Following [
16], we have
$$ s_{\delta }(f;y)-f(y)=\frac{1}{2\pi } \int _{0}^{\pi } \phi _{y}(l) \frac{ \sin (\delta +\frac{1}{2})l}{\sin (\frac{l}{2})} \,dl. $$
Denoting the Hausdorff matrix summability transform of
\(s_{\delta }(y)\) by
\(t_{\delta }^{\triangle _{H}}(y)\), we get
$$\begin{aligned} t_{\delta }^{\triangle _{H}}(y)-f(y) &= \sum_{m=0}^{\delta } h_{\delta ,m}\bigl[s_{m}(y)-f(y)\bigr] \\ &=\sum_{m=0}^{\delta }\binom{\delta }{m} \triangle ^{\delta -m}\mu _{m} \biggl\{ \frac{1}{2\pi } \int _{0}^{\pi }\phi _{y}(l) \frac{\sin (m+ \frac{1}{2})l}{\sin (\frac{l}{2})} \,dl \biggr\} \\ &=\frac{1}{2\pi } \int _{0}^{\pi }\phi _{y}(l)\sum _{m=0}^{\delta }\binom{ \delta }{m}\triangle ^{\delta -m} \biggl( \int _{0}^{1}z^{m}\,d\nu (z) \biggr) \frac{ \sin (m+\frac{1}{2})l}{\sin (\frac{l}{2})}\,dl \\ &=\frac{1}{2\pi } \int _{0}^{\pi }\phi _{y}(l)\sum _{m=0}^{\delta } \int _{0}^{1}\binom{\delta }{m}z^{m}(1-z)^{\delta -m} \,d\nu (z)\frac{\sin (m+ \frac{1}{2})l}{\sin \frac{l}{2}}\,dl. \end{aligned}$$
The
\(N_{pq}\) transform of
\(t_{\delta }^{\triangle _{H}}(y)\), denoted by
\(t_{\delta }^{N_{pq} \triangle _{H}}(y)\), is given by
$$\begin{aligned} &t_{\delta }^{N_{pq} \triangle _{H}}(y)-f(y) \\ &\quad=\frac{1}{R_{\delta }}\sum_{m=0}^{\delta }p_{\delta -k}q_{m} \Biggl(\frac{1}{2 \pi } \int _{0}^{\pi }\phi _{y}(l)\sum _{v=0}^{m} \int _{0}^{1}\binom{m}{v}z ^{v}(1-z)^{m-v}\,d\nu (z)\frac{\sin (v+\frac{1}{2})l}{\sin \frac{l}{2}}\,dl \Biggr). \end{aligned}$$
(11)
Replacing
l by
u
$$\begin{aligned} &= \int _{0}^{\pi }\phi _{y}(u) \frac{1}{2\pi R_{\delta }}\sum_{m=0}^{ \delta }p_{\delta -m}q_{m} \sum_{v=0}^{m} \int _{0}^{1}\binom{m}{v}z^{v}(1-z)^{m-v} \,d \nu (z)\frac{\sin (v+\frac{1}{2})u}{\sin (\frac{u}{2})}\,du \\ &= \int _{0}^{\pi }\phi _{y}(u) M_{\delta }(u)\,du. \end{aligned}$$
(12)
Let
$$ T_{\delta }(y)=t_{\delta }^{N_{pq} \triangle _{H}}(y)-f(y)= \int _{0} ^{\pi }\phi _{y}(u) M_{\delta }(u)\,du. $$
(13)
Using the definition of the Besov norm given by (
5), we have
$$\begin{aligned} &\bigl\Vert T_{\delta }(\cdot) \bigr\Vert _{B_{\sigma }^{\eta }(L_{\rho })}= \bigl\Vert T _{\delta }(\cdot) \bigr\Vert _{\rho }+ \bigl\Vert w_{k}(T_{\delta },\cdot) \bigr\Vert _{\eta, \sigma }. \end{aligned}$$
(14)
Now using (
6) and Lemma
4.4(iii)
$$\begin{aligned} \bigl\Vert T_{\delta }(\cdot) \bigr\Vert _{\rho } &\leq \int _{0}^{\pi } \bigl\Vert \phi _{.}(u) \bigr\Vert _{\rho } \bigl\vert M_{\delta }(u) \bigr\vert \,du \\ &\leq \int _{0}^{\pi }2w_{k}(f,u)_{\rho } \bigl\vert M_{\delta }(u) \bigr\vert \,du. \end{aligned}$$
(15)
Using Hölder’s inequality and definition of Besov space given in (
4), we get,
