A Sequence of Weighted Birman–Hardy–Rellich Inequalities with Logarithmic Refinements

Abstract

The principal aim of this paper is to extend Birman’s sequence of integral inequalities originally obtained in Mat. Sb. (N.S.) 55(97), 125–174, (1961), and containing Hardy’s and Rellich’s inequality as special cases, to a sequence of inequalities that incorporates power weights $x^{\alpha }$ for x varying in intervals $(0,\rho ), \rho \in (0,\infty ) \cup \{\infty \}$, on either side and logarithmic refinements on the right-hand side of the inequality as well. Employing a new technique of proof relying on a combination of transforms originally due to Hartman and Müller-Pfeiffer, the parameter $\alpha \in {{\mathbb {R}}}$ in the power weights is now unrestricted, considerably improving on prior results in the literature. We also discuss optimality of the constants in these inequalities. This continues a tradition of logarithmic refinements in connection with Hardy’s inequality, going back to work in oscillation theory by Kneser (Math. Ann. 42, 409–435, (1893)), Hartman (Am. J. Math. 70, 764-779 (1948)), Hille (Trans. Am. Math. Soc. 64, 234-252 (1948)), and Rellich (Math. Ann. 122, 343–368 (1951)), resulting in a sequence of sharp statements of boundedness from below by zero of a class of homogeneous 2mth order differential operators on $C_0^{\infty }((0,\rho ))$. We also prove the analogous inequalities on exterior intervals, that is, for $f \in C_0^{\infty }((\rho ,\infty ))$. Finally, we also indicate a vector-valued version of these inequalities, replacing complex-valued $f(\,\cdot \,)$ by $f(\,\cdot \,) \in {{\mathcal {H}}}$, with ${{\mathcal {H}}}$ a complex, separable Hilbert space.

Appendices

Appendix A: Optimality of $A(\ell ,\alpha )$, $1 \le \ell \le m$

In this appendix we demonstrate sharpness of the constants $A(\ell ,\alpha )$, $1 \le \ell \le m$.

Theorem A.1

The constants $A(\ell ,\alpha )$, $1 \le \ell \le m$, $\alpha \in {{\mathbb {R}}}\backslash \{2j-1\}_{1 \le j \le \ell }$, in Theorems 3.1 and 3.5 are sharp.

Proof

For simplicity, we consider the interval $(0,\rho )$ (the case $(\rho ,\infty )$ being completely analogous).

To simplify notation we assume, without loss of generality, that $\rho > 2$ for the remainder of this proof.

We first present the proof for the case $\ell = m$ and near the end indicate the necessary changes to treat the analogous cases $1 \le \ell \le m-1$, $m \ge 2$. Introducing

$$\begin{aligned} y_0(x) = x^{(2\ell - 1 - \alpha )/2}, \quad x > 0, \; \ell \in {{\mathbb {N}}}, \; \alpha \in {{\mathbb {R}}}, \end{aligned}$$

(A.1)

one notes the facts

$$\begin{aligned}&y_0^{(\ell )}(x) = 2^{- \ell } (2\ell -1-\alpha )(2\ell -3-\alpha ) \cdots (3-\alpha )(1-\alpha ) x^{-(1+\alpha )/2},\qquad \end{aligned}$$

(A.2)

$$\begin{aligned}&x^{\alpha } \big [y_0^{(\ell )}\big ]^2 = A(\ell ,\alpha ) x^{\alpha -2\ell } [y_0(x)]^2 = A(\ell ,\alpha ) x^{-1}, \end{aligned}$$

(A.3)

$$\begin{aligned}&(- 1)^{\ell } \big (x^{\alpha } y_0^{(\ell )}(x)\big )^{(\ell )} - A(\ell ,\alpha ) x^{\alpha - 2 \ell } y_0(x) = 0. \end{aligned}$$

(A.4)

Next, we also introduce the cutoff functions

$$\begin{aligned}&\phi \in C^{\infty }({{\mathbb {R}}}), \quad 0 \le \phi (x) \le 1, \, x \in {{\mathbb {R}}}, \quad \phi (x) = {\left\{ \begin{array}{ll} 0, &{} x \le 1, \\ 1, &{} x \ge 2, \end{array}\right. } \qquad \end{aligned}$$

(A.5)

$$\begin{aligned}&\phi _{\varepsilon }(x) = \phi (x/\varepsilon ), \, x \in {{\mathbb {R}}}, \, 0 < \varepsilon \text { sufficiently small},\qquad \end{aligned}$$

(A.6)

$$\begin{aligned}&\psi \in C^{\infty }({{\mathbb {R}}}), \quad 0 \le \psi (x) \le 1, \, x \in {{\mathbb {R}}}, \quad \psi (x) = {\left\{ \begin{array}{ll} 1, &{} x \le \rho -2, \\ 0, &{} x \ge \rho -1, \end{array}\right. }\qquad \end{aligned}$$

(A.7)

and mollify $y_0$ as follows,

$$\begin{aligned} y_{0,\varepsilon }(x) = y_0(x) \phi _{\varepsilon }(x) \psi (x), \; 0 \le x \le \rho , \quad y_{0,\varepsilon } \in C_0^{\infty }((0,\rho )). \end{aligned}$$

(A.8)

