1. Introduction and preliminaries
Given \(\alpha\geq0\). The harmonic analysis on \(\mathbb{X}=[0,+\infty[\times[0,+\infty[\) is generated by the following partial differential operators
\[\left\{
\begin{array}{ll}
\mathcal{D}_{1,\alpha}= \frac{\partial^{2} }{\partial t^{2}}+\frac{2\alpha}{t}\frac{\partial}{\partial t},\\
\mathcal{D}_{2,\alpha}= \frac{\partial^{2} }{\partial x^{2}}+\frac{2\alpha+1}{x}\frac{\partial }{\partial x}+x^{2}\mathcal{D}_{1,\alpha},
\end{array}
\right.\]
where for all \((x,t)\in \mathbb{X}\). For \((\lambda,m)\in [0,+\infty[\times\mathbb{N}\), the initial value problem
\[\left\{
\begin{array}{ll}
\mathcal{D}_{1,\alpha}u=-\lambda^{2} u, \\
\mathcal{D}_{2,\alpha}u=-4\lambda(m+\frac{\alpha+1}{2})u ,\\
u(0,0)=1 , \frac{\partial u}{\partial x}(0,0)=\frac{\partial u}{\partial t}(0,0)=0,
\end{array}
\right.\tag{1}\]
has a unique solution \(\psi_{\lambda,m}\) given by
\[\psi_{\lambda,m}(x,t)=j_{\alpha-\frac{1}{2}}(\lambda t)\mathfrak{L}^{\alpha}_{m}(\lambda x^{2}),~~ \forall (x,t)\in \mathbb{X},\tag{2}\]
where \(\mathfrak{L}^{\alpha}_{m}\) is the Laguerre function defined on \(\mathbb{R}_{+}\) by
\[\mathfrak{L}^{\alpha}_{m}(x)= e^{-\frac{x}{2}} \frac{L^{\alpha}_{m}(x)}{L^{\alpha}_{m}(0)},\tag{3}\]
\(L^{\alpha}_{m}\) being the Laguerre polynomial of degree \(m\) and order \(\alpha\), and \(j_{\alpha}\) is the normalized Bessel function given by
\[j_{\alpha}(x)=\Gamma(\alpha+1)\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!\Gamma(\alpha+k+1)}\left(\frac{x}{2}\right)^{2k}.\tag{4}\]
For \((x,t) \in \mathbb{X}\), the generalized translation operator \(T_{(x, t)}^{(\alpha)}\) is defined for \(\alpha=0\) by
\[\frac{1}{4 \pi} \sum_{i, j=0}^{1} \int_{0}^{\pi} f\left(\Delta_{\theta}(x, y), Y+(-1)^{i} t+(-1)^{j} s\right) d \theta,\]
and for \(\alpha>0\) by
\[b_{\alpha} \int_{[0, \pi]^{3}} f\left(\Delta_{\theta}(x, y), \Delta_{\theta}(x, y) \xi\right) d \mu_{\alpha}( \xi, \psi, \theta),\]
where \(\Delta_{\theta}(x, y)=\sqrt{x^{2}+y^{2}+2 x y \cos \theta},\; b_{\alpha}=\frac{(\alpha+1) \Gamma\left(\alpha+\frac{1}{2}\right)}{\pi^{\frac{3}{4}} \Gamma(\alpha)}\), \(Y=x y \sin \theta\) and
\[d \mu_{\alpha}(\xi, \psi, \theta)=(\sin \xi)^{2 \alpha-1}(\sin \psi)^{2 \alpha-1}(\sin \theta)^{2 \alpha} d \xi d \psi d \theta.\]
We denote by:
\(\bullet\) \(\mid x,t\mid=\mid(x,t)\mid_{\mathbb{X}}=(x^{4}+4t^{2})^{\frac{1}{4}}\) the homogeneous norm on \(\mathbb{X}\).
