Product formulas and convolutions for Laplace–Beltrami operators on product spaces: beyond the trivial case

Abstract

We introduce the notion of a family of convolution operators associated with a given elliptic partial differential operator. Such a convolution structure is shown to exist for a general class of Laplace–Beltrami operators on product manifolds endowed with warped Riemannian metrics. This structure gives rise to a convolution semigroup representation for the Markovian semigroup generated by the Laplace–Beltrami operator. We provide several examples on the product ${\mathbb {R}}^+ \times {\mathbb {T}}$, and include a study of product formulas and convolution structures generated by elliptic operators on ${\mathbb {R}}^+_0 \times I$ (I being an interval).

Appendix: Generalized convolution structures for Sturm–Liouville operators

One-dimensional convolutions associated with Sturm–Liouville operators have been extensively studied in recent work of the authors [51, 52]. For convenience, in this appendix we summarize some fundamental results on this topic.

Let

$$\begin{aligned} \ell (u)(x) := {1 \over r(x)} \Bigl ( -(pu')'(x) + q(x)u(x) \Bigr ), \qquad x \in (a,b) \subset {\mathbb {R}} \end{aligned}$$

be a Sturm–Liouville expression whose coefficients are such that $p, r > 0$ on (a, b), p, r are locally absolutely continuous and q is locally integrable on (a, b).

Consider the integrals

$$\begin{aligned}&I_a = \int _a^c \int _a^y {dx \over p(x)} \bigl (r(y) + q(y)\bigr ) dy, \quad J_a = \int _a^c \int _y^c {dx \over p(x)} \bigl (r(y) + q(y)\bigr ) dy, \\&I_b = \int _c^b \int _y^b {dx \over p(x)} \bigl (r(y) + q(y)\bigr ) dy, \quad J_b = \int _c^b \int _c^y {dx \over p(x)} \bigl (r(y) + q(y)\bigr ) dy. \end{aligned}$$

The Feller boundary classification for $\ell$ is as follows: the endpoint $e \in \{a,b\}$ is said to be:

$$\begin{aligned} \begin{array}{llll} \text {regular } &{} \text {if} &{} I_e< \infty , &{} \!\!\!J_e< \infty ; \\ \text {exit } &{} \text {if} &{} I_e< \infty , &{} \!\!\!J_e = \infty ; \end{array} \qquad \quad \begin{array}{llll} \text {entrance } &{} \text {if} &{} I_e = \infty , &{} \!\!\!J_e < \infty ; \\ \text {natural } &{} \text {if} &{} I_e = \infty , &{} \!\!\!J_e = \infty \end{array} \end{aligned}$$

(A.1)

(the classification is independent of the choice of c). Throughout the appendix we assume that the endpoint a is regular or entrance.

Lemma A.1

The boundary value problem

$$\begin{aligned} \ell (v) = \lambda v \quad (a< x < b, \; \lambda \in {\mathbb {C}}), \qquad \; v(a) = 1, \qquad \; (pv')(a) = 0 \end{aligned}$$

has a unique solution $v_\lambda (\varvec{\cdot })$. Moreover, $\lambda \mapsto v_\lambda (x)$ is, for fixed x, an entire function of exponential type.

Proof

See [52, Lemma 2.1]. (The result of [52] is stated for the case $q \equiv 0$, but the general case can be proved in a similar way.) $\square$