Then one verifies

$$\begin{aligned}&A(\ell ,\alpha ) \int _0^{\rho } dx \, x^{\alpha - 2 \ell } [y_{0,\varepsilon }(x)]^2\nonumber \\&\quad = A(\ell ,\alpha ) \int _0^{\rho } dx \, x^{-1} \phi (x/\varepsilon )^2 \psi (x)^2 \nonumber \\&\quad = A(\ell ,\alpha ) \int _{\varepsilon }^{\rho -2} dx \, x^{-1} \phi (x/\varepsilon )^2 + A(\ell ,\alpha ) \int _{\rho -2}^{\rho -1} dx \, x^{-1} \psi (x)^2 \nonumber \\&\quad = A(\ell ,\alpha ) \int _{2}^{(\rho - 2)/\varepsilon } d\xi \, \xi ^{-1} \phi (\xi )^2 + A(\ell ,\alpha ) \int _1^2 d\xi \, \xi ^{-1} \phi (\xi )^2 \nonumber \\&\qquad + A(\ell ,\alpha ) \int _{\rho -2}^{\rho -1} dx \, x^{-1} \psi (x)^2 \nonumber \\&\quad \underset{\varepsilon \downarrow 0}{=} A(\ell ,\alpha ) \ln (1/\varepsilon ) + O(1), \end{aligned}$$

(A.9)

and

$$\begin{aligned} \int _0^{\rho } dx \, x^{\alpha } \big [y_{0,\varepsilon }^{(\ell )}(x)\big ]^2= & {} \int _{\varepsilon }^{\rho -2} dx \, x^{\alpha } \big [y_{0,\varepsilon }^{(\ell )}(x)\big ]^2 + \int _{\rho -2}^{\rho -1} dx \, x^{\alpha } \big [y_{0,\varepsilon }^{(\ell )}(x)\big ]^2 \nonumber \\= & {} \int _{\varepsilon }^{\rho -2} dx \, x^{\alpha } \big \{[(y_0(x) \phi (x/\varepsilon )]^{(\ell )}\big \}^2 \nonumber \\&+ \int _{\rho -2}^{\rho -1} dx \, x^{\alpha } \big \{[y_0(x) \psi (x)]^{(\ell )}\big \}^2. \end{aligned}$$

(A.10)

Next, one employs

$$\begin{aligned}{}[y_0(x) \phi (x/\varepsilon )]^{(\ell )}= & {} \sum _{k=0}^{\ell } \begin{pmatrix} \ell \\ k \end{pmatrix} y_0^{(\ell -k)}(x) \frac{d^k}{dx^k} \phi (x/\varepsilon ) \nonumber \\= & {} x^{-(1 + \alpha )/2} \sum _{k=0}^{\ell } c_{\ell ,k,\alpha } (x/\varepsilon )^k \phi ^{(k)}(x/\varepsilon ), \end{aligned}$$

(A.11)

where

$$\begin{aligned} \begin{aligned} c_{\ell ,0,\alpha }&= 2^{- \ell } (2\ell -1-\alpha )(2\ell -3-\alpha ) \cdots (3-\alpha )(1-\alpha ), \\ c_{\ell ,0,\alpha }^2&= A(\ell ,\alpha ). \end{aligned} \end{aligned}$$

(A.12)

Thus, one can continue (A.10) as follows:

$$\begin{aligned} \,(A.10)= & {} \int _{\varepsilon }^{\rho -2} dx \, x^{-1} \Bigg [\sum _{k=0}^{\ell } c_{\ell ,k,\alpha } (x/\varepsilon )^k \phi ^{(k)}(x/\varepsilon )\Bigg ]^2 \nonumber \\&+ \int _{\rho -2}^{\rho -1} dx \, x^{\alpha } \big \{[y_0(x) \psi (x)]^{(\ell )}\big \}^2 \nonumber \\= & {} \int _{1}^{(\rho -2)/\varepsilon } d\xi \, \xi ^{-1} \Bigg [\sum _{k=0}^{\ell } c_{\ell ,k,\alpha } \xi ^k \phi ^{(k)}(\xi )\Bigg ]^2 \nonumber \\&+ \int _{\rho -2}^{\rho -1} dx \, x^{\alpha } \big \{[y_0(x) \psi (x)]^{(\ell )}\big \}^2 \nonumber \\= & {} \int _{1}^{(\rho -2)/\varepsilon } d\xi \, \xi ^{-1} \Bigg [c_{\ell ,0,\alpha } \phi (\xi ) + \sum _{k=1}^{\ell } c_{\ell ,k,\alpha } \xi ^k \phi ^{(k)}(\xi )\Bigg ]^2 \nonumber \\&+ \int _{\rho -2}^{\rho -1} dx \, x^{\alpha } \big \{[y_0(x) \psi (x)]^{(\ell )}\big \}^2 \nonumber \\= & {} A(\ell ,\alpha ) \int _{1}^{(\rho -2)/\varepsilon } d\xi \, \xi ^{-1} \phi (\xi )^2 \nonumber \\&+ \int _{1}^{2} d\xi \, \xi ^{-1} \Bigg \{2 c_{\ell ,0,\alpha } \phi (\xi ) \sum _{k=1}^{\ell } c_{\ell ,k,\alpha } \xi ^k \phi ^{(k)}(\xi ) \nonumber \\&+ \Bigg [\sum _{k=1}^{\ell } c_{\ell ,k,\alpha } \xi ^k \phi ^{(k)}(\xi )\Bigg ]^2 \Bigg \} + \int _{\rho -2}^{\rho -1} dx \, x^{\alpha } \big \{[y_0(x) \psi (x)]^{(\ell )}\big \}^2 \nonumber \\&\underset{\varepsilon \downarrow 0}{=} A(\ell ,\alpha ) \int _{1}^{(\rho -2)/\varepsilon } d\xi \, \xi ^{-1} \phi (\xi )^2 + O(1) \nonumber \\&\underset{\varepsilon \downarrow 0}{=} A(\ell ,\alpha ) \int _{2}^{(\rho -2)/\varepsilon } d\xi \, \xi ^{-1} + A(\ell ,\alpha ) \int _{1}^{2} d\xi \, \xi ^{-1} \phi (\xi )^2 + O(1) \nonumber \\&\underset{\varepsilon \downarrow 0}{=} A(\ell ,\alpha ) \ln (1/\varepsilon ) + O(1), \end{aligned}$$