\(\bullet\) \(\mid \lambda,m\mid=\mid(\lambda,m)\mid_{[0,+\infty[\times\mathbb{N}}=4\lambda(m+\frac{\alpha+1}{2})\) the quasinorm on \([0,+\infty[\times\mathbb{N}\). Let denote \(\mathbb{B}_{r}\) the ball centered \(0\) and of radius \(r\), defined by,
\[\mathbb{B}_{r}=\{(\lambda,m)\in[0,+\infty[\times\mathbb{N}; \mid\lambda ,m\mid<r \}~~ and~~\mathbb{B}_{r}^{c}=([0,+\infty[\times\mathbb{N})\backslash\mathbb{B}_{r}.~\]
\(\bullet\) \(L_{\alpha}^{p}(\mathbb{X}), p \in[1,+\infty]\), the spaces of measurable functions on \(\mathbb{X}\) such that
\[\left\{
\begin{array}{ll}
\|f\|_{p,\alpha}=\left[\int_{\mathbb{X}}|f(x,t)|^{p}dm_{\alpha}(x,t)\right]^{\frac{1}{p}}<+\infty,~~if~~p\in [1,+\infty[ ,\\
\|f\|_{\infty,\alpha}=esssup_{(x,t)\in \mathbb{X} }|f(x,t)|<+\infty,
\end{array}
\right.\]
where \(dm_{\alpha}\) the weighted Lebesgue measure on \(\mathbb{X}\), given by
\[dm_{\alpha}(x,t)=\frac{x^{2\alpha+1} t^{2\alpha}}{\Gamma(\alpha+1)\Gamma(\alpha+\frac{1}{2})}dxdt.\]
\(\bullet\) \(L_{\gamma_{\alpha}}^{p}([0,+\infty[\times\mathbb{N}), p \in[1,+\infty]\), the spaces of measurable functions on \([0,+\infty[\times\mathbb{N}\) such that
\[\left\{
\begin{array}{ll}
\|g\|_{\gamma_{\alpha, p}}=\left[\int_{[0,+\infty[\times\mathbb{N}}|g(\lambda, m)|^{p} d \gamma_{\alpha}(\lambda, m)\right]^{\frac{1}{p}}<+\infty, \text { if } p \in[1,+\infty), \\
\|g\|_{\gamma_{\alpha, \infty}}= \operatorname{esssup}_{(\lambda, m) \in [0,+\infty[\times\mathbb{N}}|g(\lambda, m)|<+\infty,
\end{array}
\right.\]
where \(d\gamma_{\alpha}\) is the positive measure defined on \([0,+\infty[\times\mathbb{N}\) by
\[\int_{[0,+\infty[\times \mathbb{N}} g(\lambda, m) d \gamma_{\alpha}(\lambda, m)=\frac{1}{2^{2 \alpha-1} \Gamma\left(\alpha+\frac{1}{2}\right)} \sum_{m=0}^{\infty} L_{m}^{\alpha}(0) \int_{0}^{+\infty} g(\lambda, m) \lambda^{3 \alpha+1} d \lambda .\]
The Fourier Laguerre-Bessel transform of a function in \(L^{1}_{\alpha}(\mathbb{X})\) is given by
\[\mathcal{F}_{LB}f(\lambda,m)=\int_{\mathbb{X}}f(x,t)\psi_{\lambda,m}(x,t)dm_{\alpha}(x,t),~~(\lambda,m)\in [0,+\infty[\times\mathbb{N}.\]
From [1], it is well known that Fourier Laguerre-Bessel transform can be inverted to
\[\mathcal{F}_{LB}^{-1}f(x,t)=\int_{[0,+\infty[\times\mathbb{N}}f(\lambda,m)\psi_{\lambda,m}(x,t)d\gamma_{\alpha}(\lambda,m),~~(x,t)\in \mathbb{X}.\]
It is well-known (see [1–4]) that the Fourier Laguerre-Bessel transform \(\mathcal{F}_{LB}\) satisfies the following properties:
The integral transform can be extended to an isometric isomorphism \(L^{2}_{\alpha}(\mathbb{X})\) to \(L_{\gamma_{\alpha}}^{2}([0,+\infty[\times\mathbb{N})\) and we have the Plancherel formula
\[\|f\|_{2,\alpha} =\|\mathcal{F}_{LB}f\|_{\gamma_{\alpha},2},~~for~~f\in L^{1}_{\alpha}(\mathbb{X})\cap L^{2}_{\alpha}(\mathbb{X}).\tag{5}\]
We also have the inverse formula of the generalized Fourier transform:
\[f(x,t)=\int_{[0,+\infty[\times\mathbb{N}}\mathcal{F}_{LB}f(\lambda,m)\psi_{\lambda,m}(x,t)d\gamma_{\alpha}(\lambda,m),~~(x,t)\in \mathbb{X},\]
provided \(\mathcal{F}_{LB}f\in L^{1}_{\gamma_{\alpha}}([0,+\infty[\times\mathbb{N})\).
