In this section, we will prove our main results.
Proofs of upper bounds
We rewrite Theorem 3.1 as follows before giving its proof.
Theorem 5.1
Assume
\(r\in [1,+\infty ), q\in [1,+\infty ]\)
and
\(s>\frac{1}{r}\), then, for
\(p\in [1,+\infty )\), the estimator
\(\widehat{f'_{{\sigma }^{2}}} ^{\mathrm{lin}}\)
in (7) with
\(2^{j_{0}}\sim n^{ \frac{1}{2s'+2(v+1)+1}}\)
satisfies
$$\begin{aligned} {\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)}} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}}, \end{aligned}$$
where
\(s'=s-(\frac{1}{r}-\frac{1}{p})_{+}\)
and
\(a_{+}=\max \{a,0\}\).
Proof
When \(r>p\), \(s'=s-(\frac{1}{r}-\frac{1}{p})_{+}=s\). Denote \(\varOmega = \operatorname{supp} (\widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}})\). Then
$$\begin{aligned} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} &=E \int \bigl\vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\vert ^{p}\,dx \\ &\leq E \biggl[ \int _{\varOmega }\bigl( \bigl\vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{ {\sigma }^{2}} \bigr\vert ^{p}\bigr) ^{\frac{r}{p}}\,dx \biggr]^{\frac{p}{r}} \biggl( \int _{\varOmega } 1\,dx\biggr)^{1-\frac{p}{r}}\lesssim E\bigl( \bigl\Vert \widehat{f'_{{\sigma }^{2}}} ^{\mathrm{lin}}-f'_{ {\sigma }^{2}} \bigr\Vert _{r}^{r}\bigr)^{\frac{p}{r}} \end{aligned}$$
due to the Hölder inequality. Furthermore, according to Jensen’s inequality and \(\frac{p}{r}<1\),
$$\begin{aligned} \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} \bigl(E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{r}^{r}\bigr)^{ \frac{p}{r}}. \end{aligned}$$
(18)
By \(s'=s-\frac{1}{r}+\frac{1}{p}\leq s\) and \(r\leq p\), one finds \(B_{r,q}^{s}\hookrightarrow {B_{p,q}^{s'}}\). Hence,
$$\begin{aligned} {\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)}} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim \sup_{f'_{{\sigma }^{2}}\in B_{p,q}^{s'}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}. \end{aligned}$$
(19)
Next, one only need estimate \(\sup_{f'_{{\sigma }^{2}}\in B_{p,q}^{s'}(M)} E\| \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}}\|_{p}^{p}\) by (18) and (19). Note that
$$\begin{aligned} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} & \leq E \bigl[ \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-P_{j_{0}}f'_{ {\sigma }^{2}} \bigr\Vert _{p}+ \bigl\Vert P_{j_{0}}f'_{{\sigma }^{2}}-f'_{ {\sigma }^{2}} \bigr\Vert _{p} \bigr]^{p} \\ &\lesssim E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-P_{j_{0}}f'_{ {\sigma }^{2}} \bigr\Vert _{p}^{p}+ \bigl\Vert P_{j_{0}}f'_{{\sigma }^{2}}-f'_{ {\sigma }^{2}} \bigr\Vert _{p}^{p}. \end{aligned}$$
(20)
Combining \(2^{j_{0}}\sim n^{\frac{1}{2s'+2(v+1)+1}}\), \(f'_{{\sigma } ^{2}}\in B_{p,q}^{s'}(M)\) with Lemma 2.1, one concludes
$$\begin{aligned} \bigl\Vert P_{j_{0}}f'_{{\sigma }^{2}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim 2^{-j _{0}s'p}\lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}}. \end{aligned}$$
(21)
On the other hand,
$$\begin{aligned} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-P_{j_{0}}f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim 2^{j_{0}(\frac{p}{2}-1)}\sum _{k\in \varOmega _{j_{0}}} E \vert \widehat{\alpha }_{j_{0},k}-{\alpha } _{j_{0},k} \vert ^{p}\lesssim n^{-\frac{p}{2}}2^{(v+1+\frac{1}{2})j_{0}p} \end{aligned}$$
(22)
thanks to Lemma 4.1 and Lemma 4.2. Then it follows
$$\begin{aligned} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-P_{j_{0}}f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}} \end{aligned}$$
from \(2^{j_{0}}\sim n^{\frac{1}{2s'+2(v+1)+1}}\). This with (20) and (21) leads to
$$\begin{aligned} \sup_{f'_{{\sigma }^{2}}\in B_{p,q}^{s'}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}}. \end{aligned}$$
(23)
Combining (23) with (18) and (19), one finds that
$$\begin{aligned} \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}}. \end{aligned}$$
The proof is done. □
Now, the upper bound of nonlinear wavelet estimator (Theorem 3.2) is restated below.
Theorem 5.2
Let
\(r\in [1,+\infty ), q\in [1,+\infty ]\)
and
\(s>\frac{1}{r}\). Then, for any
\(p\in [1,+\infty )\), the estimator
\(\widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}\)
in (9) satisfies
$$\begin{aligned} {\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)}} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim (\ln n)^{p} \bigl(n^{-1}\ln n\bigr)^{\alpha p}, \end{aligned}$$
where
\(\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}\).
Proof
When \(r>p\), similar to (18),
$$\begin{aligned} \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} \bigl(E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}-f'_{{\sigma }^{2}} \bigr\Vert _{r}^{r}\bigr)^{ \frac{p}{r}}. \end{aligned}$$
Hence, it suffices to establish the result for \(r\leq p\). According to (1), (2) and (9), \(E\| \widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}-f'_{{\sigma }^{2}}\|_{p}^{p}\lesssim A_{1}+A_{2}+A_{3}\), where
Next, one proves \(A_{1}+A_{2}+A_{3}\lesssim (\ln n)^{p}(n^{-1}\ln n)^{ \alpha p}\) for \(f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)\) and \(r\leq p\).