(A.13)

employing the fact that ${{\,\mathrm{supp}\,}}\big (\phi ^{(k)}\big ) \subseteq [1,2]$, $k \ge 1$. Thus, (A.9) and (A.13) yield

$$\begin{aligned} \frac{ \int _0^{\rho } dx \, x^{\alpha } \big [y_{0,\varepsilon }^{(\ell )}(x)\big ]^2}{A(\ell ,\alpha ) \int _0^{\rho } dx \, x^{\alpha - 2 \ell } [y_{0,\varepsilon }(x)]^2 } \underset{\varepsilon \downarrow 0}{=} 1 + O(1/\ln (1/\varepsilon )), \end{aligned}$$

(A.14)

proving sharpness of $A(\ell ,\alpha )$ for $\ell \in {{\mathbb {N}}}$ and $\alpha \in {{\mathbb {R}}}\backslash \{2j-1\}_{1 \le j \le \ell }$ on the function space $C_0^{\infty }((0,\rho ))$.

For $1 \le \ell \le m-1$, $m \ge 2$, one replaces $y_0$ by

$$\begin{aligned} \begin{aligned}&f_0(x) = [\widetilde{A}(\ell ,\alpha )/ \widetilde{A}(m,\alpha )] x^{(2m - \alpha - 1)/2}, \quad x >0, \; \alpha \in {{\mathbb {R}}}, \\&\widetilde{A}(\ell ,\alpha ) = 2^{-\ell } (2 \ell - 1 - \alpha )(2 \ell - 3 - \alpha ) \cdots (3-\alpha ) (1 - \alpha ), \quad \alpha \in {{\mathbb {R}}}, \end{aligned}\qquad \end{aligned}$$

(A.15)

and observes the facts,

$$\begin{aligned} f_0^{(m - \ell )}(x)= & {} x^{(2\ell - 1 - \alpha )/2}, \end{aligned}$$

(A.16)

$$\begin{aligned} f_0^{(m)}(x)= & {} \widetilde{A}(\ell ,\alpha ) x^{- (\alpha + 1)/2}, \end{aligned}$$

(A.17)

$$\begin{aligned} x^{\alpha } \big [f_0^{(m)}(x)\big ]^2= & {} A(\ell ,\alpha ) x^{\alpha - 2 \ell } \big [f_0^{(m-\ell )}(x)\big ]^2 = A(\ell ,\alpha ) x^{-1}, \end{aligned}$$

(A.18)

and then mollifies $f_0$ as before via

$$\begin{aligned} f_{0,\varepsilon }(x) = f_0(x) \phi _{\varepsilon }(x) \psi (x), \; 0 \le x \le \rho , \quad f_{0,\varepsilon } \in C^{\infty }_0((0, \rho )). \end{aligned}$$

(A.19)

At this point one can follow the above proof step by step arriving at

$$\begin{aligned} \frac{ \int _0^{\rho } dx \, x^{\alpha } \big [f_{0,\varepsilon }^{(m)}(x)\big ]^2}{A(\ell ,\alpha ) \int _0^{\rho } dx \, x^{\alpha - 2 \ell } \big [f_{0,\varepsilon }^{(m-\ell )}(x)\big ]^2 } \underset{\varepsilon \downarrow 0}{=} 1 + O(1/\ln (1/\varepsilon )), \end{aligned}$$

(A.20)

once more proving sharpness of $A(\ell ,\alpha )$ for $\ell \in {{\mathbb {N}}}$ and $\alpha \in {{\mathbb {R}}}\backslash \{2j-1\}_{1 \le j \le \ell }$.

Since Theorem 3.5 exhibits the same constant $A(\ell ,\alpha )$, the latter is sharp also for the larger function space $H_0^m((0,\rho ); x^{\alpha }dx)$. $\square$

Remark A.2

(i) Once more we recall that $A(\ell ,\alpha ) = 0$ if and only if $\alpha \in \{2j-1\}_{1 \le j \le \ell }$. Thus, the inequality

$$\begin{aligned} \int _0^{\rho } dx \, x^{\alpha } \big | f^{(m )}(x) \big |^{2} \ge A(\ell , \alpha ) \int _0^{\rho } dx \, x^{\alpha - 2\ell } \big |f^{(m - \ell )}(x)\big |^{2}, \quad f \in C_0^{\infty }((0,\rho )), \nonumber \\ \end{aligned}$$

(A.21)

is rendered trivial if $\alpha \in \{2j-1\}_{1 \le j \le \ell }$, with the right-hand side of (A.21) being zero. The same observation applies of course to the remaining three cases (i), (ii), and (iv) in Theorem 3.1. However, we emphasize that inequalities (3.1)–(3.4) remain valid and nontrivial with just the first terms on their right-hand sides removed.