For \(f\in L^{2}_{\alpha}(\mathbb{X}),(x,t)\in \mathbb{X}\) and \((\lambda,m)\in [0,+\infty[\times\mathbb{N}\), we have
\[\mathcal{F}_{LB}(T^{(\alpha)}_{(x,t)}f)(\lambda,m)=\psi_{\lambda,m}(x,t)\mathcal{F}_{LB}(f)(\lambda,m),\tag{6}\]
and from (6), we get
\[\mathcal{F}_{LB}(T^{(\alpha)}_{(x,t)}f-f)(\lambda,m)=(\psi_{\lambda,m}(x,t)-1)\mathcal{F}_{LB}(f)(\lambda,m).\tag{7}\]
Now we define the finite differences of order \(k\in \mathbb{N}\) and step \((x,t)\in \mathbb{X}\) by
\[\Delta^{k}_{(x,t)}f(y,s)=(T^{(\alpha)}_{(x,t)}-I)^{k}f(y,s),\]
where \(I\) denotes the unit operator and \((x,t)\neq(0,0)\).
Lemma 1. [5] For a fixed \((x,t)\neq(0,0)\), we have
\[\mathcal{F}_{LB}(\Delta^{k}_{(x,t)}f)(\lambda,m)=(\psi_{\lambda,m}(x,t)-1)^{k}\mathcal{F}_{LB}(f)(\lambda,m).\tag{8}\]
The \(k^{\text {th }}\) order generalized modulus of continuity of function \(f \in L^{2}_{\alpha}(\mathbb{X})\) is defined as
\[\Omega_{k}(f, \delta)=\sup_{0<\mid x,t\mid\leq \delta}\|\Delta^{k}_{(x,t)}f\|_{2,\alpha}.\tag{9}\]
Let \(\mathrm{W}_{2, \phi}^{r, k}(\mathcal{D}_{2,\alpha})\) denote the class of functions \(f \in L^{2}_{\alpha}(\mathbb{X})\) that have generalized derivatives satisfying the estimate
\[\Omega_{k}\left(\mathcal{D}_{2,\alpha}^{r} f, \delta\right)=O\left(\phi\left(\delta^{k}\right)\right), \delta \longrightarrow 0,\]
i.e.,
\[\mathrm{W}_{2, \phi}^{r, k}(\mathcal{D}_{2,\alpha})=\{f \in L^{2}_{\alpha}(\mathbb{X}) / \mathcal{D}_{2,\alpha}^{r} f \in L^{2}_{\alpha}(\mathbb{X}), \Omega_{k}\left(\mathcal{D}_{2,\alpha}^{r} f, \delta\right)=O\left(\phi\left(\delta^{k}\right)\right), \delta \rightarrow 0\},\tag{10}\]
where \(\phi(t)\) is any nonnegative function given on \([0, \infty[\). For the Laguerre-Bessel operator \(\mathcal{D}_{2,\alpha}\), we have \(\mathcal{D}_{2,\alpha}^{0} f=f, \mathcal{D}_{2,\alpha}^{r} f=\mathcal{D}_{2,\alpha}\left(\mathcal{D}_{2,\alpha}^{r-1} f\right), r=1,2, \ldots\).
The goal of the present paper is to prove an analogue of the results by Abilov et al. ([6,7]) for the Laguerre-Bessel transform. It is a special case of the following classical question in harmonic analysis: describe the relation between the regularity of a function and the rapidity of decay of its Fourier transform. More precisely, we have:
Theorem 1. Given \(\phi, r, k\) and \(f \in \mathrm{W}_{2, \phi}^{r, k}\left(\mathcal{D}_{2,\alpha}\right)\). Then there exists a constant \(c_{1}>0\) such that the following inequality holds, for all \(N>0\)
\[\int_{\mathbb{B}^{c}_{N}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid d\gamma_{\alpha}(\lambda,m) = O\left( N^{-2 r}\left(\phi\left(c_{1} N^{-k}\right)\right)^{2}\right),\]
as \(N \longrightarrow +\infty\), where the constant in the \(O-\)symbol depends only on \(r,k, \alpha\).