Proposition A.2

The operator ${\mathcal {L}}:{\mathcal {D}}({\mathcal {L}}) \longrightarrow L^2(r) \equiv L^2\bigl ((a,b),r(z)dz\bigr )$, where

$$\begin{aligned} {\mathcal {D}}({\mathcal {L}})&= {\left\{ \begin{array}{ll} \bigl \{ u \in L^2(r) \bigm | u, u' \!\in \mathrm {AC}_{\mathrm {loc}}(a,b), \; \ell (u) \in L^2(r), \; (pu')(a) = 0 \bigr \} &{} \text { if } b \text { is exit or natural},\\ \bigl \{ u \in L^2(r) \bigm | u, u' \!\in \mathrm {AC}_{\mathrm {loc}}(a,b), \; \ell (u) \in L^2(r), \; (pu')(a) = (pu')(b) = 0 \bigr \} &{} \text { otherwise}\\ \end{array}\right. } \\ {\mathcal {L}} u&= \ell (u), \qquad u \in {\mathcal {D}}({\mathcal {L}}) \end{aligned}$$

is a positive self-adjoint operator. There exists a locally finite positive Borel measure $\varvec{\rho }_\ell$ on ${\mathbb {R}}_0^+$ such that the integral operator ${\mathcal {F}}_\ell : L^2(r) \longrightarrow L^2({\mathbb {R}}_0^+, \varvec{\rho }_\ell )$ defined by

$$\begin{aligned} ({\mathcal {F}}_{\ell \,} f)(\lambda ) := \int _a^b f(x) \, v_\lambda (x) \, r(x) dz \end{aligned}$$

(A.2)

is an isometric isomorphism whose inverse is given by

$$\begin{aligned} ({\mathcal {F}}_\ell ^{-1} \varphi )(x) = \int _{{\mathbb {R}}_0^+} \varphi (\lambda ) \, v_\lambda (x) \, \varvec{\rho }_\ell (d\lambda ). \end{aligned}$$

(A.3)

(The convergence of the integrals above is understood with respect to the norm of $L^2({\mathbb {R}}_0^+, \varvec{\rho }_\ell )$ and $L^2(r)$ respectively.) The operator ${\mathcal {F}}_\ell$ is a spectral representation of ${\mathcal {L}}$, i.e. we have

$$\begin{aligned} {\mathcal {D}}({\mathcal {L}})&= \biggl \{f \in L^2(r) \biggm | \int _{{\mathbb {R}}_0^+} \lambda ^2 \bigl |({\mathcal {F}}_\ell f)(\lambda )\bigr |^2 \varvec{\rho }_\ell (d\lambda ) < \infty \biggr \},\\&\quad \bigl ({\mathcal {F}}_\ell ({\mathcal {L}} f)\bigr ) (\lambda ) = \lambda \,\cdot \,({\mathcal {F}}_\ell f)(\lambda ), \qquad \quad f \in {\mathcal {D}}({\mathcal {L}}). \end{aligned}$$

Moreover, if $f \in {\mathcal {D}}({\mathcal {L}})$ then $f(x) = \int _{{\mathbb {R}}_0^+} ({\mathcal {F}}_{\ell \,}f)(\lambda ) \, v_\lambda (x) \, \varvec{\rho }_\ell (d\lambda )$ for all $x \in (a,b)$, where the integral converges absolutely and locally uniformly.

If b is regular, entrance or exit, then ${\mathcal {L}}$ has purely discrete spectrum $0 = \lambda _1 < \lambda _2 \le \ldots \rightarrow \infty$, and the isomorphism (A.2)–(A.3) reduces to

$$\begin{aligned}&{\mathcal {F}}_\ell : L^2(r) \longrightarrow \ell _2\bigl (\tfrac{1}{\Vert v_{\lambda _k} \Vert ^2}\bigr ), \qquad {\mathcal {F}}_{\ell \,} f \equiv \bigl ( ({\mathcal {F}}_{\ell \,} f)(\lambda _1), ({\mathcal {F}}_{\ell \,} f)(\lambda _2), \ldots \bigr ), \\&\qquad \qquad \qquad \qquad \qquad ({\mathcal {F}}_\ell ^{-1} \{c_k\})(x) = \sum _{k = 1}^\infty {c_k \, v_{\lambda _k}(x) \over \Vert v_{\lambda _k} \Vert ^2}, \end{aligned}$$

where $\ell _2\bigl (\frac{1}{\Vert v_{\lambda _k} \Vert ^2}\bigr )$ denotes the weighted sequence space whose norm is $\Vert \{c_k\}\Vert = \bigl (\sum _{k=1}^\infty {|c_k|^2 \over \Vert v_{\lambda _k} \Vert ^2}\bigr )^{1/2}$.

