2.1 Kato class \(K_{n}^{\infty }(D)\)
Definition 2.1
(See [28])
A function ψ in \(\mathcal{B}(D)\) is said to be in the Kato class \(K_{n}^{\infty }(D)\) if
$$ \lim_{r\rightarrow 0} \biggl( \sup_{x\in D} \int _{D\cap B(x,r)} \Gamma (x,y) \bigl\vert \psi (y) \bigr\vert \,dy \biggr) =0, $$
(2.1)
and
$$ \lim_{M\rightarrow \infty } \biggl( \sup_{x\in D} \int _{D \cap ( \vert y \vert \geq M)}\Gamma (x,y) \bigl\vert \psi (y) \bigr\vert \,dy \biggr) =0, $$
(2.2)
where \(\Gamma (x,y)\) is given by (1.13).
Example 2.2
Let \(p>\frac{n}{2}\). Then, we have
$$ L^{p} ( D ) \cap L^{1} ( D ) \subset K_{n}^{ \infty }(D). $$
Indeed, for \(\psi \in L^{p} ( D ) \), by using the Hölder inequality, it is clear that (2.1) holds. Now, assume further that \(\psi \in L^{1} ( D ) \), then
$$\begin{aligned}& \int _{D\cap ( \vert y \vert \geq M)}\Gamma (x,y) \bigl\vert \psi (y) \bigr\vert \,dy \\& \quad \leq \int _{D\cap B(x,r)}\Gamma (x,y) \bigl\vert \psi (y) \bigr\vert \,dy +c_{n}r^{2-n} \int _{D\cap ( \vert y \vert \geq M)\cap ( \vert x-y \vert \geq r)} \bigl\vert \psi (y) \bigr\vert \,dy. \end{aligned}$$
Hence, ψ satisfies (2.2).
The next Lemma is due to Mâagli and Zribi, see [20, Remark 2 and Proposition 1].
Lemma 2.3
-
(i)
Let ψ be a radial function in D, then
$$ \psi \in K_{n}^{\infty }(D)\quad \textit{if and only if}\quad \int _{0}^{\infty }r \bigl\vert \psi (r) \bigr\vert \,dr< \infty . $$
-
(ii)
Let \(\psi \in \mathcal{B}(D)\) satisfying (2.1). Then, for each \(M>0\), we have
$$ \int _{D\cap ( \vert y \vert \leq M)} \bigl\vert \psi (y) \bigr\vert \,dy < \infty . $$
Remark 2.4
For all \(x,y,z\in \mathbb{R}^{n}\), we have
$$ \frac{\Gamma (x,y)\Gamma (y,z)}{\Gamma (x,z)}\leq 2^{n-3}c_{n} \bigl( \Gamma (x,y)+ \Gamma (y,z) \bigr) , $$
(2.3)
where \(c_{n}=\frac{\Gamma (\frac{n}{2}-1)}{4\pi ^{\frac{n}{2}}}\).
Proposition 2.5
Let \(\psi \in K_{n}^{\infty }(D)\), \(x_{0}\in \mathbb{R}^{n}\) and \(h\in \mathcal{S}^{+} ( D ) \). Then, we have
$$ \lim_{r\rightarrow 0} \biggl( \sup_{x\in D} \frac{1}{h(x)} \int _{D\cap B(x_{0},r)}\Gamma (x,y)h(y) \bigl\vert \psi (y) \bigr\vert \,dy \biggr) =0, $$
(2.4)
and
$$ \lim_{M\rightarrow \infty } \biggl( \sup_{x\in D} \frac{1}{h(x)}\int _{D\cap ( \vert y \vert \geq M)}\Gamma (x,y)h(y) \bigl\vert \psi (y) \bigr\vert \,dy \biggr) =0. $$
(2.5)
Proof
Since \(h\in \mathcal{S}^{+} ( D ) \), then by [23, Theorem 2.1, p. 164], there exists a sequence \((h_{k})_{k}\subset \mathcal{B}^{+}(D)\) such that
$$ h(y)=\sup _{k} \int _{D}\Gamma (y,z)h_{k}(z)\,dy , \quad \text{for }y\in D. $$
Therefore, we need to prove (2.4) and (2.5) only for \(h(y)=\Gamma (y,z)\) uniformly in \(z\in D\).