(ii) For $\alpha \in {{\mathbb {R}}}\backslash \{2j-1\}_{1 \le j \le \ell }$, inequality (A.21) extends to $\rho = \infty$, again with $A(\ell , \alpha )$ being the sharp constant for $f \in C_0^{\infty }((0,\infty ))$. In particular, the proof of Theorem A.1, suitably adapted, extends to the case $\rho = \infty$. (This observation applies of course to cases (i), (ii) (if $\rho = 0$), and (iii), (iv) (if $\rho = \infty$) in Theorem 3.1.) This is of course in accordance with the fact that $C_0^{\infty }((0,\rho ))$-functions extended by zero beyond $\rho$, $\rho \in (0,\infty )$, can be viewed as a subset of $C_0^{\infty }((0,\infty ))$. $\diamond$

Remark A.3

Regarding sharpness (optimality) of constants, we first note that the smaller the underlying function space, the larger the efforts needed to prove optimality. In particular, in connection with the proof presented in Theorem A.1, assuming $f \in C_0^{\infty }((0,\rho ))$ requires mollification of $y_0$ in (A.1) near $x=0$ and $x=\rho$ and of course analogously in the case $f \in C_0^{\infty }((\rho ,\infty ))$. Many of the results cited in the remainder of this remark, under particular restrictions on the weight parameter $\alpha$, establish sharpness for larger classes of functions f which do not automatically continue to hold in the $C_0^{\infty }((0,\rho ))$-context (the issue of dependence of optimal constants on the underlying function space is nicely illustrated in [31]). It is this simple observation that adds considerable complexity to sharpness proofs for the space $C_0^{\infty }((0,\rho ))$. (By the same token, optimality proofs obtained for $C_0^{\infty }$ 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 $A(m,\alpha ), B(m,\alpha )$.) This comment applies, in particular, to many papers that prove sharpness results in multi-dimensional situations for larger function spaces such as $C_0^{\infty }(B(0;\rho ))$ or (homogeneous, weighted) Sobolev spaces rather than $C_0^{\infty }(B(0;\rho ) \backslash \{0\})$, $B(0;\rho ) \subseteq {{\mathbb {R}}}^n$ the open ball in ${{\mathbb {R}}}^n$, $n \ge 2$, with center at the origin $x=0$ and radius $\rho > 0$. Unless $C_0^{\infty }(B(0;\rho ) \backslash \{0\})$ is dense in the appropriate norm (cf. the discussion preceding Theorem 3.5 in the one-dimensional context), one cannot a priori assume that the optimal constants $A(m, \widetilde{\alpha })$ and $B(m, \widetilde{\alpha })$ (with $\widetilde{\alpha }$ appropriately depending on n, e.g., $\widetilde{\alpha }= \alpha + n - 1$) remain the same for $C_0^{\infty }(B(0;\rho ))$ and $C_0^{\infty }(B(0;\rho ) \backslash \{0\})$, say. At least in principle, they could actually increase for the space $C_0^{\infty }(B(0;\rho ) \backslash \{0\})$. In this context we emphasize that the multi-dimensional results then naturally lead to one-dimensional results for $C_0^{\infty }((0,\rho ))$ upon specializing to radially symmetric functions in $C_0^{\infty } (B(0;\rho ) \backslash \{0\})$.

Sharpness of the constant A(m, 0), $m \in {{\mathbb {N}}}$ (i.e., in the unweighted case, $\alpha = 0$), in connection with the space $C_0^{\infty }((0,\infty ))$ has been shown by Yafaev [101]. In fact, he also established this result for fractional m (in this context we also refer to appropriate norm bounds in $L^p({{\mathbb {R}}}^n; d^nx)$ of operators of the form $|x|^{-\beta } |-i \nabla |^{-\beta }$, $1< p < n/\beta$, see [13, Sect. 1.7], [14, Sects. 1.7, 4.2], [58, 63, 65, 87, 96, 97]). Sharpness of A(2, 0) (i.e., in the unweighted Rellich case) was shown by Rellich [91, p. 91–101] in connection with the space $C_0^{\infty }((0,\infty ))$; his multi-dimensional results also yield sharpness of $A(2,n-1)$ for $n \in {{\mathbb {N}}}$, $n \ge 3$, again for $C_0^{\infty }((0,\infty ))$; in this context see also [14, Corollary 6.3.5]. An exhaustive study of optimality of $A(2, \widetilde{\alpha })$ (i.e., Rellich inequalities with power weights) for the space $C_0^{\infty }(\Omega \backslash \{0\})$ for cones $\Omega \subseteq {{\mathbb {R}}}^n$, $n \ge 2$, appeared in Caldiroli and Musina [21]. The authors, in particular, describe situations where $A(2, \widetilde{\alpha })$ 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 [83, 84]; further extensions of power weighted Rellich inequalities with sharp constants on $C_0^{\infty }({{\mathbb {R}}}^n \backslash \{0\})$ were obtained in [77]; for optimal power weighted Hardy, Rellich, and higher-order inequalities on homogeneous groups, see [92, 93]. Many of these references also discuss sharp (power weighted) Hardy inequalities, implying optimality for $A(1, \widetilde{\alpha })$. Moreover, replacing f(x) by $F(x) = \int _0^x dt \, f(t)$ (or $F(x) = \int _x^{\infty } dt \, f(t)$), optimality of the Hardy constant A(1, 0) for larger, $L^p$-based function spaces, can already be found in [55, Sect. 9.8] (see also [14, Theorem 1.2.1], [67, Ch. 3], [68, p. 5–11], [71, 80, 89], in connection with $A(1, \alpha )$).