$$\begin{aligned} \bigl\Vert T_{\delta }(\cdot) \bigr\Vert _{\rho } &\leq 2 \biggl\{ \int _{0}^{\pi } \bigl( \bigl\vert M_{\delta }(u) \bigr\vert u^{\nu +\frac{1}{\sigma }} \bigr)^{\frac{ \sigma }{\sigma -1}} \,du \biggr\} ^{1-\frac{1}{\sigma }} \biggl\{ \int _{0}^{\pi } \biggl( \frac{w_{k}(f,u)_{\rho }}{u^{\nu +\frac{1}{\sigma }}} \biggr)^{\sigma }\,du \biggr\} ^{\frac{1}{\sigma }} \\ ={}&O(1) \biggl\{ \int _{0}^{\pi } \bigl( \bigl\vert M_{\delta }(u) \bigr\vert u^{\nu +\frac{1}{\sigma }} \bigr)^{\frac{ \sigma }{\sigma -1}} \,du \biggr\} ^{1-\frac{1}{\sigma }} \\ ={}&O\biggl[ \biggl\{ \int _{0}^{\frac{1}{\delta +1}} \bigl( \bigl\vert M _{\delta }(u) \bigr\vert u^{\nu +\frac{1}{\sigma }} \bigr)^{\frac{\sigma }{ \sigma -1}} \,du \biggr\} ^{1-\frac{1}{\sigma }} \\ &{}+ \biggl\{ \int _{\frac{1}{ \delta +1}}^{\pi } \bigl( \bigl\vert M_{\delta }(u) \bigr\vert u^{\nu +\frac{1}{ \sigma }} \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\frac{1}{ \sigma }} \biggr] \\ ={}&R+S. \end{aligned}$$
(16)
Now using Lemma
4.2, we have
$$\begin{aligned} R &= O\biggl\{ \int _{0}^{\frac{1}{\delta +1}} \bigl( \bigl\vert M_{\delta }(u) \bigr\vert u^{\nu +\frac{1}{\sigma }} \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\frac{1}{\sigma }} \\ &=O \biggl[ \int _{0}^{\frac{1}{\delta +1}} \bigl\{ (\delta +1) u^{ \nu +\frac{1}{\sigma }} \bigr\} ^{\frac{\sigma }{\sigma -1}} \,du \biggr] ^{1-\frac{1}{\sigma }} \\ &=O \biggl\{ (\delta +1)^{\frac{\sigma }{\sigma -1}} \int _{0}^{\frac{1}{ \delta +1}} u^{\frac{\nu \sigma }{\sigma -1}+\frac{1}{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ &=O \biggl\{ \frac{1}{(\delta +1)^{\nu }} \biggr\} . \end{aligned}$$
(17)
Using Lemma
4.3, we have
$$\begin{aligned} S &= O\biggl\{ \int _{\frac{1}{\delta +1}}^{\pi } \bigl( \bigl\vert M_{\delta }(u) \bigr\vert u^{\nu +\frac{1}{\sigma }} \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\frac{1}{\sigma }} \\ &=O \biggl\{ \int _{\frac{1}{\delta +1}}^{\pi } \biggl( \frac{1}{(\delta +1)u^{2}} u^{\nu +\frac{1}{\sigma }} \biggr)^{ \frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\frac{1}{\sigma }} \\ &=O \biggl\{ \int _{\frac{1}{\delta +1}}^{\pi } \biggl(\frac{1}{\delta +1} \times u^{\nu +\frac{1}{\sigma }-2} \biggr)^{ \frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\frac{1}{\sigma }} \\ &=O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu >1, \\ (\delta +1)^{-\nu }, &\nu < 1, \\ (\delta +1)^{-1}[\log (\delta +1)\pi ]^{1-\sigma ^{-1}}, &\nu =1. \end{cases}\displaystyle \end{aligned}$$
(18)
Combining (
16)–(
18), we have
$$ \bigl\Vert T_{\delta }(\cdot) \bigr\Vert _{\rho } =O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu >1, \\ (\delta +1)^{-\nu }, &\nu < 1, \\ (\delta +1)^{-1}[\log (\delta +1)\pi ]^{1-\sigma ^{-1}}, &\nu =1. \end{cases} $$