In the case where \(\phi(t)=t^{v}, v>0\), we have:
Theorem 2. Let \(\phi(t)=t^{v}\). Then
\[\sqrt{\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)}=O(N^{-r-kv})~~as~N \longrightarrow +\infty \Longleftrightarrow f \in \mathrm{W}_{2, \phi}^{r, k}\left(\mathcal{D}_{2,\alpha}\right) ,\]
where \(r=0,1,…, k=1,2,…, 0<v<1\).
2. Auxiliary results
In the remainder of this paper, we refer to \(c_1 , c_2 , c_3 , …,\) as positive constants which are generally different in different places and which may depend on \(k, r, \alpha\) and other inessential parameters. To prove the main results, we need to rely on some preliminary results.
Lemma 2. [1] For all \((\lambda,m)\in [0,+\infty[\times\mathbb{N}\), the functions \(\psi_{\lambda,m}\) is infinitely differentiable on \(\mathbb{R}^{2}\), even with respect to each variable and we have
\[sup_{(x,t)\in \mathbb{X}}|\psi_{\lambda,m}(x,t)|=1.\tag{11}\]
Lemma 3. If \(f\in\mathrm{W}_{2, \phi}^{r, k}(\mathcal{D}_{2,\alpha})\), we have
(i)\[\mathcal{F}_{LB}(\mathcal{D}_{2,\alpha}^{r}f)(\lambda,m)=(-1)^{r}\mid\lambda,m\mid^{r} \mathcal{F}_{LB}(f)(\lambda,m), r\in \mathbb{N}.\tag{12}\]
(ii)\[\|\Delta^{k}_{(x,t)} (\mathcal{D}_{2,\alpha}^{r}f)\|^{2}_{2,\alpha}=\int_{ [0,+\infty[\times\mathbb{N}}\mid\psi_{\lambda,m}(x,t)-1\mid^{2k}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m),\tag{13}\]
where \(r = 0, 1, . . .,k.\)
Proof. (i) From (1), we have
\[\mathcal{F}_{LB}(\mathcal{D}_{2,\alpha}f)(\lambda,m)=-\mid\lambda,m\mid \mathcal{F}_{LB}(f)(\lambda,m).\]
The result follows easily by induction on \(r\).
(ii) From (12), (8) and (5), we obtain
\[\|\Delta^{k}_{(x,t)} (\mathcal{D}_{2,\alpha}^{r}f)\|^{2}_{2,\alpha}=\int_{ [0,+\infty[\times\mathbb{N}}\mid\psi_{\lambda,m}(x,t)-1\mid^{2k}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m),\]
where \(r = 0, 1, . . .,k.\) \(\square\)
Lemma 4. Let \(N>0\), the following assertions are verified:
(i)There exist \(c_{2}>0\) such that for all \((\lambda,m)\in [0,+\infty[\times\mathbb{N}\) and \((x,t)\in \mathbb{X}\),
\[\mid\psi_{\lambda,m}(x,t)-1\mid\leq c_{2} \mid \lambda,m\mid\mid x,t\mid.\tag{14}\]
(ii)There exist \(c_{3}>0\) such that for all \((\lambda,m)\in {\mathbb{B}_{N}^{c}}\) and \((x,t)\in \mathbb{X}\),
\[\mid\psi_{\lambda,m}(x,t)\mid \leq c_{3} \left(\mid \lambda,m\mid x^{2}\right)^{-\frac{\alpha}{2}-\frac{1}{4}}.\tag{15}\]
Proof. (i) From ([8], Lemma 3.1), we deduce that for every \((\lambda,m)\in [0,+\infty[\times\mathbb{N}\) and \((x,t)\in \mathbb{X}\), we have
\[\psi_{\lambda,m}(x,t)=1-\frac{(\lambda t)^{2}}{4(\alpha+\frac{1}{2})}-\frac{\mid \lambda,m\mid x^{2}}{4(\alpha+1)}+\kappa_{\alpha,m}\lambda^{2}x^{4}+o(\lambda^{2} \mid x,t\mid^{4}),\tag{16}\]
with \(\kappa_{\alpha,m}\) depending only on \(\alpha\) and \(m\).