Sharpness results for $A(m, \alpha )$ and $B(m, \alpha )$ together are much less frequently discussed in the literature, even under suitable restrictions on m and $\alpha$. The results we found primarily follow upon specializing multi-dimensional results for function spaces such as $C_0^{\infty }(\Omega \backslash \{0\})$, or $C_0^{\infty }(\Omega )$, $\Omega \subseteq {{\mathbb {R}}}^n$ open, and appropriate restrictions on m, $\alpha$, and $n\ge 2$, 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, 24, 27, 37, 53, 62, 73, 95, 99] (all for $N=1$), and in [10, 29, 49] (all for $N \in {{\mathbb {N}}}$); the case with power weight functions is discussed in [17, 50, 51, Ch. 6] (for $N \in {{\mathbb {N}}}$); see also [74]. The Rellich case $m=2$ with a general power weight on $C_0^{\infty }(\Omega \backslash \{0\})$ is discussed in [21] (for $N=1$); the Rellich case $m=2$, without weight function on $C_0^{\infty }(\Omega )$, is studied in [27, 28, 30] (all for $N=1$), the case $N \in {{\mathbb {N}}}$ is studied in [4]; the case of additional power weights is treated in [50, 51, Ch. 6], [79]. The general case $m \in {{\mathbb {N}}}$ is discussed in [6] (for $N=1$) and in [15, 50, 51, Ch. 6], [100] (all for $N \in {{\mathbb {N}}}$ 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 $N \in {{\mathbb {N}}}$, was proved in [46].

The special results available on sharpness of $B(m,\alpha )$ are all saddled with enormous complexity, especially, for larger values of $N \in {{\mathbb {N}}}$. In fact, a careful proof for general N will rival the length of this paper and hence has not been attempted here as briefly discussed in the following remark. $\diamond$

Remark A.4

The proof of optimality of $A(\ell , \alpha )$ in Theorem A.1 consists of two principal steps:

(i) Identify a function $y_0$ (see (A.1)) which is not in $C^{\infty }_0((\rho , \infty ))$, but which satisfies (see (A.4)),

$$\begin{aligned} \frac{\int _{\rho }^{\infty } dx \, x^{\alpha } \big |y_0^{(m)}(x)\big |^2}{\int _{\rho }^{\infty } dx \, x^{\alpha - 2\ell } \big |y_0^{(m - \ell )}(x)\big |^2} = A(\ell , \alpha ). \end{aligned}$$

(A.22)

(ii) Exhibit a family $\{y_{0, \varepsilon }\}_{\varepsilon > 0} \subset C^{\infty }_0((\rho , \infty ))$ of multiplicative mollifications of $y_0$ (see (A.8)) that approaches $y_0$ as $\varepsilon \downarrow 0$ and for which (see (A.14))

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} \frac{\int _{\rho }^{\infty } dx \, x^{\alpha } \big |y_{0, \varepsilon }^{(m)}(x)\big |^2}{\int _{\rho }^{\infty } dx \, x^{\alpha - 2\ell } dx \big |y_{0, \varepsilon }^{(m - \ell )}(x)\big |^2} = A(\ell , \alpha ). \end{aligned}$$

(A.23)

Unfortunately, due to the ensuing complexity when having to apply the product rule of differentiation again and again, this approach in connection with $A(\ell ,\alpha )$ cannot naturally be adapted to a proof of optimality of $B(\ell , \alpha )$. The proof of optimality of $B(\ell , \alpha )$ we are currently working out requires substantial modification to steps (i) and (ii) above. We sketch the new approach in the special case $\ell = m$ in inequality (3.1). We abbreviate

$$\begin{aligned} W_{m, \alpha , N}(x) = A(m,\alpha ) + B(m,\alpha ) \sum _{k=1}^{N-1} \prod _{p=1}^{k} [\ln _{p}(\gamma /x)]^{-2}. \end{aligned}$$

(A.24)

Instead of identifying one explicit functon $y_0$ which satisfies (A.22), we use a modification of the proof of [15, Theorem 2] to identify a family $\{f_{0, \varepsilon } :(\rho , \infty ) \rightarrow {\mathbb {C}}\}_{\varepsilon > 0}$ of functions which are not in $C^{\infty }_0((\rho , \infty ))$ but for which

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}\frac{\int _{\rho }^{\infty } dx \, x^{\alpha } \big | f_{0, \varepsilon }^{(m )}(x) \big |^{2} - \int _{\rho }^{\infty } dx \, x^{\alpha -2m} W_{m, \alpha , N}(x) | f_{0, \varepsilon } (x)|^2}{\int _{\rho }^{\infty } dx \, x^{\alpha - 2m} \prod _{p=1}^{N} [\ln _{p}(\gamma /x)]^{-2} | f_{0, \varepsilon } (x)|^2}= B(m, \alpha ). \nonumber \\ \end{aligned}$$

