Approximation by multivariate Baskakov–Durrmeyer operators in Orlicz spaces

Abstract

Employing some properties of multivariate Baskakov–Durrmeyer operators and utilizing modified K-functional and a decomposition technique, the authors obtain the direct theorem and weak type inverse theorem in the Orlicz spaces.

1 Preliminaries

For proceeding smoothly, we recall from [27] some definitions and related results.

A continuous convex function $\Phi (t)$ on $[0,\infty )$ is called a Young function if it satisfies

$$ \lim_{t\to 0^{+}}\frac{\Phi (t)}{t}=0\quad \text{and}\quad \lim_{t \to \infty}\frac{\Phi (t)}{t}=\infty . $$

For a Young function $\Phi (t)$, its complementary Young function is denoted by $\Psi (t)$.

A function $\varphi :[0,\infty )\to \mathbb{R}$ is said to be star-shaped if $\varphi (\nu t)\le \nu \varphi (t)$ for all $\nu \in [0,1]$ and $t\ge 0$. A real function φ defined on a set $S\subset \mathbb{R}^{n}$ is said to be super-additive if $s,t\in S$ implies $s+t\in S$ and $\varphi (s+t)\ge \varphi (s)+\varphi (t)$. See [21, Chap. 16] and [23, Sect. 3.4]. Among convex functions, star-shaped functions, and super-additive functions, the following relations hold true:

• 1) If φ is convex on $[0,\infty )$ with $\varphi (0)\le 0$, then φ is star-shaped;

• 2) If $\varphi :[0,\infty )\to \mathbb{R}$ is star-shaped, then φ is super-additive.

See [21, pp. 650–651, Section B.9], [24, p. 706], [25, pp. 616–617], or [26, Lemma 2.2]. Therefore, a Young function $\Phi (t)$ is both star-shaped and super-additive.

A Young function $\Phi (t)$ is said to satisfy the $\Delta _{2}$-condition, denoted by $\Phi \in \Delta _{2}$, if there exist $t_{0}\geq 0$ and $C\ge 1$ such that $\Phi (2t)\le C\Phi (t)$ for $t\ge t_{0}$.

Throughout the paper we shall use the following standard notations:

$$\begin{aligned}& \mathbb{N} =\{1,2,\ldots \}, \qquad \mathbb{N}_{0} =\{0,1,2,\ldots \},\qquad \binom{n}{\boldsymbol{k}} = \frac{n!}{\boldsymbol{k}!(n- \vert \boldsymbol{k} \vert )!}, \\& \boldsymbol{k} =(k_{1}, k_{2},\ldots , k_{m})\in \mathbb{N}_{0}^{m},\qquad \boldsymbol{k}! =k_{1}!k_{2}!\cdots k_{m}!,\qquad \vert \boldsymbol{k} \vert =\sum_{i=1}^{m} k_{i}, \\& \boldsymbol{x} =(x_{1}, x_{2},\ldots , x_{m})\in \mathbb{R}^{m}, \qquad \boldsymbol{x}^{\boldsymbol{k}} =x_{1}^{k_{1}}x_{2}^{k_{2}} \cdots x_{m}^{k_{m}}, \qquad \vert \boldsymbol{x} \vert =\sum _{i=1}^{m} x_{i}, \\& \sum_{\boldsymbol{k}=0}^{\infty} =\sum _{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{ \infty} \cdots \sum_{k_{m}=0}^{\infty},\qquad D^{\boldsymbol{k}} =D_{1}^{k_{1}}D_{2}^{k_{2}} \cdots D_{m}^{k_{m}}, \qquad D_{i}^{r} = \frac{\partial ^{r}}{\partial x_{i}^{r}}, \end{aligned}$$

and

$$ \mathbb{R}^{m}_{0}=\bigl\{ \boldsymbol{x}=(x_{1}, x_{2},\ldots , x_{m})\in \mathbb{R}^{m}: 0 \le x_{i}< \infty , 1\le i\le m\bigr\} $$

for $m\in \mathbb{N}$ and $r\in \mathbb{N}$.