Proof

See [52, Proposition 2.5 and Lemma 2.6], [37, Section 5]. $\square$

Proposition A.3

The self-adjoint operator $-{\mathcal {L}}$ is the generator of a Markovian semigroup $\{e^{-t{\mathcal {L}}}\}_{t \ge 0}$ on $L^2(r)$. For $t>0$, the operators $e^{-t{\mathcal {L}}}$ are given by

$$\begin{aligned} (e^{-t{\mathcal {L}}} h)(x) = \int _a^b h(y)\, p_\ell (t,x,\xi )\, r(\xi )d\xi \qquad \bigl (h \in L^2(r), \; x \in (a,b)\bigr ), \end{aligned}$$

(A.4)

where the kernel is defined by the integral

$$\begin{aligned} p_\ell (t,x_1,x_2) = \int _{{\mathbb {R}}_0^+} e^{-t\lambda } \, v_\lambda (x_1) \, v_\lambda (x_2)\, \varvec{\rho }_\ell (d\lambda ) \qquad \bigl (t > 0, \; x_1,x_2 \in (a,b)\bigr ).\qquad \end{aligned}$$

(A.5)

The right hand side of (A.5) is (for fixed $t>0$) absolutely and uniformly convergent on compact squares of $(a,b) \times (a,b)$.

Suppose also that a is regular and b is exit or natural. Then the integral in (A.5) converges absolutely and uniformly on compact squares of $[a,b) \times [a,b)$. Moreover, the restriction of $e^{-t{\mathcal {L}}}$ to $L^2(r) \cap \mathrm {C}_0[a,b)$ extends into a strongly continuous contraction semigroup on $\mathrm {C}_0[a,b)$ which can be represented by the right hand side of (A.4), which is convergent for all $h \in \mathrm {C}_0[a,b)$ and $x \in [a,b)$.

Proof

See [52, Proposition 2.7], [23]. $\square$

Next we restrict our attention to the case $q \equiv 0$ and state some further properties of the generalized eigenfunctions $v_\lambda (x)$.

Lemma A.4

(a) If $q \equiv 0$ and $x \mapsto p(x) r(x)$ is an increasing function, then $|v_\lambda (x)| \le 1$ for all $a \le x < b$ and $\lambda \ge 0$.

(b) Let $S(\xi ) := \sqrt{p(\gamma ^{-1}(\xi )) \, r(\gamma ^{-1}(\xi ))}$, where $\gamma (x) = \int _c^x\! {\sqrt{r(y) \over p(y)}} dy$ and $\gamma ^{-1}$ is its inverse function. (Here $c \in (a,b)$ is a fixed point; if $\sqrt{r(y) \over p(y)}$ is integrable near a, then we may also take $c=a$.) Assume that

$$\begin{aligned} \begin{array}{l} q \equiv 0,\\ \gamma (b) = \int _c^b\! {\sqrt{r(y) \over p(y)}} dy = \infty ,\,\\ \hbox { and there exists }\eta \in \mathrm {C}^1(\gamma (a),\infty )\hbox { such that }\eta \ge 0,\\ \hbox { the functions }\varvec{\phi }_\eta := {S' \over S} - \eta , \,\varvec{\psi }_\eta := {1 \over 2} \eta ' - {1 \over 4} \eta ^2 + {S' \over 2S} \,\cdot \,\eta \\ \hbox { are both decreasing on }(\gamma (a),\infty ) \hbox { and }\varvec{\phi }_\eta \hbox { satisfies }\lim _{\xi \rightarrow \infty } \varvec{\phi }_\eta (\xi ) = 0. \end{array} \end{aligned}$$

(A.6)

Then the following assertions are equivalent:

• $\lim _{x \uparrow b} p(x)r(x) = \infty$;

• $\lim _{x \uparrow b} v_\lambda (x) = 0$ for all $\lambda > 0$.