By the same arguments as (22),
$$\begin{aligned} A_{1}\lesssim 2^{\tau (\frac{p}{2}-1)}\sum_{k\in \varOmega _{ \tau }} E \vert \widehat{\alpha }_{\tau ,k}-{\alpha }_{\tau ,k} \vert ^{p}\lesssim n^{-\frac{p}{2}}2^{(v+1+\frac{1}{2})\tau p}\thicksim n^{-\frac{p}{2}} \lesssim \bigl(n^{-1}\ln n\bigr)^{\alpha p} \end{aligned}$$
thanks to \(\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}<\frac{1}{2}\).
Note that \(f'_{{\sigma }^{2}}\in B_{r,q}^{s}\hookrightarrow B_{p,q} ^{s-\frac{1}{r}+\frac{1}{p}}\) for \(r\leq p\). This with Lemma 2.1 and \(2^{j_{1}}\thicksim (\frac{n}{\ln n})^{ \frac{1}{2(v+1)+1}}\) shows
$$\begin{aligned} A_{3}= \bigl\Vert P_{j_{1}+1}f'_{{\sigma }^{2}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim 2^{-j_{1}(s-\frac{1}{r}+\frac{1}{p})p}\lesssim \biggl( \frac{\ln n}{n}\biggr)^{\frac{(s-\frac{1}{r}+\frac{1}{p})p}{2(v+1)+1}} \lesssim \bigl(n^{-1}\ln n\bigr)^{\alpha p}, \end{aligned}$$
because \(s>\frac{1}{r}\) and \(\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}\leq \frac{s- \frac{1}{r}+\frac{1}{p}}{2(v+1)+1}\).
To estimate \(A_{2}\), define
$$\begin{aligned} \widehat{B}_{j}=\bigl\{ k: \vert \widehat{\beta }_{j,k} \vert \geq \varUpsilon \lambda _{j} \bigr\} ;\qquad {B}_{j}= \biggl\{ k: \vert {\beta }_{j,k} \vert \geq \frac{1}{2} \varUpsilon \lambda _{j} \biggr\} \quad\text{and}\quad {C}_{j}=\bigl\{ k: \vert \beta _{j,k} \vert \geq 2\varUpsilon \lambda _{j}\bigr\} . \end{aligned}$$
Then \(E\|\sum_{j={\tau }}^{j_{1}} \sum_{k\in \varOmega _{j}}( \widetilde{\beta }_{j,k}-{\beta }_{j,k}) {\psi }_{j,k}\|_{p}^{p} \lesssim (\ln n)^{p-1}\sum_{i=1}^{4}Ee_{i}\) by Lemma 4.1, where
$$\begin{aligned} &e_{1}={\sum_{j={\tau }}^{j_{1}}} 2^{j(\frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B}_{j}\cap {B}_{j}^{c}\bigr\} ; \\ &e_{2}={\sum_{j={\tau }}^{j_{1}}} 2^{j(\frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p} I\{k\in \widehat{B}_{j}\cap {B}_{j}\}; \\ &e_{3}={\sum_{j={\tau }}^{j_{1}}} 2^{j(\frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} \vert {\beta }_{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}\bigr\} ; \\ &e_{4}={\sum_{j={\tau }}^{j_{1}}} 2^{j(\frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} \vert {\beta }_{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}^{c}\bigr\} . \end{aligned}$$
By the Hölder inequality and \(\{k\in \widehat{B}_{j}\cap {B} _{j}^{c}\}\subseteq \{|\widehat{\beta }_{j,k}- {\beta }_{j,k}|>\varUpsilon \lambda _{j}/2\}\),
$$\begin{aligned} E \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p}I\bigl\{ k\in \widehat{B}_{j} \cap {B}_{j}^{c} \bigr\} \leq \bigl(E \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{2p}\bigr)^{ \frac{1}{2}} P^{\frac{1}{2}} \bigl\{ \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert > \varUpsilon \lambda _{j}/2\bigr\} . \end{aligned}$$
This with Lemma 4.2 and Lemma 4.3 shows that
$$\begin{aligned} Ee_{1}\lesssim \sum_{j={\tau }}^{j_{1}}2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} n^{-\frac{p}{2}} 2^{j[(v+1)p-\frac{ \gamma }{2}]} \lesssim n^{-\frac{p}{2}}2^{\tau (vp+\frac{3}{2}p-\frac{ \gamma }{2})}\lesssim \biggl(\frac{\ln n}{n} \biggr)^{\alpha p}, \end{aligned}$$
where one uses \(\gamma >p(2v+3)\) and \(\alpha =\min \{ \frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s- \frac{1}{r})+2(v+1)+1}\}<\frac{1}{2}\).