Let \(r>0\). By using Remark 2.4, there exists a constant \(c>0\), such that for all \(x,y,z\in D\),
$$ \frac{1}{h(x)} \int _{D\cap B(x_{0},r)}\Gamma (x,y)h(y) \bigl\vert \psi (y) \bigr\vert \,dy \leq 2c\sup_{\xi \in D} \int _{D\cap B(x_{0},r)} \Gamma (\xi ,y) \bigl\vert \psi (y) \bigr\vert \,dy . $$
(2.6)
For \(\varepsilon >0\), by Definition 2.1, there exists \(s>0\) and \(M>0\) such that
$$ \int _{D\cap B(x_{0},r)}\Gamma (\xi ,y) \bigl\vert \psi (y) \bigr\vert \,dy \leq \varepsilon +\frac{c_{n}}{s^{n-2}} \int _{D\cap B(x_{0},r)\cap ( \vert y \vert \leq M)} \bigl\vert \psi (y) \bigr\vert \,dy . $$
Using this fact, (2.6) and Lemma 2.3(ii), we obtain (2.4) by letting \(r\rightarrow 0\).
Finally, note that assertion (2.5) follows by using similar arguments as above. □
Proposition 2.6
Let \(\psi \in K_{n}^{\infty }(D)\) and \(\varrho _{0}(x):= \frac{1+ \vert x \vert ^{n-2}}{ \vert x \vert ^{n-2}}\). Then, the function
$$ v(x):=\frac{1}{\varrho _{0}(x)} \int _{D}\Gamma (x,y) \varrho _{0}(y)\psi (y) \,dy $$
is continuous on \(\mathbb{R}^{n}\) with \(\lim _{ \vert x \vert \rightarrow \infty }v(x)=0\). That is, \(v(x)\in C_{0}(\mathbb{R}^{n})\).
Proof
Let \(\psi \in K_{n}^{\infty }(D)\) and \(x_{0}\in \mathbb{R}^{n}\). Since \(\varrho _{0}\in \mathcal{S}^{+} ( D ) \), then for \(\varepsilon >0\), by Proposition 2.5, there exists \(M>r>0\), such that the following holds:
(i) If \(x_{0}\neq 0\), then for \(x\in B(x_{0},\frac{r}{2})\cap D\), we have
$$ \bigl\vert v(x)-v(x_{0}) \bigr\vert \leq \frac{\varepsilon }{2}+ \int _{D_{0}\cap ( \vert y \vert \leq M)} \biggl\vert \frac{1}{\varrho _{0}(x)}\Gamma (x,y)- \frac{1}{\varrho _{0}(x_{0})}\Gamma (x_{0},y) \biggr\vert \varrho _{0}(y) \bigl\vert \psi (y) \bigr\vert \,dy , $$
where \(D_{0}=D\cap B^{c}(0,r)\cap B^{c}(x_{0},r)\).
Since \((x,y)\mapsto \frac{1}{\varrho _{0}(x)}\Gamma (x,y)\) is continuous on \(( B(x_{0},\frac{r}{2})\cap D ) \times (D_{0}\cap ( \vert y \vert \leq M))\), we obtain by Lemma 2.3 (ii) and Lebesgue’s dominated convergence theorem,
$$ \int _{D_{0}\cap ( \vert y \vert \leq M)} \biggl\vert \frac{1}{\varrho _{0}(x)}\Gamma (x,y)- \frac{1}{\varrho _{0}(x_{0})}\Gamma (x_{0},y) \biggr\vert \varrho _{0}(y) \bigl\vert \psi (y) \bigr\vert \,dy \rightarrow 0\quad \text{as }x\rightarrow x_{0}. $$
Hence, there exists \(\delta >0\) with \(\delta <\frac{r}{2}\) such that if \(x\in B(x_{0},\delta )\cap D\),
$$ \int _{D_{0}\cap ( \vert y \vert \leq M)} \biggl\vert \frac{1}{\varrho _{0}(x)}\Gamma (x,y)- \frac{1}{\varrho _{0}(x_{0})}\Gamma (x_{0},y) \biggr\vert \varrho _{0}(y) \bigl\vert \psi (y) \bigr\vert \,dy\leq \frac{\varepsilon }{2}. $$
Hence, for \(x\in B(x_{0},\delta )\cap D\), we have
$$ \bigl\vert v(x)-v(x_{0}) \bigr\vert \leq \varepsilon . $$
That is,
$$ \lim_{x\rightarrow x_{0}}v(x)=v(x_{0}). $$
(ii) If \(x_{0}=0\) and \(x\in B(0,\frac{r}{2})\cap D\), then we have
$$ \bigl\vert v(x) \bigr\vert \leq \frac{\varepsilon }{2}+ \int _{D\cap B^{c}(0,r)\cap ( \vert y \vert \leq M)} \frac{1}{\varrho _{0}(x)}\Gamma (x,y)\varrho _{0}(y) \bigl\vert \psi (y) \bigr\vert \,dy . $$
Now, since \(\lim _{ \vert x \vert \rightarrow 0}\frac{1}{\varrho _{0}(x)}\Gamma (x,y)\varrho _{0}(y)=0\), for all \(y\in D\cap B^{c}(0,r)\cap ( \vert y \vert \leq M)\), we deduce by similar arguments as above that
$$ \lim _{ \vert x \vert \rightarrow 0} v(x)=0=v(x_{0}). $$
(iii) It remains to prove that \(\lim _{ \vert x \vert \rightarrow \infty }v(x)=0\).