(19)
Using the generalized Minkowski inequality [
21] repeatedly and Lemma
4.5, we get
$$\begin{aligned} \bigl\Vert w_{k}(T_{\delta },\cdot) \bigr\Vert _{\eta,\sigma }={} &\biggl[ \int _{0}^{ \pi } \biggl(\frac{w_{k}(T_{\delta },l)_{\rho }}{l^{\eta }} \biggr) ^{\sigma } \frac{dl}{l} \biggr]^{\frac{1}{\sigma }} \\ ={}& \biggl[ \int _{0}^{\pi } \biggl(\frac{ \Vert \varUpsilon _{\delta }(\cdot,l) \Vert _{\rho }}{l^{\eta }} \biggr)^{\sigma } \frac{dl}{l} \biggr] ^{\frac{1}{\sigma}} \\ \leq{}& \int _{0}^{\pi } \bigl\vert M_{\delta }(u) \bigr\vert \,du \biggl( \int _{0}^{\pi } \frac{ \Vert \Phi (\cdot,l,u) \Vert _{\rho }^{\sigma }}{l^{\eta \sigma }} \frac{dl}{l} \biggr)^{\sigma ^{-1}} \\ \leq{}& \biggl[ \int _{0}^{\pi } \bigl\vert M_{\delta }(u) \bigr\vert \,du \biggl\{ \int _{0}^{u}\frac{ \Vert \Phi (\cdot,l,u) \Vert _{\rho }^{\sigma }}{l^{\eta \sigma }} \frac{dl}{l} \biggr\} ^{\sigma ^{-1}} \biggr] \\ &{}+ \biggl[ \int _{0}^{\pi } \bigl\vert M_{\delta }(u) \bigr\vert \,du \biggl\{ \int _{u}^{\pi }\frac{ \Vert \Phi (\cdot,l,u) \Vert _{\rho }^{\sigma }}{l^{\eta \sigma }} \frac{dl}{l} \biggr\} ^{\sigma ^{-1}} \biggr] \\ ={}&O(1) \biggl\{ \int _{0}^{\pi }\bigl(u^{\nu -\eta } \bigl\vert M_{\delta }(u) \bigr\vert \bigr)^{\frac{ \sigma }{\sigma -1}} \,du \biggr\} ^{1-\frac{1}{\sigma }} \\ &{}+O(1) \biggl\{ \int _{0}^{\pi }\bigl(u^{\nu -\eta +\frac{1}{\sigma }} \bigl\vert M _{\delta }(u) \bigr\vert \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\frac{1}{\sigma }} \\ ={}&O(1) (R_{1}+S_{1}). \end{aligned}$$
(20)
Since
\((a+b)^{\rho} \leq a^{\rho}+b^{\rho}\) for positive
a,
b and
\(0 < \rho \leq 1\) for
\(\rho=1-\frac{1}{\sigma} <1\), then
$$\begin{aligned} R_{1}={}& \biggl\{ \int _{0}^{\pi }\bigl(u^{\nu -\eta } \bigl\vert M_{\delta }(u) \bigr\vert \bigr)^{\frac{ \sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ \leq{}& \biggl\{ \int _{0}^{\frac{1}{\delta +1}}\bigl(u^{\nu -\eta } \bigl\vert M_{\delta }(u) \bigr\vert \bigr)^{\frac{ \sigma }{\sigma -1}}\,du \biggr\} ^{1-\sigma ^{-1}} \\ &{}+ \biggl\{ \int _{\frac{1}{\delta +1}}^{\pi }\bigl(u^{\nu -\eta } \bigl\vert M_{\delta }(u) \bigr\vert \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ ={}&R_{11}+R_{12}. \end{aligned}$$
(21)
Using Lemma
4.2, we have
$$\begin{aligned} R_{11} &= O\biggl\{ \int _{0}^{\frac{1}{\delta +1}} \bigl( (\delta +1)u^{\nu - \eta } \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ &=O\bigl\{ (\delta +1)^{-\nu +\eta +\frac{1}{\sigma }} \bigr\} . \end{aligned}$$
(22)
Using Lemma
4.3, we have