From \(k\in \widehat{B}_{j}^{c}\cap {C}_{j}\), one finds \(| \widehat{\beta }_{j,k}- {\beta }_{j,k}|>\varUpsilon \lambda _{j}\) and \(|\beta _{j,k}|\leq |\widehat{\beta }_{j,k}- {\beta }_{j,k}|+| \widehat{\beta }_{j,k}|\leq 2|\widehat{\beta }_{j,k}- {\beta }_{j,k}|\). On the other hand, \(\{k\in \widehat{B}_{j}^{c}\cap {C}_{j}\}\subseteq \{|\widehat{\beta }_{j,k}- {\beta }_{j,k}|>\varUpsilon \lambda _{j}\} \subseteq \{|\widehat{\beta }_{j,k}- {\beta }_{j,k}|>\varUpsilon \lambda _{j}/2\}\). Therefore, it follows from the same arguments as \(Ee_{1}\) that
$$\begin{aligned} Ee_{3}\lesssim \sum_{j={\tau }}^{j_{1}} 2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p}I \bigl\{ k\in \widehat{B}_{j}^{c}\cap {C}_{j}\bigr\} \lesssim \biggl(\frac{\ln n}{n}\biggr)^{ \alpha p}. \end{aligned}$$
Next, one estimates \(Ee_{2}\) and \(Ee_{4}\). Define
$$\begin{aligned} \begin{aligned} &\omega =sr+\biggl(v+\frac{3}{2}\biggr)r-\biggl(v+ \frac{3}{2}\biggr)p,\qquad 2^{j_{0}^{*}}\sim \biggl(\frac{n}{ \ln n} \biggr)^{\frac{1}{2s+2(v+1)+1}},\\ & 2^{j_{1}^{*}}\sim \biggl(\frac{n}{\ln n} \biggr)^{\frac{1}{2(s- \frac{1}{r})+2(v+1)+1}}. \end{aligned} \end{aligned}$$
(24)
Then, by \(s>\frac{1}{r}\) and \(2^{j_{1}}\thicksim (\frac{n}{\ln n})^{ \frac{1}{2(v+1)+1}}\),
$$\begin{aligned} 0< \frac{1}{2s+2(v+1)+1},~\frac{1}{2(s-\frac{1}{r})+2(v+1)+1}< \frac{1}{2(v+1)+1} \quad\text{and}\quad \tau < j_{0}^{*}, j_{1}^{*}< j_{1}. \end{aligned}$$
When \(\omega \geq 0\), one writes down
$$\begin{aligned} e_{2}=\Biggl(\sum_{j={\tau }}^{j_{0}^{*}}+ \sum_{j=j_{0}^{*}} ^{j_{1}}\Biggr) 2^{j(\frac{p}{2}-1)}\sum _{k\in \varOmega _{j}} \vert \widehat{\beta }_{j,k}-{ \beta }_{j,k} \vert ^{p}I\{k\in \widehat{B}_{j} \cap {B}_{j}\} :=e_{21}+e_{22}. \end{aligned}$$
(25)
According to (22),
$$\begin{aligned} Ee_{21}\lesssim \sum_{j={\tau }}^{j_{0}^{*}}2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} E \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p} \lesssim n^{-\frac{p}{2}}2^{(v+1+\frac{1}{2})j_{0}^{*}p}. \end{aligned}$$
(26)
Note that \(\frac{2|\beta _{jk}|}{\varUpsilon \lambda _{j}}\geq 1\), \(\sum_{k}|\beta _{jk}|^{r}\lesssim 2^{-j(s+\frac{1}{2}-\frac{1}{r})r}\) from \(k\in B_{j}\), \(f'_{\sigma ^{2}}\in B_{r,q}^{s}(M)\) and Lemma 2.1; On the other hand, Lemma 4.2 tells \(E|\widehat{\beta }_{j,k}-{\beta }_{j,k}|^{p}\lesssim 2^{(v+1)jp} n ^{-\frac{p}{2}}\). These with \(\lambda _{j}=2^{(v+1)j}\sqrt{ \frac{j}{n}}\) and \(\omega =sr+(v+\frac{3}{2})r-(v+\frac{3}{2})p\geq 0\) lead to
$$\begin{aligned} Ee_{22} &\lesssim \sum_{j=j_{0}^{*}}^{j_{1}} 2^{j( \frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} E \vert \widehat{\beta }_{j,k}- {\beta }_{j,k} \vert ^{p}\biggl( \frac{ \vert \beta _{jk} \vert }{\varUpsilon \lambda _{j}}\biggr)^{r} \\ &\lesssim \sum_{j=j_{0}^{*}}^{j_{1}}2^{j(\frac{p}{2}-1)}2^{(v+1)jp} n^{-\frac{p}{2}}\lambda _{j}^{-r}\sum _{k} \vert \beta _{jk} \vert ^{r} \lesssim 2^{-j _{0}^{*}\omega }n^{-\frac{p-r}{2}}. \end{aligned}$$
(27)
Combining (25)–(27) with \(2^{j_{0}^{*}}\sim (\frac{n}{ \ln n})^{\frac{1}{2s+2(v+1)+1}}\), one obtains
$$\begin{aligned} Ee_{2}=Ee_{21}+Ee_{22}\lesssim n^{-\frac{p}{2}}2^{(v+1+\frac{1}{2})j _{0}^{*}p}+2^{-j_{0}^{*}\omega }n^{-\frac{p-r}{2}} \lesssim \biggl(\frac{ \ln n}{n}\biggr)^{\frac{sp}{2s+2(v+1)+1}}=\biggl(\frac{\ln n}{n} \biggr)^{\alpha p} \end{aligned}$$
because of \(\omega \geq 0\) and \(\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}= \frac{s}{2s+2(v+1)+1}\).
When \(\omega =sr+(v+\frac{3}{2})r-(v+\frac{3}{2})p<0\), \(\alpha = \min \{\frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s- \frac{1}{r})+2(v+1)+1}\}=\frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s- \frac{1}{r})+2(v+1)+1}\). Define \(p_{1}=(1-2\alpha )p\). Then \(r\leq p_{1}\leq p\) follows from
$$\begin{aligned} \omega < 0 \quad\text{and}\quad r\leq p_{1}=(1-2\alpha )p= \frac{2(v+1)p+p-2}{2(s-\frac{1}{r})+2(v+1)+1}\leq p. \end{aligned}$$
Moreover, \(\sum_{k}|\beta _{jk}|^{p_{1}}\leq (\sum_{k}|\beta _{jk}|^{r})^{\frac{p _{1}}{r}} \lesssim 2^{-j(s+\frac{1}{2}-\frac{1}{r})p_{1}} \) thanks to \(r\leq p_{1}\), \(f'_{\sigma ^{2}}\in B_{r,q}^{s}(M)\) and Lemma 2.1. This with (27) and \(\lambda _{j}=2^{(v+1)j}\sqrt{ \frac{j}{n}}\) shows
$$\begin{aligned} Ee_{2} &\lesssim \sum_{j=\tau }^{j_{1}} 2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} E \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p}\biggl( \frac{ \vert \beta _{jk} \vert }{\varUpsilon \lambda _{j}}\biggr)^{p_{1}} \\ &\lesssim \sum_{j=\tau }^{j_{1}}2^{j(\frac{p}{2}-1)}2^{(v+1)jp} n^{-\frac{p}{2}}\lambda _{j}^{-p_{1}}\sum _{k} \vert \beta _{jk} \vert ^{p_{1}} \\ &\lesssim n^{-\frac{p-p_{1}}{2}}\sum_{j=\tau }^{{j_{1}}} 2^{j[ \frac{p}{2}-1+(v+1)(p-p_{1})-(s+\frac{1}{2}-\frac{1}{r})p_{1}]} \lesssim \ln n \biggl(\frac{\ln n}{n}\biggr)^{\alpha p} \end{aligned}$$
due to \(\frac{p-p_{1}}{2}=\alpha p\) and \(\frac{p}{2}-1+(v+1)(p-p_{1})-(s+ \frac{1}{2}-\frac{1}{r})p_{1}=0\).