To this end, let \(x\in D\) such that \(\vert x \vert \geq M+1\). Using Proposition 2.5 and Lemma 2.3 (ii), we deduce that
$$\begin{aligned} \bigl\vert v(x) \bigr\vert \leq &\frac{\varepsilon }{2}+ \frac{1+M^{n-2}}{r^{n-2}} \int _{D\cap B^{c}(0,r)\cap ( \vert y \vert \leq M)}\Gamma (x,y) \bigl\vert \psi (y) \bigr\vert \,dy \\ \leq &\frac{\varepsilon }{2}+ \frac{c}{( \vert x \vert -M)^{n-2}}, \end{aligned}$$
where c is some positive constant.
This implies that \(\lim _{ \vert x \vert \rightarrow \infty }v(x)=0\). □
2.2 Karamata regular variation theory
Let us recall some basic properties of Karamata regular variation theory (see [2, 14, 21, 24, 25]).
The following result concerns operations that preserve slow variation.
Proposition 2.7
If \(L_{1}(t)\), \(L_{2}(t)\) are slowly varying at infinity (resp., at zero), then the same holds for \(L_{1}(t)+L_{2}(t)\), \(L_{1}(t)L_{2}(t)\), and \((L_{1}(t))^{\nu }\) for any \(\nu \in \mathbb{R}\).
Proposition 2.8
-
(i)
If \(L(t)\in \mathcal{NSV}_{\infty }\), then for any \(\varepsilon >0\)
$$ \lim _{t\rightarrow \infty }t^{\varepsilon }L(t)=\infty ,\qquad \lim _{t\rightarrow \infty }t^{-\varepsilon }L(t)=0. $$
-
(ii)
If \(L(t)\in \mathcal{NSV}_{0}\), then for any \(\varepsilon >0 \)
$$ \lim _{t\rightarrow 0^{+}}t^{\varepsilon }L(t)=0 \quad \textit{and}\quad \lim_{t\rightarrow 0^{+}} t^{-\varepsilon }L(t)=\infty . $$
The following result, termed Karamata’s integration theorem, will be used later.
Proposition 2.9
Let \(L(t)\in \mathcal{NSV}_{\infty }\). Then,
-
(i)
if \(\nu >-1\),
$$ \int _{A}^{t}s^{\nu }L(s)\,ds\sim \frac{1}{\nu +1}t^{\nu +1}L(t),\quad t\rightarrow \infty ; $$
-
(ii)
if \(\nu <-1\),
$$ \int _{t}^{\infty }s^{\nu }L(s)\,ds\sim - \frac{1}{\nu +1}t^{\nu +1}L(t),\quad t\rightarrow \infty ; $$
-
(iii)
if \(\nu =-1\),
$$ l(t)= \int _{A}^{t}s^{-1}L(s)\,ds\in \mathcal{NSV}_{\infty }\quad \textit{and}\quad \lim _{t\rightarrow \infty } \frac{L(t)}{l(t)}=0; $$
and if \(\int _{A}^{\infty }s^{-1}L(s)\,ds<\infty \),
$$ m(t)= \int _{t}^{\infty }s^{-1}L(s)\,ds\in \mathcal{NSV}_{\infty } \quad \textit{and}\quad \lim _{t\rightarrow \infty } \frac{L(t)}{m(t)}=0. $$
The following is an analog of Proposition 2.9 for L defined at zero instead of ∞.