$$\begin{aligned} R_{12} &= O\biggl\{ \int _{\frac{1}{\delta +1}}^{\pi } \biggl(u^{\nu -\eta } \frac{1}{( \delta +1)u^{2}} \biggr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ &=O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu -\eta -\sigma ^{-1} >1, \\ (\delta +1)^{-\nu +\eta +\sigma ^{-1}}, &\nu -\eta -\sigma ^{-1}< 1, \\ (\delta +1)^{-1}[\log (\delta +1)\pi ]^{1-\sigma ^{-1}}, &\nu -\eta -\frac{1}{ \sigma }=1. \end{cases}\displaystyle \end{aligned}$$
(23)
Now from (
21) to (
23), we get
$$ R_{1}=O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu -\eta -\sigma ^{-1} >1, \\ (\delta +1)^{-\nu +\eta +\sigma ^{-1}}, &\nu -\eta -\sigma ^{-1}< 1, \\ (\delta +1)^{-1}[\log (\delta +1)\pi ]^{1-\sigma ^{-1}}, &\nu -\eta -\frac{1}{ \sigma }=1. \end{cases} $$
(24)
Since
\((a+b)^{\rho} \leq a^{\rho}+b^{\rho}\) for positive
a,
b and
\(0 < \rho \leq 1\) for
\(\rho=1-\frac{1}{\sigma} <1\), then
$$\begin{aligned} S_{1}={}& \biggl\{ \int _{0}^{\pi }\bigl(u^{\nu -\eta +\sigma ^{-1}} \bigl\vert M_{\delta }(u) \bigr\vert \bigr)^{\frac{ \sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ \leq {}& \biggl\{ \int _{0}^{\frac{1}{\delta +1}}\bigl(u^{\nu -\eta +\sigma ^{-1}} \bigl\vert M_{\delta }(u) \bigr\vert \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ &{}+ \biggl\{ \int _{\frac{1}{\delta +1}}^{\pi }\bigl(u^{\nu -\eta +\sigma ^{-1}} \bigl\vert M _{\delta }(u) \bigr\vert \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ ={}&S_{11}+S_{12} \quad\text{say.} \end{aligned}$$
(25)
Using Lemma
4.2, we have
$$\begin{aligned} S_{11} &= O\biggl\{ \int _{0}^{\frac{1}{\delta +1}} \bigl(u^{\nu -\eta + \sigma ^{-1}} \bigl\vert M_{\delta }(u) \bigr\vert \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ &= O\biggl\{ \int _{0}^{\frac{1}{\delta +1}} \bigl(u^{\nu -\eta +\sigma ^{-1}} (\delta +1) \bigr)^{\frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ &=O\bigl\{ (\delta +1)^{-\nu +\eta }\bigr\} . \end{aligned}$$
(26)
Using Lemma
4.3, we have
$$\begin{aligned} S_{12} &= O\biggl\{ \int _{\frac{1}{\delta +1}}^{\pi } \bigl(u^{\nu - \eta +\frac{1}{\sigma }} \bigl\vert M_{\delta }(u) \bigr\vert \bigr)^{\frac{\sigma }{ \sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ &= O\biggl\{ \int _{\frac{1}{\delta +1}}^{\pi } \biggl(u^{\nu -\eta +\frac{1}{ \sigma }} \frac{1}{(\delta +1)u^{2}} \biggr)^{ \frac{\sigma }{\sigma -1}} \,du \biggr\} ^{1-\sigma ^{-1}} \\ &=O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu -\eta >1, \\ (\delta +1)^{-\nu +\eta }, &\nu -\eta < 1, \\ (\delta +1)^{-1}[\log (\delta +1)\pi ]^{1-\sigma ^{-1}}, &\nu -\eta =1. \end{cases}\displaystyle \end{aligned}$$
(27)
Now, from (
25)-(
27), we get