Finally, one estimates \(Ee_{4}\). When \(\omega =sr+(v+\frac{3}{2})r-(v+ \frac{3}{2})p\geq 0\), \(\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}= \frac{s}{2s+2(v+1)+1}\). Furthermore,
$$\begin{aligned} e_{4}=\Biggl(\sum_{j={\tau }}^{j_{0}^{*}}+ \sum_{j={j_{0}^{*}}+1} ^{j_{1}}\Biggr) 2^{j(\frac{p}{2}-1)}\sum _{k\in \varOmega _{j}} \vert {\beta } _{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B}_{j}^{c}\cap {C}_{j}^{c}\bigr\} :=e_{41}+e _{42}, \end{aligned}$$
(28)
where \(j_{0}^{*}\) is given by (24).
Since \(|\beta _{j,k}|\leq 2\varUpsilon \lambda _{j}\lesssim 2^{(v+1)j}(jn ^{-1})^{\frac{1}{2}}\) holds by \(k\in C_{j}^{c}\) and \(\lambda _{j}=2^{(v+1)j}\sqrt{ \frac{j}{n}}\), one concludes that
$$\begin{aligned} Ee_{41} &= E\sum_{j={\tau }}^{j_{0}^{*}}2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} \vert \beta _{j,k} \vert ^{p}I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}^{c}\bigr\} \\ &\lesssim \sum_{j={\tau }}^{j_{0}^{*}}2^{j(\frac{p}{2}-1)}2^{j} 2^{(v+1)jp}\bigl(jn^{-1}\bigr)^{\frac{p}{2}}\lesssim 2^{(v+1+\frac{1}{2})j_{0} ^{*}p}\biggl(\frac{\ln n}{n}\biggr)^{\frac{p}{2}}. \end{aligned}$$
(29)
On the other hand,
$$\begin{aligned} Ee_{42} &= E\sum_{j={j_{0}^{*}+1}}^{j_{1}} 2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} \vert \beta _{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}^{c}\bigr\} \\ &\lesssim \sum_{j=j_{0}^{*}+1}^{j_{1}}2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} \vert \beta _{j,k} \vert ^{p}\bigl(\lambda _{j} \vert \beta _{j,k} \vert ^{-1}\bigr)^{p-r} \\ &\lesssim \sum_{j={j_{0}^{*}+1}}^{j_{1}}2^{j(\frac{p}{2}-1)} \lambda _{j}^{p-r}\sum_{k} \vert \beta _{j,k} \vert ^{r} \end{aligned}$$
due to \(|\beta _{j,k}|\leq 2\varUpsilon \lambda _{j}\) and \(r\leq p\).
Clearly, \(\|{\beta }_{j,\cdot }\|_{l_{r}} \lesssim 2^{-j(s-\frac{1}{r}+ \frac{1}{2})}\) by \(f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)\) and Lemma 2.1. This with \(\lambda _{j}=2^{(v+1)j}\sqrt{\frac{j}{n}}\) implies that
$$\begin{aligned} Ee_{42} \lesssim \sum_{j={j_{0}^{*}+1}}^{j_{1}}2^{j( \frac{p}{2}-1)} 2^{(v+1)(p-r)j}2^{-j(sr+\frac{r}{2}-1)}\biggl( \frac{\ln n}{n}\biggr)^{\frac{p-r}{2}} \lesssim \biggl(\frac{\ln n}{n}\biggr)^{ \frac{p-r}{2}}2^{-j_{0}^{\ast }\omega }, \end{aligned}$$
(30)
because \(\omega =sr+(v+\frac{3}{2})r-(v+\frac{3}{2})p\geq 0\).
According to (28)–(30) and \(2^{j_{0}^{*}}\sim (\frac{n}{ \ln n})^{\frac{1}{2s+2(v+1)+1}}\), one obtains
$$\begin{aligned} Ee_{4}=Ee_{41}+Ee_{42}\lesssim 2^{(v+1+\frac{1}{2})j_{0}^{*}p} \biggl(\frac{ \ln n}{n}\biggr)^{\frac{p}{2}}+\biggl(\frac{\ln n}{n} \biggr)^{\frac{p-r}{2}}2^{-j_{0} ^{\ast }\omega } \lesssim \biggl(\frac{\ln n}{n} \biggr)^{\alpha p} \end{aligned}$$
by \(\alpha =\frac{s}{2s+2(v+1)+1}\).
For the case of \(\omega =sr+(v+\frac{3}{2})r-(v+\frac{3}{2})p<0\). Let
$$\begin{aligned} e_{4}=\Biggl(\sum_{j={\tau }}^{j_{1}^{*}}+ \sum_{j={j_{1}^{*}}+1} ^{j_{1}}\Biggr) 2^{j(\frac{p}{2}-1)}\sum _{k\in \varOmega _{j}} \vert {\beta } _{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B}_{j}^{c}\cap {C}_{j}^{c}\bigr\} :=e_{41}'+e _{42}', \end{aligned}$$
(31)
where \(j_{1}^{*}\) is given by (24). Similar to (30),
$$\begin{aligned} Ee_{41}'=\sum _{j={\tau }}^{j_{1}^{*}} 2^{j(\frac{p}{2}-1)} \sum _{k\in \varOmega _{j}} \vert {\beta }_{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}^{c} \bigr\} \lesssim \biggl(\frac{\ln n}{n}\biggr)^{\frac{p-r}{2}}2^{-j _{1}^{\ast }\omega } \end{aligned}$$
(32)
thanks to \(\omega <0\).