Proof

See [52, Lemma 2.3 and Proposition 3.6]. $\square$

Theorem A.5

[Product formula for $v_\lambda$] Assume that (A.6) holds. Then there exists a family of measures $\{\pi _{x_1,x_2}\}_{x_1,x_2 \in [a,b)} \subset {\mathcal {P}}[a,b)$ such that we have

$$\begin{aligned} v_\lambda (x_1) \, v_\lambda (x_2) = \int _{[a,b)} v_\lambda (x_3)\, \pi _{x_1,x_2}(dx_3) \qquad \text {for all }\, x_1, x_2 \in [a,b), \; \lambda \in {\mathbb {C}}. \end{aligned}$$

If $p \equiv r$ and $a = \gamma (a) = 0$, then $\mathrm {supp}(\pi _{x_1,x_2}) \subset [|x_1-x_2|,x_1+x_2]$.

Proof

See [52, Section 4 and Subsection 5.3]. $\square$

Proposition A.6

Assume that (A.6) holds. Let ${\mathcal {F}}_\ell$ be the $\ell$-Fourier transform of measures defined by

$$\begin{aligned} ({\mathcal {F}}_{\ell \,} \mu )(\lambda ) := \int _{[a,b)} v_\lambda (x) \, \mu (dx) \qquad (\mu \in {\mathcal {P}}[a,b),\, \lambda \ge 0). \end{aligned}$$

(A.7)

Then:

(i):

${\mathcal {F}}_{\ell \,} \mu$ is continuous on ${\mathbb {R}}_0^+$. Moreover, if the family $\{\mu _j\} \subset {\mathcal {M}}_{\mathbb {C}}[a,b)$ is tight and uniformly bounded, then $\{{\mathcal {F}}_{\ell \,} \mu _j\}$ is equicontinuous on ${\mathbb {R}}_0^+$.

(ii):

Let $\mu _1, \mu _2 \in {\mathcal {M}}_{\mathbb {C}}[a,b)$. If ${\mathcal {F}}_{\ell \,} \mu _1 \equiv {\mathcal {F}}_{\ell \,} \mu _2$, then $\mu _1 = \mu _2$.

(iii):

Let $\{\mu _n\} \subset {\mathcal {M}}_+[a,b)$, $\mu \in {\mathcal {M}}_+[a,b)$, and suppose that $\mu _n \overset{w}{\longrightarrow }\mu$. Then

$$\begin{aligned} {\mathcal {F}}_{\ell \,} \mu _n \xrightarrow [\,n \rightarrow \infty \,]{} {\mathcal {F}}_{\ell \,} \mu \qquad \text {uniformly on compact sets.} \end{aligned}$$

Proof

See [52, Proposition 5.2]. $\square$

The $\ell$-convolution and the $\ell$-translation operator are respectively defined by

$$\begin{aligned} (\mu {\mathop {\diamond }\limits _{\ell }} \nu )(d\xi )&:= \int _{[a,b)} \int _{[a,b)} \pi _{x,y}(d\xi ) \, \mu (dx) \, \nu (dy), \quad \mu , \nu \in {\mathcal {M}}_{\mathbb {C}}[a,b), \\ ({\mathcal {T}}_\ell ^\mu f)(x)&:= \int _{[a,b)} f \, d(\delta _x {\mathop {\diamond }\limits _{\ell }} \mu ), \quad \mu \in {\mathcal {M}}_{\mathbb {C}}[a,b), \; x \in [a,b), \; f \in L^p(r), \end{aligned}$$

where $L^p(r) \equiv L^p\bigl ((a,b),r(z)dz\bigr )$  ($1 \le p \le \infty$).

Proposition A.7

Assume that (A.6) holds.

