−1∏ p=1 [lnp(t)] −2 ]−1 − B(m,α) ∣∣∣∣ 6 η. (4.36) Introducing ϕ̃(ξ) = ξ(2m−1−α)/2u(ln(ξ)), ξ ∈ (1/ρ,∞), (4.37) Corollary 4.4 implies ∣∣∣∣ [ ˆ ∞ 1/ρ dξ ξα ∣∣ϕ̃(m)(ξ) ∣∣2 − A(m,α) ˆ ∞ 1/ρ dξ ξα−2m|ϕ̃(ξ)|2 − B(m,α) ˆ ∞ 1/ρ dξ ξα−2m|ϕ̃(ξ)|2 N−1∑ k=1 k∏ p=1 [lnp(ξ)] −2 ] × [ ˆ ∞ 1/ρ dξ ξα−2m|ϕ̃(ξ)|2 N∏ p=1 [lnp(ξ)] −2 ]−1 − B(m,α) ∣∣∣∣ 6 η. (4.38) Putting ϕ(x) = ϕ̃(x/Γ), x ∈ (Γ/ρ,∞) = (r,∞), (4.39) 158 F. Gesztesy, I. Michael & M. M. H. Pang CUBO 24, 1 (2022) one infers ∣∣∣∣ [ Γ2m−α−1 { ˆ ∞ r dxxα ∣∣ϕ(m)(x) ∣∣2 − A(m,α) ˆ ∞ r dxxα−2m|ϕ(x)|2 − B(m,α) ˆ ∞ r dxxα−2m|ϕ(x)|2 N−1∑ k=1 k∏ p=1 [lnp(x/Γ)] −2 }] × [ Γ2m−α−1 ˆ ∞ r dxxα−2m|ϕ(x)|2 N∏ p=1 [lnp(x/Γ)] −2 ]−1 − B(m,α) ∣∣∣∣ 6 η, (4.40) finishing the proof since ϕ ∈ C∞0 ((r,∞)). Remark 4.9. (i) Theorem 4.1 (resp., Theorem 4.7) extends to ρ = ∞ (resp., r = 0) upon disregarding all logarithmic terms (i.e., upon putting B(m,α) = 0), we omit the details. (ii) The sequence of logarithmically refined power-weighted Birman–Hardy–Rellich inequalities underlying Theorems 4.1, 4.2, 4.7, and 4.8, extend from C∞0 −functions to functions in ap- propriately weighted (homogeneous) Sobolev spaces as shown in detail in [41, Sect. 3]. In the course of this extension, the constants A(m,α) and the N constants B(m,α) remain the same and hence optimal. (iii) We note once more that Theorems 4.1 and 4.7 were proved in [41, Theorem A.1] using a different method. 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