To estimate \(Ee_{42}'\), one observes that \(\|\beta _{j,\cdot }\|_{l_{p}} \lesssim \|\beta _{j,\cdot }\|_{l_{r}}\lesssim 2^{-j(s-\frac{1}{r}+ \frac{1}{2})}\) by \(r\leq p\), \(f'_{\sigma ^{2}}\in B_{r,q}^{s}(M)\) and Lemma 2.1. Hence,
$$\begin{aligned} Ee_{42}' &=\sum _{j={j_{1}^{*}+1}}^{j_{1}}2^{j(\frac{p}{2}-1)} \sum _{k\in \varOmega _{j}} \vert \beta _{j,k} \vert ^{p}I \bigl\{ k\in \widehat{B}_{j} ^{c}\cap {C}_{j}^{c} \bigr\} \lesssim \sum_{j={j_{1}^{*}+1}}^{j_{1}}2^{j( \frac{p}{2}-1)} \Vert \beta _{j,\cdot } \Vert _{l_{r}}^{p} \\ &\lesssim \sum_{j={j_{1}^{*}+1}}^{j_{1}}2^{j(\frac{p}{2}-1)}2^{-j(s- \frac{1}{r}+\frac{1}{2})p} \lesssim 2^{-j_{1}^{*}(s-\frac{1}{r}+ \frac{1}{p})p} \end{aligned}$$
(33)
because of \(s>\frac{1}{r}\).
Combining (31)–(33) with \(2^{j_{1}^{*}}\sim (\frac{n}{ \ln n})^{\frac{1}{2(s-\frac{1}{r})+2(v+1)+1}}\), one knows
$$\begin{aligned} Ee_{4}=Ee_{41}'+Ee_{42}' \lesssim \biggl(\frac{\ln n}{n}\biggr)^{\frac{p-r}{2}}2^{-j _{1}^{\ast }\omega } +2^{-j_{1}^{*}(s-\frac{1}{r}+\frac{1}{p})p} \lesssim \biggl(\frac{\ln n}{n}\biggr)^{\alpha p} \end{aligned}$$
thanks to \(\omega <0\) and \(\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}=\frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\). This completes the proof of Theorem 5.2. □
Proof of lower bound
Finally, we are in a position to state and prove the lower bound estimation.
Theorem 5.3
Assume
\(s>0\)
and
\(r,q\in [1,+\infty ]\), then, for any
\(p\in [1,+ \infty )\),
$$\begin{aligned} \inf_{\widehat{f}'_{{\sigma }^{2}}}\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} E \bigl\Vert \widehat{f}'_{{\sigma } ^{2}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \gtrsim n^{-\frac{(s-\frac{1}{r}+ \frac{1}{p})p}{2(s-\frac{1}{r})+2(v+1)+1}}, \end{aligned}$$
where
\(\widehat{f}'_{{\sigma }^{2}}\)
runs over all possible estimators of
\(f'_{{\sigma }^{2}}\).
Proof
It is sufficient to construct density functions \(h_{k}\) such that \(h'_{k}\in B_{r,q}^{s}(M)\) and
$$\begin{aligned} \sup_{k} E \bigl\Vert \widehat{f}'_{{\sigma }^{2}}-h'_{k} \bigr\Vert _{p}^{p} \gtrsim n^{-\frac{(s-\frac{1}{r}+ \frac{1}{p})p}{2(s-\frac{1}{r})+2(v+1)+1}}. \end{aligned}$$
Define \(g(x)=Cm(x)\), where \(m\in C_{0}^{\infty }\) with \(\operatorname{supp} m \subseteq [0,1]\), \(\int _{\mathbb{R}} m(x)\,dx=0\) and \(C>0\) is a constant. Let \(C_{0}^{\infty }\) stand for the set of all infinitely many times differentiable and compactly supported functions. Furthermore, one chooses a density function \(h_{0}\) satisfying \(h_{0}\in B_{r,q}^{s+1}( \frac{M}{2}), \operatorname{supp} h_{0}\subseteq [0,1]\) and \(h_{0}(x)\geq M _{1}>0\) for \(x\in [\frac{1}{2},\frac{3}{4}]\).
Take \(a_{j}=2^{-j(s-\frac{1}{r}+\frac{1}{2}+v+1)}\) and
$$\begin{aligned} h_{1}(x)=h_{0}(x)+a_{j}G_{v}(g_{j,l}) (x), \end{aligned}$$
(34)
where G is given by (5) and \(g_{j,l}(x)=2^{\frac{j}{2}}g(2^{j}x-l)\) with \(l=2^{j-1}\).