(A.25)

Instead of a family of multiplicative mollifications as in (A.8), for each $\varepsilon >0$ we employ a family $\{f_{0, \varepsilon , \nu }\}_{\nu >0} \subset C^{\infty }_0((\rho , \infty ))$ of mollifications of $f_{0, \varepsilon }$ using convolution with an approximate identity which has the properties:

$$\begin{aligned} \lim _{\nu \downarrow 0} \int _{\rho }^{\infty } dx\, x^{\alpha } \big | f_{0, \varepsilon , \nu }^{(m)}(x)\big |^2 = \int _{\rho }^{\infty } dx\, x^{\alpha } \big |f_{0, \varepsilon }^{(m)}(x)\big |^2, \end{aligned}$$

(A.26)

and for $k = 0, 1, \cdot \cdot \cdot , N$,

$$\begin{aligned} \begin{aligned}&\lim _{\nu \downarrow 0} \int _{\rho }^{\infty } dx \, x^{\alpha -2m} \prod _{p=1}^{k} [\ln _{p}(\gamma /x)]^{-2} | f_{0, \varepsilon , \nu } (x)|^2 \\&\quad = \int _{\rho }^{\infty } dx \, x^{\alpha -2m} \prod _{p=1}^{k} [\ln _{p}(\gamma /x)]^{-2} \big | f_{0, \varepsilon } (x)\big |^2. \end{aligned} \end{aligned}$$

(A.27)

Thus, roughly speaking, one gets

$$\begin{aligned} \lim _{\varepsilon , \nu \downarrow 0} \frac{\int _{\rho }^{\infty } dx \, x^{\alpha } \big | f_{0, \varepsilon , \nu }^{(m )}(x) \big |^{2} - \int _{\rho }^{\infty } dx \, x^{\alpha - 2m} W_{m, \alpha , N}(x) |f_{0, \varepsilon , \nu }(x)|^2}{\int _{\rho }^{\infty } dx \, x^{\alpha - 2m} \prod _{p=1}^{N} [\ln _{p}(\gamma /x)]^{-2} | f_{0, \varepsilon , \nu } (x)|^2}= B(m, \alpha ). \nonumber \\ \end{aligned}$$

(A.28)

This approach is that much longer than the proof of Theorem A.1, that we felt we had no choice but to write a separate paper [44] for the proof of optimality of $B(\ell , \alpha )$. $\diamond$

Appendix B: The Interval Case $(\rho ,\infty )$ in Theorem 3.5

We recall the space

$$\begin{aligned} H_{m,\alpha }([\rho ,\infty ))= & {} \big \{f: [\rho ,\infty ) \rightarrow {{\mathbb {C}}}\, \big | \, \text {for all }R > \rho , \, f^{(k)} \in AC([\rho ,R]), \nonumber \\&f^{(k)}(\rho ) = 0, \, 0 \le k \le m-1; \, f^{(m)} \in L^2\big ((\rho ,\infty ); x^{\alpha }dx\big )\big \},\nonumber \\ \end{aligned}$$

(B.1)

and introduce the bilinear form $\langle \, \cdot \,,\, \cdot \, \rangle _{m,\alpha }$ on $H_{m,\alpha }([\rho ,\infty ))$ by

$$\begin{aligned} \langle f,g \rangle _{m,\alpha } = \int _{\rho }^{\infty } x^{\alpha } dx \, \overline{f^{(m)}(x)} g^{(m)}(x), \quad f, g \in H_{m,\alpha }([\rho ,\infty )). \end{aligned}$$

(B.2)

Proposition B.1

The bilinear form $\langle \, \cdot \,,\, \cdot \, \rangle _{m,\alpha }$ is an inner product on the space $H_{m,\alpha }([\rho ,\infty ))$, in fact, $(H_{m,\alpha }([\rho ,\infty )), \langle \, \cdot \,,\, \cdot \, \rangle _{m,\alpha })$ is a Hilbert space.

Proof

Assuming $\langle f,f \rangle _{m,\alpha } = 0$, $f \in H_{m,\alpha }([\rho ,\infty ))$, one obtains

$$\begin{aligned} \int _{\rho }^{\infty } x^{\alpha } dx \, \big |f^{(m)}(x)\big |^2 = 0, \end{aligned}$$

(B.3)

and hence $f^{(m)}=0$ a.e. on $(\rho ,\infty )$. Thus, employing $f^{(m-1)}(\rho ) = 0$, one concludes that

$$\begin{aligned} f^{(m-1)}(x) = \int _{\rho }^x dt \, f^{(m)}(t) = 0, \quad x \ge \rho . \end{aligned}$$

(B.4)

Similarly, as $f^{(m-2)}(\rho ) = 0$,

$$\begin{aligned} f^{(m-2)}(x) = \int _{\rho }^x dt \, f^{(m-1)}(t) =0, \quad x \ge \rho , \end{aligned}$$

(B.5)

and hence inductively,

$$\begin{aligned} f^{(k)}(x) = 0, \quad 0 \le k \le m - 1, \; x \ge \rho . \end{aligned}$$

(B.6)

Thus, $\langle \, \cdot \,,\, \cdot \, \rangle _{m,\alpha }$ is an inner product on $H_{m,\alpha }([\rho ,\infty ))$.