Let $\Phi (t)$ be a Young function. We define the Orlicz class $L_{\Phi}(\mathbb{R}^{m}_{0})$ as the collection of all Lebesgue measurable functions $f(\boldsymbol{x})$ on $\mathbb{R}^{m}_{0}$ for which

$$ \rho (f,\Phi )= \int _{\mathbb{R}^{m}_{0}}\Phi \bigl( \bigl\vert f(\boldsymbol{x}) \bigr\vert \bigr) \,\mathrm {d}\boldsymbol{x}< \infty . $$

We also define the Orlicz space $L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$ as the set of all Lebesgue measurable functions $f(\boldsymbol{x})$ on $\mathbb{R}^{m}_{0}$, such that $\int _{\mathbb{R}^{m}_{0}}\Phi (|\alpha f(\boldsymbol{x})|)\,\mathrm {d}\boldsymbol{x}<\infty$ for some $\alpha >0$. The Orlicz space is a Banach space under the Luxemburg norm

$$ \Vert f \Vert _{(\Phi )}=\inf_{\lambda >0} \biggl\{ \lambda : \rho \biggl( \frac{f}{\lambda},\Phi \biggr)\le 1 \biggr\} . $$

The Orlicz norm, an equivalence of the Luxemburg norm on $L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$, is given by

$$ \Vert f \Vert _{\Phi}= \sup_{\rho (g,\Psi )\le 1} \biggl\vert \int _{\mathbb{R}^{m}_{0}}f( \boldsymbol{x})g(\boldsymbol{x})\,\mathrm {d}\boldsymbol{x} \biggr\vert $$

and satisfies

$$ \Vert f \Vert _{(\Phi )}\le \Vert f \Vert _{\Phi}\le 2 \Vert f \Vert _{(\Phi )}. $$

(1)

If $\Phi (u)=\frac{u^{p}}{p}$ for $1< p<\infty$, then the complementary function becomes $\Psi (u)=\frac{|u|^{q}}{q}$ with $\frac{1}{p}+\frac{1}{q}=1$, and then $L_{\Phi}^{*}(\mathbb{R}^{m}_{0})=L_{p}(\mathbb{R}^{m}_{0})$. So the Orlicz spaces $L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$ are more general than the classical $L_{p}(\mathbb{R}^{m}_{0})$ spaces which are composed of measurable functions $f(\boldsymbol{x})$ such that $|f(\boldsymbol{x})|^{p}$ are integrable.

Throughout this paper we use C to denote a constant independent of n and x, which may be not necessarily the same in different cases.

For $\boldsymbol{x}\in \mathbb{R}^{m}_{0}$, we introduce weight functions

$$ \varphi (x)=\sqrt{x(1+x)} $$

for $m=1$ and

$$ \varphi _{i}(\boldsymbol{x})=\sqrt{x_{i}\bigl(1+ \vert \boldsymbol{x} \vert \bigr)} $$

for $m>1$ and $1\le i\le m$. We also define the weighted Sobolev space

$$ W_{\varphi}^{r,\Phi}\bigl(\mathbb{R}^{m}_{0} \bigr)= \bigl\{ f\in L_{\Phi}^{*}\bigl( \mathbb{R}^{m}_{0}\bigr): D^{\boldsymbol{k}}f\in A.C.loc \bigl( \overset{\circ}{\mathbb{R}^{m}_{0}} \bigr),\, \varphi _{i}^{r} D_{i}^{r} f \in L_{\Phi}^{*}\bigl(\mathbb{R}^{m}_{0} \bigr) \bigr\} , $$

where $|\boldsymbol{k}|\le r$ and ${\overset{\circ}{\mathbb{R}^{m}_{0}}}$ is the interior of $\mathbb{R}^{m}_{0}$.

The modified Peetre K-functionals are defined by

$$ \bar{K}_{r,\varphi} \bigl(f,t^{r} \bigr)_{\Phi}=\inf \Biggl\{ \Vert f-g \Vert _{ \Phi}+t^{r}\sum _{i=1}^{m} \bigl\Vert \varphi _{i}^{r} D_{i}^{r} g \bigr\Vert _{\Phi }+t^{2r} \sum_{i=1}^{m} \bigl\Vert D_{i}^{r} g \bigr\Vert _{\Phi}: g\in W_{\varphi}^{r,\Phi}\bigl( \mathbb{R}^{m}_{0}\bigr) \Biggr\} $$

and

$$ \tilde{K}_{r,\varphi} \bigl(f,t^{r} \bigr)_{\Phi}= \inf \Bigl\{ \Vert f-g \Vert _{ \Phi}+t^{r}\max _{1\leq i\leq m} \bigl\Vert \varphi _{i}^{r} D_{i}^{r} g \bigr\Vert _{ \Phi}: g\in W_{\varphi}^{r,\Phi}\bigl(\mathbb{R}^{m}_{0} \bigr) \Bigr\} $$

for $t>0$.