First, one checks that \(h_{1}\) is a density function. Since \(\operatorname{supp} g_{j,l}\subseteq [\frac{1}{2},\frac{3}{4}]\) by \(\operatorname{supp} m\subseteq [0,1]\) and j large enough, one finds \(h_{1}(x)\geq 0\) for \(x\notin [\frac{1}{2},\frac{3}{4}]\). It is easy to calculate that
$$\begin{aligned} G_{v}(g_{j,l}) (x)=(-1)^{v}\sum _{u=1}^{v}C_{u}x^{u}(g_{j,l})^{(u)}(x), \end{aligned}$$
(35)
where \(C_{u}>0\) is a constant. Then, for \(x\in [\frac{1}{2}, \frac{3}{4}]\) and large j,
$$\begin{aligned} h_{1}(x) &\geq M_{1}- \Biggl\vert a_{j}\sum_{u=1}^{v}C_{u}x^{u}(g_{j,l})^{(u)}(x) \Biggr\vert \\ &\geq M_{1}-a_{j}2^{\frac{j}{2}}\sum _{u=1}^{v}C_{u}2^{uj} \bigl\Vert {g}^{(u)}\bigl(2^{j}\cdot -l\bigr) \bigr\Vert _{\infty } \\ &\geq M_{1}-2^{-j(s-\frac{1}{r}+1)}\sum_{u=1}^{v}C_{u} \bigl\Vert g ^{(u)} \bigr\Vert _{\infty }\geq 0 \end{aligned}$$
(36)
thanks to \(h_{0}(x)|_{[\frac{1}{2},\frac{3}{4}]}\geq M_{1}\) and \(a_{j}=2^{-j(s-\frac{1}{r}+\frac{1}{2}+v+1)}\). On the other hand, \(\int _{\mathbb{R}} g(x)\,dx=\int _{\mathbb{R}} Cm(x)\,dx=0\) and \(\operatorname{supp} g_{j,l}\subseteq [\frac{1}{2},\frac{1}{2}+2^{-j}]\) by \(\operatorname{supp} m\subseteq [0,1]\) and \(l=2^{j-1}\). This with \(m\in C_{0}^{\infty }\) and \(g(x)=Cm(x)\) shows
$$\begin{aligned} \int x^{u}(g_{j,l})^{(u)}(x) \,dx&=x^{u}(g_{j,l})^{(u-1)} (x) | _{\frac{1}{2}}^{\frac{1}{2}+2^{-j}}-u \int x^{u-1}(g_{j,l})^{(u-1)}(x)\,dx \\ &=\cdots =(-1)^{m}\frac{u!}{(u-m)!} \int x^{u-m}(g_{j,l})^{(u-m)} (x) \,dx\\ &=(-1)^{u}u! \int g_{j,l}(x)\,dx=0 \end{aligned}$$
for any \(u\in \{1,\ldots , v\}\). Therefore,
$$\begin{aligned} \int h_{1}(x)\,dx= \int h_{0}(x)\,dx+(-1)^{v}a_{j}\sum _{u=1}^{v}C _{u} \int x^{u}(g_{j,l})^{(u)}(x)\,dx=1. \end{aligned}$$
From this with (36) one concludes that \(h_{1}\) is a density function.
Next, one shows \(h'_{0}, h'_{1}\in B_{r,q}^{s}(M)\). Clearly, \(h'_{0}\in B_{r,q}^{s}(M)\) by \(h_{0}\in B_{r,q}^{s+1}(\frac{M}{2})\). Hence, one only needs prove \(h'_{1}\in B_{r,q}^{s}(M)\).
By (34) and (35),
$$\begin{aligned} \Vert h_{1} \Vert _{B_{r,q}^{s+1}}\leq \Vert h_{0} \Vert _{B_{r,q}^{s+1}}+a_{j}\sum _{u=1}^{v}C_{u} \bigl\Vert x^{u}(g_{j,l})^{(u)}(x) \bigr\Vert _{B_{r,q}^{s+1}}. \end{aligned}$$
(37)
On the other hand, for each \(\tau \in \{0,\ldots ,u\}\),
$$\begin{aligned} \biggl\Vert \biggl[2^{j}\biggl(x-\frac{1}{2}\biggr) \biggr]^{u-\tau }g^{(u)}\biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \biggr\Vert _{B_{r,q}^{s+1}} \leq 2^{j(s+1-\frac{1}{r})} \bigl\Vert x^{u-\tau }g^{(u)}(x) \bigr\Vert _{B_{r,q}^{s+1}} \end{aligned}$$
because of Lemma 4.4. Combining this with \(l=2^{j-1}\) and \([2^{j}(x-\frac{1}{2}+\frac{1}{2})]^{u}= \sum_{\tau =0}^{u}C ^{\tau }_{u}[2^{j}(x-\frac{1}{2})]^{u-\tau }2^{-\tau }2^{\tau j}\), one obtains
$$\begin{aligned} \bigl\Vert x^{u}(g_{j,l})^{(u)}(x) \bigr\Vert _{B_{r,q}^{s+1}} &=2^{\frac{j}{2}} \biggl\Vert \biggl[2^{j}\biggl(x- \frac{1}{2}+\frac{1}{2}\biggr) \biggr]^{u}g^{(u)}\biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \biggr\Vert _{B_{r,q} ^{s+1}} \\ &\leq 2^{(u+\frac{1}{2})j}\sum_{\tau =0}^{u}C^{\tau }_{u} \biggl\Vert \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr) \biggr]^{u-\tau }g^{(u)}\biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \biggr\Vert _{B_{r,q}^{s+1}} \\ &\leq 2^{j(s-\frac{1}{r}+\frac{1}{2}+u+1)}\sum_{\tau =0}^{u}C ^{\tau }_{u} \bigl\Vert x^{u-\tau }g^{(u)}(x) \bigr\Vert _{B_{r,q}^{s+1}}. \end{aligned}$$
(38)
Denote \(M'=\max \{C_{u},C_{u}^{\tau }: \tau =0,\ldots ,u; u=1,\ldots ,v\}\). Then there exists a constant \(C>0\) such that
$$\begin{aligned} x^{u-\tau }g^{(u)}\in B_{r,q}^{s+1}\biggl( \frac{M}{2v^{2}M^{\prime 2}}\biggr) \end{aligned}$$
thanks to \(g(x)=Cm(x)\) and \(m\in C_{0}^{\infty }\subseteq B_{r,q}^{s+1}\). This with (37) and (38) leads to
$$\begin{aligned} \Vert h_{1} \Vert _{B_{r,q}^{s+1}}\leq \Vert h_{0} \Vert _{B_{r,q}^{s+1}}+a_{j}\sum_{u=1}^{v} uM^{\prime 2}2^{j(s-\frac{1}{r}+\frac{1}{2}+u+1)}\frac{M}{2v ^{2}M^{\prime 2}}\leq \frac{M}{2}+ \frac{M}{2}=M, \end{aligned}$$
because \(h_{0}\in B_{r,q}^{s+1}(\frac{M}{2})\) and \(a_{j}=2^{-j(s- \frac{1}{r}+\frac{1}{2}+v+1)}\). Therefore, \(h_{1}\in B_{r,q}^{s+1}(M)\) and \(h'_{1}\in B_{r,q}^{s}(M)\).