(a):

$\mu = \mu _1 {\mathop {\diamond }\limits _{\ell }} \mu _2$ if and only if $({\mathcal {F}}_{\ell \,}\mu )(\lambda ) = ({\mathcal {F}}_{\ell \,}\mu _1)(\lambda ) \,\cdot \,({\mathcal {F}}_{\ell \,}\mu _2)(\lambda )$ for all $\lambda \ge 0$    $(\mu , \mu _1, \mu _2 \in {\mathcal {M}}_{\mathbb {C}}[a,b))$.

(b):

The ${\mathop {\diamond }\limits _{\ell }}$ convolution is weakly continuous: if $\mu _n \overset{w}{\longrightarrow }\mu$ and $\nu _n \overset{w}{\longrightarrow }\nu$, then $\mu _n {\mathop {\diamond }\limits _{\ell }} \nu _n \overset{w}{\longrightarrow }\mu {\mathop {\diamond }\limits _{\ell }} \nu$.

(c):

If $f \in \mathrm {C}_\mathrm {c}^2[a,b)$ with $f' \in \mathrm {C}_\mathrm {c}(a,b)$, then

$$\begin{aligned}&\int _{[a,b)} f \, d(\delta _x {\mathop {\diamond }\limits _{\ell }} \mu ) \\&\quad = \int _{{\mathbb {R}}_0^+} ({\mathcal {F}}_{\ell \,} f)(\lambda ) \, ({\mathcal {F}}_{\ell \,} \mu )(\lambda ) \, v_\lambda (x) \, \varvec{\rho }_\ell (d\lambda ) \qquad \text {for all } \mu {\in } {\mathcal {M}}_{\mathbb {C}}[a,b),\, x {\in } (a,b). \end{aligned}$$

(d):

Suppose that $\lim _{x \uparrow b} p(x) r(x) = \infty$, and let $\mu \in {\mathcal {M}}_{\mathbb {C}}[a,b)$. Then $\delta _x {\mathop {\diamond }\limits _{\ell }} \mu \overset{v}{\longrightarrow }\varvec{0}$ as $x \uparrow b$, where $\varvec{0}$ is the zero measure.

(e):

Let $1 \le p \le \infty$ and $\mu \in {\mathcal {M}}_+[a,b)$. The $\ell$-translation ${\mathcal {T}}_\ell ^{\,\,\mu }$ is a bounded operator on $L^p(r)$ such that

$$\begin{aligned} \Vert {\mathcal {T}}_\ell ^{\,\,\mu } f\Vert _{L^p(r)} \le \Vert \mu \Vert \,\cdot \,\Vert f\Vert _{L^p(r)} \qquad \text {for all } f \in L^p(r). \end{aligned}$$

(f):

Let $1 \le p_1, p_2 \le \infty$ with ${1 \over p_1} + {1 \over p_2} \ge 1$, and let $f \in L^{p_1\!}(r)$, $g \in L^{p_2\!}(r)$. Then the $\ell$-convolution

$$\begin{aligned} (f {\mathop {\diamond }\limits _{\ell }} g)(x) := \int _a^b ({\mathcal {T}}_\ell ^{\,\,y} f)(x)\, g(y)\, r(y) dy \end{aligned}$$

(where ${\mathcal {T}}_\ell ^{\,\,y} \equiv {\mathcal {T}}_\ell ^{\,\,\delta _y}$) is well-defined and satisfies

$$\begin{aligned} \Vert f {\mathop {\diamond }\limits _{\ell }} g \Vert _{L^s(r)} \le \Vert f \Vert _{L^{p_1\!}(r)} \Vert g \Vert _{L^{p_2\!}(r)}, \qquad \text {where }\, s = {1 \over 1/p_1 + 1/p_2 - 1}. \end{aligned}$$

Proof

See [51, Corollary 6.3 and Proposition 6.9], [52, Proposition 5.4 and Lemma 5.5]. $\square$