Proposition 2.10
Let \(L(t)\in \mathcal{NSV}_{0}\). Then,
-
(i)
if \(\nu >-1\),
$$ \int _{0}^{t}s^{\nu }L(s)\,ds\sim \frac{1}{\nu +1}t^{\nu +1}L(t),\quad t\rightarrow 0^{+}; $$
-
(ii)
if \(\nu <-1\),
$$ \int _{t}^{a}s^{\nu }L(s)\,ds\sim - \frac{1}{\nu +1}t^{\nu +1}L(t),\quad t\rightarrow 0^{+}; $$
-
(iii)
if \(\nu =-1\),
$$ l_{0}(t)= \int _{t}^{a}s^{-1}L(s)\,ds\in \mathcal{NSV}_{0}\quad \textit{and} \quad \lim _{t\rightarrow 0^{+}} \frac{L(t)}{l_{0}(t)}=0; $$
and if \(\int _{0}^{a}s^{-1}L(s)\,ds<\infty \),
$$ m_{0}(t)= \int _{0}^{t}s^{-1}L(s)\,ds\in \mathcal{NSV}_{0}\quad \textit{and}\quad \lim_{t\rightarrow 0^{+}}\frac{L(t)}{m_{0}(t)}=0. $$
The following result, will play a central role in establishing our main result in Sect. 3.
Proposition 2.11
For \(\alpha \leq n\) and \(\beta \geq 2\), set
$$ b(x)= \vert x \vert ^{-\alpha }L_{0}\bigl( \vert x \vert \wedge 1\bigr) \bigl( \vert x \vert +1\bigr)^{\alpha -\beta }L_{ \infty } \bigl( \vert x \vert \vee 1\bigr), \quad x\in D, $$
where \(L_{0}\in \mathcal{NSV}_{0}\) defined on \((0,a]\), for some \(a>1\) and \(L_{\infty }\in \mathcal{NSV}_{\infty }\), defined on \([1,\infty )\) such that
$$ \int _{0}^{a}s^{n-\alpha -1}L_{0}(s)\,ds< \infty \quad \textit{and}\quad \int _{1}^{ \infty }s^{1-\beta }L_{\infty }(s)\,ds< \infty . $$
(2.7)
Then,
$$ \mathcal{N}b(x)\approx \vert x \vert ^{\min (0,2-\alpha )}\widetilde{L}_{0}\bigl( \vert x \vert \wedge 1\bigr) \bigl( \vert x \vert +1\bigr)^{\max (2-n,2-\beta )-\min (0,2-\alpha )} \widetilde{L}_{\infty }\bigl( \vert x \vert \vee 1\bigr),\quad \textit{on }D, $$
where for \(t\in (0,a)\),
$$ \widetilde{L}_{0}(t):=\textstyle\begin{cases} 1, & \textit{if }\alpha < 2, \\ \int _{t}^{a}\frac{L_{0}(s)}{s}\,ds, & \textit{if }\alpha =2, \\ L_{0}(t), & \textit{if }2< \alpha < n, \\ \int _{0}^{t}\frac{L_{0} ( s ) }{s}\,ds, & \textit{if }\alpha =n,\end{cases} $$
and for \(t\geq 1\),
$$ \widetilde{L}_{\infty }(t):=\textstyle\begin{cases} 1, & \textit{if }\beta >n, \\ \int _{1}^{t+1}\frac{L_{\infty }(s)}{s}\,ds, & \textit{if }\beta =n, \\ L_{\infty }(t+1), & \textit{if }2< \beta < n, \\ \int _{t+1}^{\infty }\frac{L_{\infty } ( s ) }{s}\,ds, & \textit{if }\beta =2.\end{cases} $$
Proof
Since b is a nonnegative radial measurable function on D, it follows from [23, Proposition 1.7], that
$$ \mathcal{N}b(x):= \int _{D}\Gamma (x,y)b(y)\,dy =c \int _{0}^{\infty } \frac{r^{n-1}}{ ( \vert x \vert \vee r ) ^{n-2}}b(r)\,dr=:cJ\bigl( \vert x \vert \bigr), $$
where the function J is defined on \([0,\infty )\) by
$$ J(t)= \int _{0}^{\infty } \frac{r^{n-\alpha -1}}{ ( t\vee r ) ^{n-2}}(r+1)^{\alpha-\beta }L_{0}(r \wedge 1)L_{\infty }(r\vee 1)\,dr. $$
We need to estimate \(J(t)\). Note that, under condition (2.7), \(J(t)<\infty \).