To prove completeness of $(H_{m,\alpha }([\rho ,\infty )), \langle \, \cdot \,,\, \cdot \, \rangle _{m,\alpha })$, one assumes that $\{f_n\}_{n \in {{\mathbb {N}}}}$ is a Cauchy sequence in $H_{m,\alpha }([\rho ,\infty ))$. Then $\big \{f_n^{(m)}\big \}_{n \in {{\mathbb {N}}}}$ is a Cauchy sequence in $L^2\big ((\rho ,\infty ); x^{\alpha }dx\big )$. Hence, there exists $g \in L^2\big ((\rho ,\infty ); x^{\alpha }dx\big )$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \big \Vert f_n^{(m)} - g\big \Vert _{L^2((\rho ,\infty ); x^{\alpha }dx)} = 0. \end{aligned}$$

(B.7)

Introducing $f: [\rho ,\infty ) \rightarrow {{\mathbb {C}}}$ by

$$\begin{aligned} f(x) = \int _{\rho }^x \int _{\rho }^{t_1} \cdots \int _{\rho }^{t_{m-1}} dt_1 \cdots dt_{m-1} d u \, g(u), \quad x \ge \rho , \end{aligned}$$

(B.8)

then $f^{(k)} \in AC([\rho ,R])$ for all $R > \rho$ and $f^{(k)} (\rho ) = 0$, $0 \le k \le m-1$, and $f^{(m)} = g \in L^2\big ((\rho ,\infty ); x^{\alpha }dx\big )$, and hence $f \in H_{m,\alpha }([\rho ,\infty ))$. In addition,

$$\begin{aligned} \begin{aligned} |||f_n - f|||_{H_{m,\alpha }([\rho ,\infty ))}&= \big \Vert f_n^{(m)} - f^{(m)}\big \Vert _{L^2((\rho ,\infty ); x^{\alpha }dx)} \\&= \big \Vert f_n^{(m)} - g\big \Vert _{L^2((\rho ,\infty ); x^{\alpha }dx)} \underset{n \rightarrow \infty }{\longrightarrow } 0, \end{aligned} \end{aligned}$$

(B.9)

completing the proof. $\square$

We recall that the norm in Hilbert space $H_{m,\alpha }([\rho ,\infty ))$ is denoted by $|||\, \cdot \,|||_{m,\alpha }$. The fact that $C_0^{\infty }((\rho ,\infty )) \subset H_{m,\alpha }([\rho ,\infty ))$ then leads to the introduction of the homogeneous weighted Sobolev space

$$\begin{aligned} \dot{H_0^{m}} \big ((\rho ,\infty ); x^{\alpha } dx\big ) = \overline{C_0^{\infty }((\rho ,\infty ))}^{|||\, \cdot \,|||_{m,\alpha }}, \end{aligned}$$

(B.10)

that is, the closure of $C_0^{\infty }((\rho ,\infty ))$ in $H_{m,\alpha }([\rho ,\infty ))$. Proposition B.1 then yields the following result.

Corollary B.2

Assume that $f \in \dot{H_0^{m}} \big ((\rho ,\infty ); x^{\alpha } dx\big )$, $\alpha \in {{\mathbb {R}}}$. Then there exists a sequence $\{f_n\}_{n \in {{\mathbb {N}}}} \subset C_0^{\infty }((\rho ,\infty ))$ such that for $0 \le k \le m$,

$$\begin{aligned} \lim _{n \rightarrow \infty } f_n^{(k)} (x) = f^{(k)} (x) \, \text { for a.e. }x > \rho . \end{aligned}$$

(B.11)

Proof

Since $f \in \dot{H_0^{m}} \big ((\rho ,\infty ); x^{\alpha } dx\big )$, there exists a sequence $\{f_n\}_{n \in {{\mathbb {N}}}} \subset C_0^{\infty }((\rho ,\infty ))$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \big \Vert f_n^{(m)} - f^{(m)}\big \Vert _{L^2((\rho ,\infty ); x^{\alpha }dx)} = 0. \end{aligned}$$

(B.12)

By taking a subsequence, if necessary, one can assume that

$$\begin{aligned} \lim _{n \rightarrow \infty } f_n^{(m)} (x) = f^{(m)} (x) \, \text { for a.e. }x > \rho . \end{aligned}$$

(B.13)

Since $f^{(k)}(\rho ) = 0$, $0 \le k \le m-1$,

$$\begin{aligned}&\big |f_n^{(m-1)} (x) - f^{(m-1)} (x)\big | \nonumber \\&\quad = \bigg |\int _{\rho }^x dt \, f_n^{(m)} (t) - \int _{\rho }^x dt \, f^{(m)} (t)\bigg | \nonumber \\&\quad \le \int _{\rho }^x dt \, \big |f_n^{(m)} (t) - f^{(m)} (t)\big | \nonumber \\&\quad \le \bigg [\int _{\rho }^x dt \, \big |f_n^{(m)} (t) - f^{(m)} (t) \big |^2\bigg ]^{1/2} (x - \rho )^{1/2} \nonumber \\&\quad \le \max \big (\rho ^{-\alpha /2}, x^{-\alpha /2}\big ) \bigg [\int _{\rho }^x dt \, t^{\alpha } \big |f_n^{(m)} (t) - f^{(m)} (t) \big |^2\bigg ]^{1/2} (x - \rho )^{1/2} \nonumber \\&\quad \underset{n \rightarrow \infty }{\longrightarrow } 0,\quad x \ge \rho . \end{aligned}$$