Proposition A.8

If (A.6) holds with $p \equiv r$ and $a = \gamma (a) = 0$, then $\bigl ({\mathbb {R}}_0^+,{\mathop {\diamond }\limits _{\ell }}\bigr )$ is a commutative hypergroup (in the sense of [6, 31]) with identity element $\delta _0$ and trivial involution, i.e. the following axioms hold:

• $\bigl ({\mathbb {R}}_0^+,{\mathop {\diamond }\limits _{\ell }}\bigr )$, equipped with the total variation norm, is a commutative Banach algebra over ${\mathbb {C}}$ whose identity element is the measure $\delta _0$;

• If $\mu , \nu \in {\mathcal {P}}({\mathbb {R}}_0^+)$, then $\mu {\mathop {\diamond }\limits _{\ell }} \nu \in {\mathcal {P}}({\mathbb {R}}_0^+)$;

• $(\mu ,\nu ) \mapsto \mu {\mathop {\diamond }\limits _{\ell }} \nu$ is continuous in the weak topology of measures;

• $(x_1,x_2) \mapsto \mathrm {supp}(\delta _{x_1} {\mathop {\diamond }\limits _{\ell }} \delta _{x_2})$ is continuous from ${\mathbb {R}}_0^+ \times {\mathbb {R}}_0^+$ into the space of compact subsets of ${\mathbb {R}}_0^+$, and we have $0 \in \mathrm {supp}(\delta _{x_1} {\mathop {\diamond }\limits _{\ell }} \delta _{x_2})$ if and only if $x_1 = x_2$.

Proof

See [52, Subsection 5.3]. $\square$

The measures $\{\mu _t\}_{t \ge 0} \subset {\mathcal {P}}[a,b)$ are said to be an $\ell$-convolution semigroup if

$$\begin{aligned} \mu _s {\mathop {\diamond }\limits _{\ell }} \mu _t = \mu _{s+t} \text { for all } s, t \ge 0, \qquad \mu _0 = \delta _a \qquad \text { and } \;\; \mu _t \overset{w}{\longrightarrow }\delta _a \text { as } t \downarrow 0. \end{aligned}$$

Proposition A.9

Let $\ell$ be a Sturm–Liouville expression with $p \equiv r$, $q \equiv 0$ and $a = \gamma (a) = 0$. Suppose that (A.6) holds and that $\lim _{x \uparrow b} p(x) = \infty$.

(a):

For $t > 0$, let $\alpha _t^\ell$ be the measure defined by $\alpha _t^\ell (dx) := p_\ell (t,0,x) r(x) dx$, where $p_\ell (t,x_1,x_2)$ is the kernel (A.5). Set $\alpha _0^\ell = \delta _0$. Then $\{\alpha _t^\ell \}_{t \ge 0}$ is an $\ell$-convolution semigroup such that

$$\begin{aligned} \lim _{t \downarrow 0} {1 \over t} \alpha _t^\ell [\varepsilon , \infty ) = 0 \qquad \text {for every }\, \varepsilon > 0. \end{aligned}$$

(b):

Let $\psi (\lambda )$ be a function which can be written as

$$\begin{aligned} \psi (\lambda ) = c\lambda + \int _{{\mathbb {R}}^+} (1-v_\lambda (x)) \, \tau (dx) \qquad (\lambda \ge 0) \end{aligned}$$

for some $c \ge 0$ and some positive measure $\tau$ on ${\mathbb {R}}^+$ such that $\tau$ is finite on the complement of any neighbourhood of 0 and satisfies $\int _{{\mathbb {R}}^+} (1-v_\lambda (x)) \, \tau (dx) < \infty$ for $\lambda \ge 0$. Then there exists an $\ell$-convolution semigroup $\{\mu _t\}_{t \ge 0}$ such that $({\mathcal {F}}_{\ell \,} \mu _t)(\lambda ) = e^{-t\psi (\lambda )}$. Moreover, there exists a constant $C > 0$ independent of $\lambda$ such that

$$\begin{aligned} \psi (\lambda ) \le C(1+\lambda ) \qquad \text {for all } \lambda \ge 0. \end{aligned}$$

(A.8)

Proof

Part (a) and the first statement in part (b) follow from [51, Theorem 6.5 and Propositions 6.13–6.14].