Let \(a>1\), then we have
$$\begin{aligned} J(t) \approx & \int _{0}^{a} \frac{r^{n-\alpha -1}}{ ( t\vee r ) ^{n-2}}L_{0}(r)\,dr+ \int _{a}^{\infty } \frac{r^{n-\beta -1}}{ ( t\vee r ) ^{n-2}}L_{\infty }(r)\,dr \\ :=&J_{1}(t)+J_{2}(t). \end{aligned}$$
We discuss the following cases:
Case 1. \(0< t\leq 1\). Clearly from (2.7), we have
$$ J_{2}(t)= \int _{a}^{\infty }r^{1-\beta }L_{\infty }(r)\,dr \approx 1. $$
On the other hand, by writing
$$ J_{1}(t)=t^{2-n} \int _{0}^{t}r^{n-\alpha -1}L_{0}(r)\,dr+ \int _{t}^{a}r^{1- \alpha }L_{0}(r)\,dr, $$
we deduce that
$$ J(t)\approx t^{2-n} \int _{0}^{t}r^{n-\alpha -1}L_{0}(r)\,dr+ \biggl(1+ \int _{t}^{a}r^{1- \alpha }L_{0}(r)\,dr \biggr). $$
Therefore, by (2.7) and Propositions 2.8 and 2.10, we obtain
$$ J(t)\approx \phi _{0}(t):=\textstyle\begin{cases} 1, & \text{if } \alpha < 2, \\ \int _{t}^{a}\frac{L_{0}(r)}{r}\,dr, & \text{if } \alpha =2, \\ t^{2-\alpha }L_{0}(t), & \text{if } 2< \alpha < n, \\ t^{2-\alpha }\int _{0}^{t}\frac{L_{0} ( r ) }{r}\,dr, & \text{if } \alpha =n. \end{cases} $$
That is,
$$ J(t)\approx t^{\min (0,2-\alpha )}\widetilde{L}_{0}(t),\quad \text{for }0< t \leq 1. $$
(2.8)
Case 2. \(t\geq a+1\). From (2.7), we derive that
$$ J_{1}(t)\approx t^{2-n} \int _{0}^{a}r^{n-\alpha -1}L_{0}(r)\,dr \approx t^{2-n}. $$
On the other hand, since
$$ J_{2}(t)=t^{2-n} \int _{a}^{t}r^{n-\beta -1}L_{\infty }(r)\,dr+ \int _{t}^{ \infty }r^{1-\beta }L_{\infty }(r)\,dr, $$
we deduce that
$$ J(t)\approx t^{2-n}\biggl(1+ \int _{a}^{t}r^{n-\beta -1}L_{\infty }(r)\,dr \biggr)+ \int _{t}^{\infty }r^{1-\beta }L_{\infty }(r)\,dr. $$
Hence, by (2.7) and Propositions 2.8 and 2.9, we obtain
$$ J(t)\approx \textstyle\begin{cases} t^{2-n}, & \text{if }\beta >n, \\ t^{2-n}\int _{a}^{t}\frac{L_{\infty }(s)}{s}\,ds, & \text{if }\beta =n, \\ t^{2-\beta }L_{\infty }(t), & \text{if }2< \beta < n, \\ \int _{t}^{\infty }\frac{L_{\infty } ( s ) }{s}\,ds, & \text{if }\beta =2.\end{cases} $$
Therefore, by using Proposition 2.9 and [5, Lemma 2.3], we conclude that
$$ J(t)\approx \phi _{\infty }(t):=\textstyle\begin{cases} (t+1)^{2-n}, & \text{if }\beta >n, \\ (t+1)^{2-n}\int _{1}^{t+1}\frac{L_{\infty }(s)}{s}\,ds, & \text{if } \beta =n, \\ (t+1)^{2-\beta }L_{\infty }(t+1), & \text{if }2< \beta < n, \\ \int _{t+1}^{\infty }\frac{L_{\infty } ( s ) }{s}\,ds, & \text{if }\beta =2.\end{cases} $$
Hence,
$$ J(t)\approx (t+1)^{\max (2-n,2-\beta )}\widetilde{L}_{\infty }(t),\quad \text{for }t \geq a+1. $$
(2.9)
Finally, since \(J(t)\), \(\phi _{0}(t)\) and \(\phi _{\infty }(t)\) are positive continuous functions on \([1,a+1]\), we deduce that
$$ J(t)\approx \phi _{0}(t)\phi _{\infty }(t),\quad \text{on }[1,a+1]. $$
(2.10)
Hence, by combining (2.8), (2.9) and (2.10), we obtain
$$ J(t)\approx t^{\min (0,2-\alpha )}\widetilde{L}_{0}(t\wedge 1) (t+1)^{ \max (2-n,2-\beta )-\min (0,2-\alpha )}\widetilde{L}_{\infty }(t\vee 1), \quad \text{on }[0,\infty ). $$
This completes the proof. □
Proposition 2.12
Assume that p satisfies hypothesis (H), then
$$ \mathcal{N}\bigl(p\theta ^{\gamma }\bigr) (x)\approx \theta (x),\quad \textit{on }D, $$
where \(\gamma <1\) and θ is given in (1.10).