According to (35),
$$\begin{aligned} \bigl[G_{v}(g_{j,l})(x)\bigr] '=(-1)^{v} \sum_{u=0}^{v}C'_{u}x^{u}(g_{j,l})^{(u+1)}(x), \end{aligned}$$
where \(C'_{u}>0\) is a constant and \(l=2^{j-1}\). Hence,
$$\begin{aligned} \bigl\Vert h'_{1}-h'_{0} \bigr\Vert _{p} &= a_{j} \bigl\Vert \bigl[G_{v}(g_{j,l})\bigr]' \bigr\Vert _{p} \\ &= a_{j}2^{\frac{3j}{2}} \Biggl\Vert \sum _{u=0}^{v}C'_{u} \biggl[2^{j}\biggl(x- \frac{1}{2}+\frac{1}{2}\biggr) \biggr]^{u}g^{(u+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p}. \end{aligned}$$
(39)
On the other hand, by using \([2^{j}(x-\frac{1}{2}+\frac{1}{2})]^{u}= \sum_{\tau =0}^{u} C^{\tau }_{u}[2^{j}(x-\frac{1}{2})]^{u- \tau }2^{-\tau }2^{\tau j}\), one concludes that
$$\begin{aligned} &\Biggl\Vert \sum_{u=0}^{v}C'_{u} \biggl[2^{j}\biggl(x-\frac{1}{2}+\frac{1}{2}\biggr) \biggr]^{u}g ^{(u+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p} \\ &\quad\geq \Biggl\Vert \sum_{u=0}^{v}C'_{u} 2^{-u}2^{uj}g^{(u+1)}\biggl[2^{j} \biggl(x-\frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p} \\ &\qquad{}- \Biggl\Vert \sum_{u=0}^{v}C'_{u} \Biggl\{ \sum_{\tau =0}^{u-1}C^{\tau } _{u}\biggl[2^{j} \biggl(x-\frac{1}{2}\biggr) \biggr]^{u-\tau }2^{-\tau } 2^{\tau j}\Biggr\} g^{(u+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p} \end{aligned}$$
(40)
and
$$\begin{aligned} &\Biggl\Vert \sum_{u=0}^{v}C'_{u}2^{-u}2^{uj}g^{(u+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p} \\ &\quad \geq \biggl\Vert C'_{v}2^{-v}2^{vj}g^{(v+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \biggr\Vert _{p} \\ &\qquad{}- \Biggl\Vert \sum_{u=0}^{v-1}C'_{u} 2^{-u}2^{uj}g^{(u+1)}\biggl[2^{j} \biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p}. \end{aligned}$$
(41)
Let \(x'=2^{j}(x-\frac{1}{2})\). Then there exists a constant \(C'>0\) such that
$$\begin{aligned} \bigl\Vert h'_{1}-h'_{0} \bigr\Vert _{p} \geq {}& a_{j}2^{\frac{3}{2}j}2^{-\frac{j}{p}} \Biggl\{ C'_{v}2^{-v}2^{vj} \bigl\Vert g^{(v+1)} \bigr\Vert _{p}-2^{(v-1)j}\sum _{u=0} ^{v-1}C'_{u}2^{-u} \bigl\Vert g^{(u+1)} \bigr\Vert _{p} \\ &{}-2^{(v-1)j}\sum_{u=0}^{v}C'_{u} \sum_{\tau =0}^{u-1}C ^{\tau }_{u}2^{-\tau } \bigl\Vert \bigl(x'\bigr)^{u-\tau }g^{(u+1)} \bigr\Vert _{p} \Biggr\} \\ \geq {}& C'a_{j}2^{\frac{3}{2}j}2^{-\frac{j}{p}}2^{jv} =C'2^{-j(s- \frac{1}{r}+\frac{1}{p})}:=\delta _{j} \end{aligned}$$
thanks to (39)–(41), \(g\in C_{0}^{\infty }\) and \(a_{j}=2^{-j(s-\frac{1}{r}+\frac{1}{2}+v+1)}\).
Define \(A_{k}=\{ \Vert \widehat{f}'_{{\sigma }^{2}}-h'_{k} \Vert _{p}<\frac{\delta _{j}}{2}\}\ (k\in \{0,1\})\). Then \(A_{0}\cap A_{1}= \emptyset \). According to Lemma 4.5,
$$\begin{aligned} \sup_{k\in \{0,1\}} P_{f_{s_{k}}}^{n} \bigl(A_{k}^{c}\bigr)\geq \min \biggl\{ \frac{1}{2}, \exp \bigl(-3e^{-1}-\kappa _{1}\bigr)\biggr\} , \end{aligned}$$
(42)
where \(P_{f}^{n}\) stands for the probability measure corresponding to the density function \(f^{n}(x):=f(x_{1})f(x_{2})\cdots f(x_{n})\). Hence,
$$\begin{aligned} E \bigl\Vert \widehat{f}'_{\sigma ^{2}}-h'_{k} \bigr\Vert _{p}^{p} \geq \biggl( \frac{\delta _{j}}{2} \biggr)^{p}P_{f_{s_{k}}}^{n}\biggl( \bigl\Vert \widehat{f}'_{{\sigma } ^{2}}-h'_{k} \bigr\Vert _{p}\geq \frac{\delta _{j}}{2}\biggr) =\biggl(\frac{\delta _{j}}{2} \biggr)^{p}P _{f_{s_{k}}}^{n}\bigl(A_{k}^{c} \bigr). \end{aligned}$$
This with (42) implies