In consequence, there exist \(c_{4} > 0\) and \(\eta > 0\) such that for all \((x,t)\in \mathbb{X}\),
\[|\lambda, m| \mid x,t\mid^{2} <\eta \Rightarrow | \psi_{\lambda,m}(x, t)-1|^{2} \leq c_{4} |\lambda, m|^{2} \mid x,t\mid^{2}.\]
On the other hand, From ([9],Lemma 4.3) then
\[\lim_{\mid\lambda,m\mid\rightarrow +\infty} \varphi_{\lambda,m}(x,t)=0,\]
where \(\varphi_{\lambda,m}(x,t)=e^{i\lambda t}\mathfrak{L}^{\alpha}_{m}(\lambda x^{2})\) the Laguerre Kernel, and from [10], we have the asymptotic formula for the normalized Bessel function \(j_{\alpha}\) when \(x \longrightarrow+\infty\) :
\[j_{\alpha}(x)=\frac{\Gamma(\alpha+1)}{\Gamma\left(\frac{1}{2}\right)}\left(\frac{2}{x}\right)^{\alpha+\frac{1}{2}} \cos \left(x-(2 \alpha+1) \frac{\pi}{4}\right)+o\left(\frac{1}{x^{\frac{3}{2}}}\right).\tag{17}\]
Hence as
\[\psi_{\lambda,m}(x,t)=j_{\alpha-\frac{1}{2}}(\lambda t) \frac{1}{e^{i\lambda t}} \psi_{\lambda,m}(x,t),\]
then \(\lim_{\mid\lambda,m\mid\rightarrow +\infty} \psi_{\lambda,m}(x,t)=0,\) we get
\[\lim _{|\lambda, m| \rightarrow +\infty}\left(\frac{| \psi_{\lambda,m}(x, t)-1|}{|\lambda, m| \mid x,t\mid}\right)=0.\]
Hence, there exist \(c_{5} > 0\) and \(A > 0,\) such that
\[|\lambda, m| > A \Rightarrow | \psi_{\lambda,m}(x, t)-1|^{2} \leq c_{5} |\lambda, m|^{2} \mid x,t\mid^{2}.\]
If \(\frac{\eta}{\mid x,t\mid^{2}}< A\). Take
\[M=\max_{\frac{\eta}{\mid x,t\mid^{2}}\leq |\lambda, m| \leq A} \frac{| \psi_{\lambda,m}(x, t)-1|^{2}}{|\lambda, m|^{2} \mid x,t\mid^{2}}.\]
Therefore for all \((\lambda,m)\in B^{c}_{\frac{\eta}{\mid x,t\mid^{2}}}\) , we have
\[| \psi_{\lambda,m}(x, t)-1| \leq c_{6} |\lambda, m| \mid x,t\mid ,\]
where \(c_{6} = min(\sqrt{c_{5}}, \sqrt{M})\). Hence we have the result where \(c_{2} = max( \sqrt{c_{4}}, c_{6})\).
(ii) From ([11], Page 87), we have the asymptotic formula
\[L_{n}^{\alpha}(x) \approx \frac{\Gamma(n+\alpha+1)}{n !} e^{x / 2}(\kappa_{m} x)^{-\alpha / 2} J_{\alpha}(2 \sqrt{\kappa_{m} x}), \quad m \rightarrow \infty,\tag{18}\]
where \(\kappa_{m}=m+\frac{\alpha+1}{2}\), for all \(0\leq x\leq a\) and arbitrary finite \(a>0\).
On the other hand, it was shown in [12] and also in ([13], p. 355], the following estimate
\[\sqrt{ x} J_{p}(x)=O(1), x \geq 0,\tag{19}\]
where \(J_{p}(x)\) is the Bessel function of the first kind. Therefore, it follows from (3), (17), (18) and (19) that \(\mathfrak{L}^{\alpha}_{m}( \lambda x^{2})=O((|\lambda, m| x^{2})^{-\frac{\alpha}{2}-\frac{1}{4}})\), the proof is completed. \(\square\)
3. Proofs of Theorems 1 and 2
Proof of Theorem 1. For a given \(f \in \mathrm{W}_{2, \phi}^{r, k}\left(\mathcal{D}_{2,\alpha}\right)\) and \(N>0\), we have
\[\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m) \leq I_{1}+I_{2},\tag{20}\]
where
\[I_{1}=\int_{\mathbb{B}_{N}^{c}} \mid\psi_{\lambda,m}(x,t)\mid\mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m),\]
and
\[I_{2}=\int_{\mathbb{B}_{N}^{c}} \mid\psi_{\lambda,m}(x,t)-1\mid\mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m).\]
From (15), we have
\[I_{1}\leq c_{3} (Nx^{2})^{-\frac{\alpha}{2}-\frac{1}{4}}\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m).\]
Choose a constant \(c_{7}\) such that the number \(c_{8}=1-c_{3}c_{7}^{-\frac{\alpha}{2}-\frac{1}{4}}\) is positive.