$$\begin{aligned} S_{1}=O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu -\eta >1, \\ (\delta +1)^{-\nu +\eta }, &\nu -\eta < 1, \\ (\delta +1)^{-1}[\log (\delta +1)\pi ]^{1-\sigma ^{-1}}, &\nu -\eta =1. \end{cases}\displaystyle \end{aligned}$$
(28)
Combining (
20), (
24) and (
28), we get
$$ w_{k}(T_{\delta },\cdot)\Vert _{\eta,\sigma }=O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu -\eta -\sigma ^{-1} >1, \\ (\delta +1)^{-\nu +\eta +\sigma ^{-1}}, &\nu -\eta -\sigma ^{-1}< 1, \\ (\delta +1)^{-1}[\log (\delta +1)\pi ]^{1-\sigma ^{-1}}, &\delta - \eta -\sigma^{-1}=1. \end{cases} $$
(29)
From (
14), (
19) and (
29), we get
$$ \bigl\Vert T_{\delta }(\cdot) \bigr\Vert _{B_{\sigma }^{\eta }(L_{\rho })}=O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu -\eta -\sigma ^{-1} >1, \\ (\delta +1)^{-\nu +\eta +\sigma ^{-1}}, &\nu -\eta -\sigma ^{-1}< 1, \\ (\delta +1)^{-1}[\log (\delta +1)\pi ]^{1-\sigma ^{-1}}, &\nu -\eta -\frac{1}{ \sigma }=1. \end{cases} $$
(30)
Case II: For
\(\sigma=\infty,0\leq \eta <\nu <2\).
$$\begin{aligned} \bigl\Vert T_{\delta }(\cdot) \bigr\Vert _{B_{\infty }^{\eta }(L_{\rho })}&= \bigl\Vert T _{\delta }(\cdot) \bigr\Vert _{\rho }+ \bigl\Vert w_{k}(T_{\delta },\cdot) \bigr\Vert _{\eta, \infty }. \end{aligned}$$
(31)
Using (
2) in (
15),
$$\begin{aligned} \bigl\Vert T_{\delta }(\cdot) \bigr\Vert _{\rho } &\leq \int _{0}^{\pi }2w_{k}(f,u)_{ \rho } \bigl\vert M_{\delta }(u) \bigr\vert \,du \\ &=O(1) \biggl\{ \int _{0}^{\frac{1}{\delta +1}} \bigl\vert M_{\delta }(u) \bigr\vert u^{\nu } \,du + \int _{\frac{1}{\delta +1}}^{\pi } \bigl\vert M_{ \delta }(u) \bigr\vert u^{\nu } \,du \biggr\} \\ &=O(1)[R_{2}+S_{2}]. \end{aligned}$$
(32)
Using Lemma
4.2, we get
$$\begin{aligned} R_{2} &= \int _{0}^{\frac{1}{\delta +1}} u^{\nu } \bigl\vert M_{\delta }(u) \bigr\vert \,du \\ &\leq \int _{0}^{\frac{1}{\delta +1}} u^{\nu }(\delta +1) \,du=( \delta +1)^{-\nu }. \end{aligned}$$
(33)
Using Lemma
4.3, we get
$$\begin{aligned} S_{2} &= \int _{\frac{1}{\delta +1}}^{\pi } u^{\nu } \bigl\vert M_{\delta }(u) \bigr\vert \,du \\ &\leq \frac{1}{\delta +1} \int _{\frac{1}{\delta +1}}^{\pi } u^{ \nu -2}\,du \\ &= \textstyle\begin{cases} (\delta +1)^{-1}, &\nu >1, \\ (\delta +1)^{-\nu }, &\nu < 1, \\ (\delta +1)^{-1}\log( \delta +1)\pi, & \nu =1. \end{cases}\displaystyle \end{aligned}$$
(34)
From (
32) and (
34), we get
$$ \bigl\Vert T_{\delta}(\cdot) \bigr\Vert _{\rho}=O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu >1, \\ (\delta +1)^{-\nu }, &\nu < 1, \\ (\delta +1)^{-1}\log(\delta +1)\pi, & \nu =1. \end{cases} $$
(35)
Using the generalized Minkowski inequality [
21] and Lemma
4.6, we get