$$\begin{aligned} \sup_{k\in \{0,1\}} E \bigl\Vert \widehat{f}'_{\sigma ^{2}}-h'_{k} \bigr\Vert _{p} ^{p}\geq \sup_{k\in \{0,1\}}\biggl( \frac{\delta _{j}}{2}\biggr)^{p}P_{f_{s _{k}}}^{n} \bigl(A_{k}^{c}\bigr)\geq \biggl(\frac{\delta _{j}}{2} \biggr)^{p}\min \biggl\{ \frac{1}{2},\exp \bigl(-3e^{-1}- \kappa _{1}\bigr)\biggr\} . \end{aligned}$$
(43)
Next, one shows \(\kappa _{1}\leq C_{0}na_{j}^{2}\). Recall that
(44)
Then
$$\begin{aligned} K\bigl(P_{f_{s_{1}}}^{n}, P_{f_{s_{0}}}^{n}\bigr) \leq \sum_{i=1}^{n} \int f_{s_{1}}(x_{i})\ln \frac{f_{s_{1}}(x_{i})}{f_{s_{0}}(x_{i})}\,dx _{i}=n \int f_{s_{1}}(x)\ln \frac{f_{s_{1}}(x)}{f_{s_{0}}(x)}\,dx \end{aligned}$$
due to \(f_{s_{0}}^{n}(x)=\prod_{j=1}^{n}f_{s_{0}}(x_{j})\) and \(f_{s_{1}}^{n}(x)=\prod_{j=1}^{n}f_{s_{1}}(x_{j})\). Since \(\ln u \leq u-1\) holds for \(u>0\), one knows
$$\begin{aligned} K\bigl(P_{f_{s_{1}}}^{n}, P_{f_{s_{0}}}^{n}\bigr) & \leq n \int f_{s_{1}}(x) \biggl( \frac{f _{s_{1}}(x)}{f_{s_{0}}(x)}-1\biggr)\,dx \\ &= n \int f^{-1}_{s_{0}}(x)\bigl| f_{s_{1}}(x)-f_{s_{0}}(x) \bigr| ^{2} \,dx. \end{aligned}$$
(45)
According to Chesneau’s work in Ref. [8], \(f_{s_{k}}(x)=\frac{1}{(v-1)!}\int _{x}^{1}(\ln y-\ln x)^{v-1}h_{k}(y) \frac{1}{y}\,dy\). Then
$$\begin{aligned} f_{s_{1}}(x)-f_{s_{0}}(x) &=\frac{a_{j}}{(v-1)!} \int _{x}^{\frac{1}{2}+2^{-j}}( \ln y-\ln x)^{v-1}G_{v}(g_{j,l}) (y)\frac{1}{y}\,dy \\ &=-\frac{a_{j}}{(v-1)!} \int _{x}^{\frac{1}{2}+2^{-j}}(\ln y-\ln x)^{v-1}\bigl[G _{v-1}(g_{j,l}) (y)\bigr]'\,dy \end{aligned}$$
because of (34) and \(G_{v}(g_{j,l})(x)=-x[G_{v-1}(g_{j,l})(x)]'\).
By the formula of integration by parts,
$$\begin{aligned} f_{s_{1}}(x)-f_{s_{0}}(x) &=-\frac{a_{j}}{(v-2)!} \int _{x}^{ \frac{1}{2}+2^{-j}} (\ln y-\ln x)^{v-2} \bigl[G_{v-2}(g_{j,l}) (y)\bigr]'\,dy= \cdots \\ &=-\frac{a_{j}}{(v-m)!} \int _{x}^{\frac{1}{2}+2^{-j}}(\ln y-\ln x)^{v-m} \bigl[G_{v-m}(g_{j,l}) (y)\bigr]'\,dy=\cdots \\ &=-a_{j} \int _{x}^{\frac{1}{2}+2^{-j}}(g_{j,l})'(y) \,dy=a_{j}g_{j,l}(x), \end{aligned}$$
(46)
because \(l=2^{j-1}\) and \((\ln y-\ln x)^{v-m}G_{v-m}(g_{j,l})(y)| _{x}^{\frac{1}{2}+2^{-j}}=0\) for any \(m\in \{1,\ldots ,v-1\}\). On the other hand, for each \(x\in [\frac{1}{2}, \frac{1}{2}+2^{-j}]\) and large j,
$$\begin{aligned} f_{s_{0}}(x) &\geq \frac{M_{1}}{(v-1)!} \int _{x}^{\frac{3}{4}}(\ln y- \ln x)^{v-1} \frac{1}{y}\,dy=\frac{M_{1}}{v!}\biggl(\ln \frac{3}{4}-\ln x \biggr)^{v} \\ &\geq \frac{M_{1}}{v!}\biggl[\ln \frac{3}{4}-\ln \biggl( \frac{1}{2}+2^{-j}\biggr)\biggr]^{v} \geq M_{2}>0 \end{aligned}$$
(47)
thanks to \(f_{s_{0}}(x)=\frac{1}{(v-1)!}\int _{x}^{1}(\ln y-\ln x)^{v-1}h _{0}(y) \frac{1}{y}\,dy\) and \(h_{0}(x)|_{[\frac{1}{2},\frac{3}{4}]} \geq M_{1}\). Combining with (44)–(47), one obtains
$$\begin{aligned} \kappa _{1}\leq M_{2}^{-1}n \int\bigl| a_{j}g_{j,l}(x)\bigr| ^{2} \,dx \leq C _{0}na_{j}^{2}, \end{aligned}$$
where \(C_{0}>0\) is a constant.
Choose \(2^{j}\thicksim n^{\frac{1}{2(s-\frac{1}{r})+2(v+1)+1}}\) and recall \(a_{j}=2^{-j(s-\frac{1}{r}+\frac{1}{2}+v+1)}\). Then
$$\begin{aligned} \kappa _{1}\lesssim na_{j}^{2}=n2^{-j[2(s-\frac{1}{r})+2(v+1)+1]} \thicksim 1 \quad \text{and} \quad e^{-\kappa _{1}}\gtrsim 1. \end{aligned}$$
Substituting \(\delta _{j}\thicksim 2^{-j(s-\frac{1}{r}+\frac{1}{p})}\), \(2^{j}\thicksim n^{\frac{1}{2(s-\frac{1}{r})+2(v+1)+1}}\) into (43), one obtains
$$\begin{aligned} \sup_{k\in \{0,1\}} E \bigl\Vert \widehat{f}'_{\sigma ^{2}}-h'_{k} \bigr\Vert _{p} ^{p}\gtrsim \delta _{j}^{p} \gtrsim n^{-\frac{(s-\frac{1}{r} + \frac{1}{p})p}{2(s-\frac{1}{r})+2(v+1)+1}}, \end{aligned}$$
which is the desired conclusion. □