Setting \(\mid x,t\mid=\frac{c_{7}}{N}\) in the inequality (20), we have
\[c_{8} \int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\leq I_{2}.\tag{21}\]
By Hölder inequality the second term in (21) satisfies
\[\begin{aligned}
I_{2} \leq& \left(\int_{\mathbb{B}_{N}^{c}} \mid\psi_{\lambda,m}(x,t)-1\mid^{2k}\mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\right)^{\frac{1}{2k}} \left(\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\right)^{1-\frac{1}{2k}}\\
\leq& \left(\int_{\mathbb{B}_{N}^{c}}|\lambda,m|^{-2r} \mid\psi_{\lambda,m}(x,t)-1\mid^{2k}|\lambda,m|^{2r}\mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\right)^{\frac{1}{2k}} \\
& \times\left(\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\right)^{1-\frac{1}{2k}} \\
\leq& N^{-\frac{r}{k}}\left(\int_{\mathbb{B}_{N}^{c}} \mid\psi_{\lambda,m}(x,t)-1\mid^{2k}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\right)^{\frac{1}{2k}} \\
& \times\left(\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\right)^{1-\frac{1}{2k}}.
\end{aligned}\]
We conclude that
\[\int_{\mathbb{B}_{N}^{c}} \mid\psi_{\lambda,m}(x,t)-1\mid^{2k}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m) \leq \|\Delta^{k}_{(x,t)} (\mathcal{D}_{2,\alpha}^{r}f)\|^{2}_{2,\alpha}.\]
Therefore
\[I_{2} \leq N^{-\frac{r}{k}} (\|\Delta^{k}_{(x,t)} (\mathcal{D}_{2,\alpha}^{r}f)\|_{2,\alpha})^{\frac{1}{k}}\left(\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\right)^{1-\frac{1}{2k}}.\]
For \(f \in \mathrm{W}_{2, \phi}^{r, k}\left(\mathcal{D}_{2,\alpha}\right)\) there exist a constant \(c_{9} > 0\) such that
\[\|\Delta^{k}_{(x,t)} (\mathcal{D}_{2,\alpha}^{r}f)\|^{2}_{2,\alpha} \leq c_{9} (\phi(\delta^{k}))^{2}~~as~\delta\rightarrow 0,\]
by virtue of (9) and (10). For \(\delta=\frac{c_{7}}{N}\), we obtain
\[c_{8} \int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\leq N^{-\frac{r}{k}} \left(c_{9} \phi\left((\frac{c_{7}}{N})^{k}\right)\right)^{\frac{1}{k}}\left(\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\right)^{1-\frac{1}{2k}},\]
then
\[c_{8} \left(\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)\right)^{\frac{1}{2k}}\leq N^{-\frac{r}{k}} \left(c_{9} \phi\left((\frac{c_{7}}{N})^{k}\right)\right)^{\frac{1}{k}}.\]
Therefore
\[\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)=O\left(N^{-2r} \left( \phi\left((\frac{c_{7}}{N})^{k}\right)\right)^{2}\right) ,\]
for all \(N > 0\). The theorem is proved with \(c_{1}=c_{7}^{k}\). \(\square\)
Proof of Theorem 2. We prove sufficiency by using Theorem 1, let \(f \in \mathrm{W}_{2, \phi}^{r, k}\left(\mathcal{D}_{2,\alpha}\right)\) then
\[\sqrt{\int_{\mathbb{B}_{N}^{c}} \mid\mathcal{F}_{LB}f(\lambda,m)\mid^{2} d\gamma_{\alpha}(\lambda,m)}=O(N^{-r-kv}).\tag{22}\]
It is easy to show, that there exists a function \(f \in L^{2}_{\alpha}(\mathbb{X})\) such that \(\mathcal{D}_{2,\alpha}^{r}f \in L^{2}_{\alpha}(\mathbb{X})\), from relation (13), we obtain