Proof
Using (1.7) and (1.10), we obtain
$$\begin{aligned}& p(x)\theta ^{\gamma }(x) \\& \quad \thickapprox \vert x \vert ^{- \alpha }\mathcal{L}_{0}\bigl( \vert x \vert \wedge 1\bigr) \bigl( \widetilde{\mathcal{L}}_{0}\bigl(\min \bigl( \vert x \vert ,1\bigr)\bigr) \bigr) ^{ \frac{\gamma }{1-\gamma }}\bigl( \vert x \vert +1 \bigr)^{\alpha -\beta }\mathcal{L}_{\infty }\bigl( \vert x \vert \vee 1 \bigr) \bigl( \widetilde{\mathcal{L}}_{ \infty }\bigl( \vert x \vert \vee 1\bigr) \bigr) ^{ \frac{\gamma }{1-\gamma }}, \end{aligned}$$
where \(\alpha :=\mu -\gamma \min (0,\frac{2-\mu }{1-\gamma })\) and \(\beta :=\lambda -\gamma \max (2-n,\frac{2-\lambda }{1-\gamma })\).
From the fact that \(\mu \leq n+(2-n)\gamma \) and \(\lambda \geq 2\), we derive that \(\alpha \leq n\) and \(\beta \geq 2\).
By using the basic properties of Karamata regular variation theory and Proposition 2.11 with \(L_{0}=\mathcal{L}_{0}( \vert x \vert \wedge 1) ( \widetilde{\mathcal{L}}_{0}(\min ( \vert x \vert ,1)) ) ^{\frac{\gamma }{1-\gamma }}\in \mathcal{NSV}_{0}\) and \(L_{\infty }=\mathcal{L}_{\infty }( \vert x \vert \vee 1) ( \widetilde{\mathcal{L}}_{\infty }( \vert x \vert \vee 1) ) ^{\frac{\gamma }{1-\gamma }}\in \mathcal{NSV}_{\infty }\), we deduce that
$$ \mathcal{N}\bigl(p\theta ^{\gamma }\bigr) (x)\approx \vert x \vert ^{ \min (0,2-\alpha )}\widetilde{L}_{0}\bigl( \vert x \vert \wedge 1 \bigr) \bigl( \vert x \vert +1\bigr)^{\max (2-n,2-\beta )-\min (0,2- \alpha )}\widetilde{L}_{\infty } \bigl( \vert x \vert \vee 1\bigr). $$
Since \(\min (0,2-\alpha )=\min (0,\frac{2-\mu }{1-\gamma }):=\xi \) and \(\max (2-n,2-\beta )=\max (2-n,\frac{2-\lambda }{1-\gamma }):=\zeta \), we deduce that
$$ \mathcal{N}\bigl(p\theta ^{\gamma }\bigr) (x)\thickapprox \vert x \vert ^{\xi }\widetilde{L}_{0}\bigl( \vert x \vert \wedge 1 \bigr) \bigl( \vert x \vert +1\bigr)^{\zeta -\xi }\widetilde{L}_{\infty } \bigl( \vert x \vert \vee 1\bigr)\thickapprox \theta (x). $$
This completes the proof. □