For any vector $\boldsymbol{e}\in \mathbb{R}^{m}$, we write

$$ \Delta _{h\boldsymbol{e}}^{r}f(\boldsymbol{x})= \textstyle\begin{cases} \sum_{i=0}^{r}\binom{r}{i}(-1)^{i} f(\boldsymbol{x}+ih\boldsymbol{e}), & \boldsymbol{x},\boldsymbol{x}+rh\boldsymbol{e}\in \mathbb{R}^{m}_{0}, \\ 0, & \text{otherwise} \end{cases} $$

for the rth forward difference of a function f in the direction of e. We define the modulus of smoothness of $f\in L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$ as

$$ \omega _{r,\varphi}(f,t)_{\Phi}=\sup_{0< h\le t} \sum_{i=1}^{m} \bigl\Vert \Delta _{h\varphi _{i}\boldsymbol{e}_{i}}^{r}f \bigr\Vert _{\Phi}. $$

2 Motivations and main results

Between the modulus of smoothness and the K-functional there exists the following equivalent theorems.

Theorem A

([13])

Let $f\in L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$ and $r\in \mathbb{N}$. Then there exist some constants C and $t_{0}$ such that

$$ \frac{\omega _{r,\varphi}(f,t)_{\Phi}}{C}\le \bar{K}_{r,\varphi} \bigl(f,t^{r}\bigr)_{ \Phi}\le C\omega _{r,\varphi}(f,t)_{\Phi}, \quad 0< t\le t_{0}. $$

(2)

Theorem B

([31])

Let $f\in L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$ and $r\in \mathbb{N}$. Then there exist some constants C and $t_{0}$ such that

$$ \frac{\omega _{r,\varphi}(f,t)_{\Phi}}{C}\le \tilde{K}_{r,\varphi} \bigl(f,t^{r}\bigr)_{ \Phi}\le C\omega _{r,\varphi}(f,t)_{\Phi}, \quad 0< t\le t_{0}. $$

(3)

Let

$$ p_{n,k}(x)=\binom{n+k-1}{k}\frac{x^{k}}{(1+x)^{n+k}},\quad x\in [0, \infty ), n\in \mathbb{N}. $$

The well-known Baskakov operators were defined in [2] as

$$ B_{n}(f,x)=\sum_{k=0}^{\infty}p_{n,k}(x)f \biggl(\frac{k}{n} \biggr). $$

These operators can be used to approximate any function f defined on $[0,\infty )$. For $f\in L_{p}[0,\infty )$ and $1\leq p<\infty$, the Baskakov–Durrmeyer operators were defined in [17] as

$$ V_{n,1}(f,x)=\sum_{k=0}^{\infty}p_{n,k}(x) (n-1) \int _{0}^{\infty}p_{n,k}(t)f(t) \,\mathrm {d}t,\quad x\in [0,\infty ). $$

For a function f defined on $\mathbb{R}^{m}_{0}$, the multivariate Baskakov operators were defined in [5] as

$$ B_{n,m}(f,\boldsymbol{x})=\sum_{\boldsymbol{k}=0}^{\infty}f \biggl( \frac{\boldsymbol{k}}{n} \biggr)p_{n,\boldsymbol{k}}(\boldsymbol{x}), $$

where

$$ p_{n,\boldsymbol{k}}(\boldsymbol{x})= \binom{n+ \vert \boldsymbol{k} \vert -1}{\boldsymbol{k}} \frac{\boldsymbol{x}^{k}}{(1+ \vert \boldsymbol{x} \vert )^{n+ \vert \boldsymbol{k} \vert }}. $$

The multivariate Baskakov–Durrmeyer operators were defined in [4] as

$$ V_{n,m}(f,\boldsymbol{x})=\sum_{\boldsymbol{k}=0}^{\infty}p_{n, \boldsymbol{k}}( \boldsymbol{x})Q_{n,\boldsymbol{k},m}(f),\quad f\in L_{p}\bigl( \mathbb{R}^{m}_{0}\bigr), $$

where

$$ Q_{n,\boldsymbol{k},m}(f)= \frac{\int _{\mathbb{R}^{m}_{0}} p_{n,\boldsymbol{k}}(\boldsymbol{u}) f(\boldsymbol{u})\,\mathrm {d}\boldsymbol{u}}{\int _{\mathbb{R}^{m}_{0}} p_{n,\boldsymbol{k}}(\boldsymbol{u})\,\mathrm {d}\boldsymbol{u}} =\prod _{k=1}^{m}(n-k) \int _{\mathbb{R}^{m}_{0}} p_{n,\boldsymbol{k}}( \boldsymbol{u})f( \boldsymbol{u})\,\mathrm {d}\boldsymbol{u}. $$

There are many approximation results about one variable operator of the Baskakov type in $C[0,\infty )$ or $L_{p}[0,\infty )$, see [1, 2, 7–9, 15, 17–19, 29, 30]. But there are few approximation results about multivariate Baskakov type operators (see [4, 5, 13, 22]) or multivariate Durrmeyer type operators (see [3, 20]).