(B.14)

Next, fix $R > \rho$. Then for all $n \in {{\mathbb {N}}}$ sufficiently large, and for all $x \in [\rho ,R]$, there exists $C(\rho , \alpha , R) \in (0,\infty )$ such that

$$\begin{aligned} \big |f_n^{(m-1)} (x)\big |\le & {} \int _{\rho }^x dt \, \big |f_n^{(m)} (t)\big | \nonumber \\\le & {} \bigg [\int _{\rho }^x dt \, \big |f_n^{(m)} (t)\big |^2\bigg ]^{1/2} (x - \rho )^{1/2} \nonumber \\\le & {} \max \big (\rho ^{- \alpha /2}, x^{-\alpha /2}\big ) \bigg [\int _{\rho }^x dt \, t^{\alpha } \big |\big [f_n^{(m)} (t) - f^{(m)}(t)\big ] + f^{(m)}(t)\big |^2\bigg ]^{1/2} \nonumber \\&\times (x - \rho )^{1/2} \nonumber \\\le & {} \max \big (\rho ^{- \alpha /2}, x^{-\alpha /2}\big ) \bigg [2\int _{\rho }^x dt \, t^{\alpha } \big |f_n^{(m)} (t) - f^{(m)}(t)\big |^2 \nonumber \\&+ 2 \int _{\rho }^x dt \, t^{\alpha } \big |f^{(m)}(t)\big |^2\bigg ]^{1/2} (x - \rho )^{1/2} \nonumber \\\le & {} \max \big (\rho ^{- \alpha /2}, x^{-\alpha /2}\big ) \bigg [o(1) + 2 \int _{\rho }^x dt \, t^{\alpha } \big |f^{(m)}(t)\big |^2\bigg ]^{1/2} (x - \rho )^{1/2} \nonumber \\\le & {} C(\rho , \alpha , R) \big \Vert f^{(m)}\big \Vert _{L^2((\rho ,\infty ); x^{\alpha } dx)}, \quad x \in [\rho ,R]. \end{aligned}$$

(B.15)

Thus, (B.14), (B.15), and an application of Lebesgue’s dominated convergence theorem implies

$$\begin{aligned} \lim _{n \rightarrow \infty } \Big \Vert f_n^{(m-1)}\big |_{[\rho ,R]} - f^{(m-1)}\big |_{[\rho ,R]}\Big \Vert _{L^1((\rho ,R); dt)} = 0. \end{aligned}$$

(B.16)

Next, one infers that

$$\begin{aligned} \big |f_n^{(m-2)}(x) - f^{(m-2)}(x)\big |= & {} \bigg |\int _{\rho }^x dt \, \big [f_n^{(m-1)}(t) - f^{(m-1)}(t)\big ]\bigg | \nonumber \\\le & {} \int _{\rho }^x dt \, \big |f_n^{(m-1)}(t) - f^{(m-1)}(t)\big | \nonumber \\\le & {} \int _{\rho }^R dt \, \big |f_n^{(m-1)}(t) - f^{(m-1)}(t)\big | \nonumber \\= & {} \Big \Vert f_n^{(m-1)}\big |_{[\rho ,R]} - f^{(m-1)}\big |_{[\rho ,R]}\Big \Vert _{L^1((\rho ,R); dt)} \nonumber \\&\underset{n \rightarrow \infty }{\longrightarrow } 0,\quad x \in [\rho ,R], \end{aligned}$$

(B.17)

by (B.16). Similarly, for all $n \in {{\mathbb {N}}}$ sufficiently large, and for all $x \in [\rho ,R]$, one has

$$\begin{aligned} \big |f_n^{(m-2)}(x)\big |\le & {} \int _{\rho }^x dt \, \big |f_n^{(m - 1)}(t)| \le \int _{\rho }^R dt \, \big |f_n^{(m - 1)}(t)| \nonumber \\= & {} \Big \Vert \big [f_n^{(m-1)} - f^{(m-1)}\big ] + f^{(m-1)}\big |_{[\rho ,R]}\Big \Vert _{L^1((\rho ,R); dt)} \nonumber \\\le & {} o(1) + \Big \Vert f^{(m-1)}\big |_{[\rho ,R]}\Big \Vert _{L^1((\rho ,R); dt)} \nonumber \\\le & {} 2 \Big \Vert f^{(m-1)}\big |_{[\rho ,R]}\Big \Vert _{L^1((\rho ,R); dt)}, \quad x \in [\rho ,R]. \end{aligned}$$

(B.18)

Thus, (B.17), (B.18), and an application of Lebesgue’s dominated convergence theorem yields

$$\begin{aligned} \lim _{n \rightarrow \infty } \Big \Vert f_n^{(m-2)}\big |_{[\rho ,R]} - f^{(m-2)}\big |_{[\rho ,R]}\Big \Vert _{L^1((\rho ,R); dt)} = 0. \end{aligned}$$

(B.19)

Iterating these arguments proves that for all $0 \le k \le m-1$,

$$\begin{aligned} \lim _{n \rightarrow \infty } f_n^{(k)} (x) = f^{(k)} (x) \, \text { for a.e. }x \in [\rho ,R]. \end{aligned}$$

(B.20)

Since $R > \rho$ was arbitrary, this concludes the proof of Corollary B.2. $\square$