To prove the estimate (A.8), start by picking $\lambda _1 > 0$. We know that $\lim _{x \uparrow b} v_{\lambda _1}(x) = 0$ (Lemma A.4(b)), hence there exists $\beta \in (a,b)$ such that $|v_{\lambda _1}(x)| \le {1 \over 2}$ for all $\beta \le x < b$. Combining this with Lemma A.4(a), we deduce that for all $\lambda \ge 0$ we have

$$\begin{aligned} \begin{aligned} n \int _{[\beta ,b)\!} \bigl (1-v_\lambda (x)\bigr ) \mu _{1/n}(dx)&\le 2n \int _{[\beta ,b)\!} \mu _{1/n}(dx) \\&\le 4n \int _{[\beta ,b)\!}\bigl ( 1-v_{\lambda _1}(x) \bigr ) \mu _{1/n}(dx) \\&\le 4n \bigl (1-({\mathcal {F}}_\ell \mu _{1/n})(\lambda _1)\bigr ) \le 4\psi (\lambda _1). \end{aligned} \end{aligned}$$

(A.9)

Next, choose $\lambda _2 > 0$ such that $1 - v_\lambda (x) < {1 \over 2}$ for all $0 \le \lambda \le \lambda _2$ and all $a < x \le \beta$. (This is possible because of the boundedness of the family of derivatives $\{\partial _\lambda v_{(\cdot )}(x)\}_{x \in (a, \beta ]}$, cf. [52, pp. 5–6].) Defining $\eta _1(x) := \int _a^x {1 \over p(y)} \int _a^y r(\xi ) d\xi \, dy$, we obtain

$$\begin{aligned} 1 - v_{\lambda _2}(x) = \lambda _2 \int _a^x {1 \over p(y)} \int _a^y v_{\lambda _2}(\xi ) r(\xi ) d\xi \, dy \ge {\lambda _2 \over 2} \eta _1(x) \qquad \text {for all } a \le x \le \beta . \end{aligned}$$

On the other hand, by Lemma A.4(a) we have $1 - v_\lambda (x) \le \lambda \int _a^x {1 \over p(y)} \int _a^y |v_\lambda (\xi )| r(\xi ) d\xi \, dy \le \lambda \eta _1(x)$ for all $x \in [a,b)$ and $\lambda \ge 0$. Consequently,

$$\begin{aligned} \begin{aligned} n \int _{[a,\beta )\!} \bigl (1-v_\lambda (x)\bigr ) \mu _{1/n}(dx)&\le \lambda n \int _{[a,\beta )\!} \eta _1(x)\, \mu _{1/n}(dx) \\&\le {2\lambda n \over \lambda _2} \int _{[a,\beta )\!} \bigl (1 - v_{\lambda _2}(x)\bigr ) \mu _{1/n}(dx) \\&\le {2\lambda n \over \lambda _2} \bigl (1-({\mathcal {F}}_\ell \mu _{1/n})(\lambda _2)\bigr ) \le {2\lambda \over \lambda _2} \psi (\lambda _2). \end{aligned} \end{aligned}$$

(A.10)

Combining (A.9) and (A.10) one sees that for all $n \in {\mathbb {N}}$ and $\lambda \ge 0$ we have $n(1-e^{-\psi (\lambda )/n}) \le C (1+\lambda )$, where $C = \max \bigl \{4\psi _\mu (\lambda _1), {2 \over \lambda _2} \psi _\mu (\lambda _2)\bigr \}$. The conclusion follows by taking the limit as $n \rightarrow \infty$. $\square$