\[\begin{aligned}
\|\Delta^{k}_{(x,t)} (\mathcal{D}_{2,\alpha}^{r}f)\|^{2}_{2,\alpha} =& \int_{ [0,+\infty[\times\mathbb{N}}\mid\psi_{\lambda,m}(x,t)-1\mid^{2k}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m), \\
=& A_{1}+A_{2},
\end{aligned}\]
where
\[A_{1}=\int_{\mathbb{B}_{N}}\mid\psi_{\lambda,m}(x,t)-1\mid^{2k}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m),\]
and
\[A_{2}=\int_{\mathbb{B}_{N}^{c}}\mid\psi_{\lambda,m}(x,t)-1\mid^{2k}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m),\]
and \(N=E\left(\frac{1}{|x, t|}\right)\), the integer part of the number \(\frac{1}{|x, t|}\). From Lemma 2, we have the estimate
\[\begin{aligned}
A_{2} \leq& c_{10} \int_{\mathbb{B}_{N}^{c}}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m),\\
=& c_{10} \sum_{l=0}^{+\infty}\int_{\mathbb{B}_{N+l+1}\backslash\mathbb{B}_{N+l}}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m),\\
\leq& c_{10} \sum_{l=0}^{+\infty} (N+l+1)^{2r} \int_{\mathbb{B}_{N+l+1}\backslash\mathbb{B}_{N+l}}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m),\\
=& c_{10} \sum_{l=0}^{+\infty} a_{l}\left(\mathfrak{I}_{l}-\mathfrak{I}_{l+1} \right),
\end{aligned}\]
with \(a_{l} = (N + l + 1)^{2r}\) and \(\mathfrak{I}_{l}= \int_{\mathbb{B}_{N+l}^{c}}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m).\) For all integers \(M \geq 1\), the Abel transformation shows
\[\begin{aligned}
\sum_{l=0}^{M} a_{l}(\mathfrak{I}_{l}-\mathfrak{I}_{l+1})=& a_{0}\mathfrak{I}_{0}-a_{M}\mathfrak{I}_{M+1}+ \sum_{l=1}^{M} ( a_{l}-a_{l-1})\mathfrak{I}_{l} \\
\leq& a_{0}\mathfrak{I}_{0} + \sum_{l=1}^{M} ( a_{l}-a_{l-1})\mathfrak{I}_{l},
\end{aligned}\]
because \(a_{M}\mathfrak{I}_{M+1}\geq0\). Moreover by the finite increments theorem, we have \(a_{l}-a_{l-1}\leq 2r(N+l+1)^{2r-1}\). On the other hand, by (22), there exists \(c_{11} > 0\) such that, for all \(N > 0\)
\[\mathfrak{I}_{l}\leq c_{11} (N+l)^{-2r-2kv}.\]
For \(N \geq 1\), we have
\[\begin{aligned}
\sum_{l=0}^{M} a_{l}(\mathfrak{I}_{l}-\mathfrak{I}_{l+1}) \leq& a_{0}\mathfrak{I}_{0} + \sum_{l=1}^{M} ( a_{l}-a_{l-1})\mathfrak{I}_{l}, \\
\leq& c_{11} \left(1+\frac{1}{N}\right)^{2r} N^{-2kv}+2r c_{11} \sum_{l=1}^{M} \left(1+\frac{1}{N+l}\right)^{2r-1} (N+l)^{-1-2kv}\\
\leq& c_{11} 2^{2r} N^{-2kv} +2^{2r}r c_{11} \sum_{l=1}^{M} (N+l)^{-1-2kv}.
\end{aligned}\]
Finally, by the integral comparison test, we have
\[\begin{aligned}
\sum_{l=1}^{M} (N+l)^{-1-2kv} \leq& \sum_{\mu=N+1}^{+\infty} (\mu)^{-1-2kv} \\
\leq& \int_{N}^{+\infty} t^{-1-2kv} dt =\frac{1}{2kv} N^{-2kv}.
\end{aligned}\]
Letting \(M\rightarrow +\infty\), we see that, for \(r \geq 0\) and \(k,v> 0\), there exists a constant \(c_{12}\) such that, for all \(N\geq 1\),
\[A_{2} \leq c_{12} N^{-2kv}.\]
Consequently, for all \(|x, t|> 0,\) we get \(A_{2} \leq c_{12} |x, t|^{2kv},~ as~ |x, t|\rightarrow0\).