In the paper [4], Cao and An obtained the strong direct inequality

$$ \bigl\Vert V_{n,m}(f)-f \bigr\Vert _{p}\leq C \biggl(\omega _{2,\varphi}\biggl(f, \frac{1}{\sqrt{n}} \biggr)_{p}+\frac{1}{n} \Vert f \Vert _{p} \biggr) $$

in $L_{p}(\mathbb{R}^{m}_{0})$. In [10–12, 14–16], we obtained approximation properties for positive and linear operators in Orlicz space. In particular, we acquired the direct theorem of multivariate Baskakov–Kantorovich operators in Orlicz space in [13].

In this paper, we will discover not only the direct theorem, but also the weak type inverse theorem for the multivariate Baskakov–Durrmeyer operators $V_{n,m}(f,\boldsymbol{x})$.

Our main results can be stated in the following two theorems.

Theorem 1

(Direct theorem)

Let $f\in L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$, $\Psi \in \Delta _{2}$, and $n>m$ for $n,m\in \mathbb{N}$. Then

$$ \bigl\Vert V_{n,m}(f)-f \bigr\Vert _{\Phi}\le C \biggl( \omega _{2,\varphi} \biggl(f, \frac{1}{\sqrt{n}} \biggr)_{\Phi} +\frac{ \Vert f \Vert _{\Phi}}{n} \biggr). $$

Theorem 2

(Weak type inverse theorem)

Let $f\in L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$ and $n>m$ for $n,m\in \mathbb{N}$. Then

$$ \omega _{2,\varphi} \biggl(f,\frac{1}{n} \biggr)_{\Phi} \leq \frac{C}{n} \sum_{k=1}^{n} \bigl\Vert V_{k,m}(f)-f \bigr\Vert _{\Phi}. $$

Remark 1

Theorem 1 is a generalization of [4, Theorem 2.2].

3 Proof of direct theorem

In order to prove the direct theorem, we need several lemmas.

Lemma 1

Let $f\in L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$ and $n>m$ for $n,m\in \mathbb{N}$. Then

$$ \bigl\Vert V_{n,m}(f) \bigr\Vert _{\Phi}\leq 2 \Vert f \Vert _{\Phi}. $$

Proof

Employing the decomposition formula

$$\begin{aligned} V_{n,m}(f,\boldsymbol{x})&=\sum_{k_{1}=0}^{\infty}p_{n,k_{1}}(x_{1}) (n-1) \int _{0}^{\infty }p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{} \times \sum_{k_{2}=0}^{\infty}p_{n+k_{1},k_{2}} \biggl( \frac{x_{2}}{1+x_{1}} \biggr) (n-2) \int _{0}^{\infty }p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad{} \times \sum_{k_{m}=0}^{\infty} p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl(\frac{x_{m}}{1+\sum_{\ell =1}^{m-1}x_{\ell}} \biggr) (n-m) \\ &\quad{} \times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr)f(u_{1}, \ldots ,u_{m}) \,\mathrm {d}u_{m} \end{aligned}$$

(4)