$$\begin{aligned} \bigl\Vert w_{k}(T_{\delta },\cdot) \bigr\Vert _{\eta,\infty } &=\sup_{l>0}\bigl(l^{- \eta }w_{k}(T_{\delta },l)_{\rho } \bigr) \\ &=\sup_{l>0}\bigl(l^{-\eta } \bigl\Vert \varUpsilon _{\delta }(\cdot,l) \bigr\Vert _{\rho }\bigr) \\ &=\sup_{l>0} \biggl[l^{-\eta } \biggl( \frac{1}{2\pi } \int _{0}^{2 \pi } \biggl\vert \int _{0}^{\pi } \bigl\vert M_{\delta }(u) \bigr\vert \Phi (y,l,u) \,du \biggr\vert ^{\rho } \,dy \biggr)^{\frac{1}{\rho }} \biggr] \\ &\leq \sup_{l>0} \biggl[l^{-\eta } \biggl( \frac{1}{2\pi } \biggr) ^{\frac{1}{\rho }} \int _{0}^{\pi } \biggl\{ \int _{0}^{2\pi } \bigl\vert M_{ \delta }(u) \bigr\vert ^{\rho } \bigl\vert \Phi (y,l,u) \bigr\vert ^{\rho } \,dy \biggr\} ^{\frac{1}{ \rho }} \,du \biggr] \\ &=\sup_{l>0} \biggl[l^{-\eta } \int _{0}^{\pi } \bigl\Vert \Phi (\cdot,l,u) \bigr\Vert _{\rho } \bigl\vert M_{\delta }(u) \bigr\vert \,du \biggr] \\ &= \int _{0}^{\pi }\Bigl(\sup_{l>0} l^{-\eta } \bigl\Vert \Phi (\cdot,l,u) \bigr\Vert _{\rho } \Bigr) \bigl\vert M_{\delta }(u) \bigr\vert \,du \\ &= O(1) \int _{0}^{\pi } u^{\nu -\eta } \bigl\vert M_{\delta }(u) \bigr\vert \,du \\ &= O(1) \biggl[ \int _{0}^{\frac{1}{\delta +1}} u^{\nu -\eta } \bigl\vert M_{ \delta }(u) \bigr\vert \,du + \int _{\frac{1}{\delta +1}}^{\pi } u^{\nu -\eta } \bigl\vert M_{\delta }(u) \bigr\vert \,du \biggr] \\ &=O(1)[R_{3}+S_{3}]. \end{aligned}$$
(36)
Using Lemma
4.2, we get
$$\begin{aligned} R_{3} &= \int _{0}^{\frac{1}{\delta +1}} u^{\nu -\eta } \bigl\vert M_{\delta }(u) \bigr\vert \,du \\ &=O\bigl((\delta +1)^{\eta -\nu }\bigr). \end{aligned}$$
(37)
Using Lemma
4.3, we get
$$\begin{aligned} S_{3} &= \int _{\frac{1}{\delta +1}}^{\pi } u^{\nu -\eta } \bigl\vert M_{\delta }(u) \bigr\vert \,du \\ &=O(1) \frac{1}{\delta +1} \int _{\frac{1}{\delta +1}}^{\pi } u^{ \nu -\eta -2} \,du \\ &= O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu -\eta >1, \\ (\delta +1)^{-\nu +\eta }, &\nu -\eta < 1, \\ (\delta +1)^{-1} \log (\delta +1)\pi, &\nu -\eta =1. \end{cases}\displaystyle \end{aligned}$$
(38)
From (
36) to (
38), we get
$$ \bigl\Vert w_{k}(T_{\delta },\cdot) \bigr\Vert _{\eta,\infty }= O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu -\eta >1, \\ (\delta +1)^{-\nu +\eta }, &\nu -\eta < 1, \\ (\delta +1)^{-1} \log (\delta +1)\pi, &\nu -\eta =1. \end{cases} $$
(39)
Combining (
31), (
35) and (
39) we obtain
$$ \bigl\Vert T_{\delta }(\cdot) \bigr\Vert _{B_{\sigma }^{\eta }(L_{ \rho })}=O(1) \textstyle\begin{cases} (\delta +1)^{-1}, &\nu -\eta -\sigma ^{-1} >1, \\ (\delta +1)^{-\nu +\eta +\sigma ^{-1}}, &\nu -\eta -\sigma ^{-1}< 1, \\ (\delta +1)^{-1}[\log (\delta +1)\pi ]^{1-\sigma ^{-1}}, &\nu -\eta -\frac{1}{ \sigma }=1. \end{cases} $$
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