We estimate \(A_{1}\), by relation (14).
\[\begin{aligned}
A_{1} \leq& c_{2} \mid x,t\mid^{2k}\int_{\mathbb{B}_{N}}\mid\lambda,m\mid^{2k}\mid\lambda,m\mid^{2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m), \\
\leq& c_{2} \mid x,t\mid^{2k}\int_{\mathbb{B}_{N}}\mid\lambda,m\mid^{2k+2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m), \\
\leq& c_{2}\mid x,t\mid^{2k} \sum_{l=0}^{N-1} \int_{\mathbb{B}_{l+1}\backslash \mathbb{B}_{l} }\mid\lambda,m\mid^{2k+2r}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m), \\
\leq& c_{2} \mid x,t\mid^{2k} \sum_{l=0}^{N-1} (l+1)^{2k+2r} \int_{\mathbb{B}_{l+1}\backslash \mathbb{B}_{l} }\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m),\\
=& c_{2} \mid x,t\mid^{2k} \sum_{l=0}^{N-1} b_{l} (\mathcal{I}_{l}-\mathcal{I}_{l+1}),
\end{aligned}\]
with \(b_{l} = ( l + 1)^{2k+2r}\) and \(\mathcal{I}_{l}= \int_{\mathbb{B}_{l}^{c}}\mid\mathcal{F}_{LB}(f)\mid^{2}d\gamma_{\alpha}(\lambda,m).\) Using a summation by parts transforms and proceeding as with \(A_{2}\) and the fact that \(\mathcal{I}_{l} \leq c_{11} l^{-2r-2kv}\) by hypothesis, we obtain
\[\begin{aligned}
A_{1} \leq& c_{2} \mid x,t\mid^{2k} \sum_{l=0}^{N-1} b_{l}(\mathcal{I}_{l}-\mathcal{I}_{l+1}) \\
\leq& c_{2} \mid x,t\mid^{2k}\left(b_{0}\mathcal{I}_{0}+ \sum_{l=1}^{N-1} \mathcal{I}_{l}( b_{l}- b_{l-1})\right) \\
\leq& c_{2} \mid x,t\mid^{2k}\left(\mathcal{I}_{0}+ c_{11}(2r+2k)\sum_{l=1}^{N-1} (l+1)^{2r+2k-1}l^{-2r-2kv}\right).
\end{aligned}\]
From the inequality \(l + 1 \leq 2l\), we conclude that \(A_{1}\leq c_{2} \mid x,t\mid^{2k}\left(\mathcal{I}_{0}+ c_{13} \sum_{l=1}^{N-1} l^{2k-2kv-1}\right).\) As a consequence of a series comparison, we have the inequality,
\[\mu \sum_{l=1}^{N-1} l^{\mu-1}\leq N^{\mu} ~~for~~\mu>0~~and~~N\geq2.\]
If \(\mu=2k-2kv>0\) for \(0<v<1\), then we obtain
\[A_{1}\leq c_{2} \mid x,t\mid^{2k}\left(\mathcal{I}_{0}+ c_{14} N^{2k-2kv}\right)\leq c_{2} \mid x,t\mid^{2k}\left(\mathcal{I}_{0}+ c_{14} \mid x,t\mid^{2kv-2k}\right),\]
since \(N\leq \frac{1}{|x,t|}\). If \(|x,t|\) is sufficiently small then \(\mathcal{I}_{0} \leq c_{14} |x,t|^{2kv-2k}\) . Then we have \(A_{1}\leq c_{15} \mid x,t\mid^{2kv} .\)
Combining the estimates for \(A_{1}\) and \(A_{2}\) gives
\[\|\Delta^{k}_{(x,t)} (\mathcal{D}_{2,\alpha}^{r}f)\|_{2,\alpha}=O(\mid x,t\mid^{kv}), as~~|x,t|\rightarrow0.\]
Consequently
\[\Omega_{k}\left(\mathcal{D}_{2,\alpha}^{r} f, \delta\right)=O(\delta^{kv})=O\left(\phi\left(\delta^{k}\right)\right), \delta \longrightarrow 0.\]
Therefore the necessity is proved and the proof of this theorem is completed. \(\square\)