and Jensen’s inequality, we obtain

$$\begin{aligned} \bigl\Vert V_{n,m}(f) \bigr\Vert _{(\Phi )} &=\inf_{\lambda >0} \Biggl\{ \lambda : \int _{ \mathbb{R}^{m}_{0}} \Phi \Biggl(\frac {1}{\lambda} \Biggl\vert \sum_{k_{1}=0}^{ \infty}p_{n,k_{1}}(x_{1}) (n-1) \int _{0}^{\infty }p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times \sum_{k_{2}=0}^{\infty}p_{n+k_{1},k_{2}} \biggl( \frac{x_{2}}{1+x_{1}} \biggr) (n-2) \int _{0}^{\infty }p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad{}\times \sum_{k_{m}=0}^{\infty} p_{n+\sum _{\ell =1}^{m-1}k_{ \ell},k_{m}} \biggl(\frac{x_{m}}{1+\sum_{\ell =1}^{m-1}x_{\ell}} \biggr) (n-m) \\ &\quad{}\times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \\ &\quad{}\times f(u_{1},\ldots ,u_{m})\,\mathrm {d}u_{m} \Biggr\vert \Biggr)\,\mathrm {d}\boldsymbol{x}\leq 1 \Biggr\} \\ &\leq \inf_{\lambda >0} \Biggl\{ \lambda : \int _{\mathbb{R}^{m}_{0}} \sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) \sum_{k_{2}=0}^{\infty }p_{n+k_{1},k_{2}} \biggl(\frac{x_{2}}{1+x_{1}} \biggr)\cdots \\ &\quad{}\times \sum_{k_{m}=0}^{\infty} p_{n+\sum _{\ell =1}^{m-1}k_{ \ell},k_{m}} \biggl(\frac{x_{m}}{1+\sum_{\ell =1}^{m-1}x_{\ell}} \biggr) \\ &\quad{}\times \Phi \Biggl(\prod_{k=1}^{m}(n-k) \int _{0}^{\infty}p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \int _{0}^{\infty} p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad{}\times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \frac{ \vert f(u_{1},u_{2},\ldots ,u_{m}) \vert }{\lambda} \,\mathrm {d}u_{m} \Biggr)\,\mathrm {d}\boldsymbol{x}\leq 1 \Biggr\} \\ &=\inf_{\lambda >0} \Biggl\{ \lambda : \sum_{k_{1}=0}^{\infty} \int _{0}^{ \infty }p_{n,k_{1}}(x_{1}) (1+x_{1})\,\mathrm {d}x_{1} \sum _{k_{2}=0}^{\infty} \int _{0}^{\infty }p_{n+k_{1},k_{2}} \biggl( \frac{x_{2}}{1+x_{1}} \biggr) (1+x_{1} \\ &\quad{}+x_{2})\,\mathrm {d}\biggl(\frac{x_{2}}{1+x_{1}} \biggr)\cdots \sum_{k_{m-1}=0}^{ \infty} \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-2}k_{\ell},k_{m-1}} \biggl( \frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr) \\ &\quad{}\times \Biggl(1+\sum_{k=1}^{m-1}x_{k} \Biggr)\,\mathrm {d}\biggl( \frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr) \\ &\quad{}\times \sum_{k_{m}=0}^{\infty} \int _{0}^{\infty }p_{n+\sum _{ \ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{x_{m}}{1+\sum_{\ell =1}^{m-1}x_{\ell}} \biggr) \,\mathrm {d}\biggl( \frac{x_{m}}{1+\sum_{\ell =1}^{m-1}x_{\ell}} \biggr) \\ &\quad{}\times \Phi \Biggl(\prod_{k=1}^{m}(n-k) \int _{0}^{\infty}p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \int _{0}^{\infty} p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad{}\times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \frac{ \vert f(u_{1},u_{2},\ldots ,u_{m}) \vert }{\lambda} \,\mathrm {d}u_{m} \Biggr)\leq 1 \Biggr\} \\ &=\inf_{\lambda >0} \Biggl\{ \lambda : \sum_{k_{1}=0}^{\infty}\sum _{k_{2}=0}^{ \infty}\cdots \sum _{k_{m}=0}^{\infty} \frac{1}{\prod_{\ell =1}^{m}(n-\ell )} \Phi \Biggl(\prod _{\ell =1}^{m}(n- \ell ) \int _{0}^{\infty}p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times \int _{0}^{\infty}p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \int _{0}^{\infty }p_{n+ \sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \\ &\quad{}\times \frac{ \vert f(u_{1},u_{2},\ldots ,u_{m}) \vert }{\lambda} \,\mathrm {d}u_{m} \Biggr)\leq 1 \Biggr\} \\ &=\inf_{\lambda >0} \Biggl\{ \lambda : \sum_{k_{1}=0}^{\infty}\cdots \sum _{k_{m}=0}^{\infty}\frac{1}{\prod_{\ell =1}^{m}(n-\ell )} \Phi \Biggl(\prod _{\ell =1}^{m}(n-\ell ) \int _{0}^{\infty}p_{n,k_{1}}(u_{1}) (1+u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times \int _{0}^{\infty} p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr) (1+u_{1}+u_{2})\,\mathrm {d}\biggl( \frac{u_{2}}{1+u_{1}} \biggr)\cdots \int _{0}^{\infty } \Biggl(1+\sum _{k=1}^{m-1}u_{k} \Biggr) \\ &\quad{}\times p_{n+\sum _{\ell =1}^{m-2}k_{\ell},k_{m-1}} \biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr) \,\mathrm {d}\biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr) \\ &\quad{}\times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \\ &\quad{}\times \frac{ \vert f(u_{1},u_{2},\ldots ,u_{m}) \vert }{\lambda} \,\mathrm {d}\biggl(\frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \Biggr) \leq 1 \Biggr\} \\ &\leq \inf_{\lambda >0} \Biggl\{ \lambda :\sum_{k_{1}=0}^{\infty} \cdots \sum _{k_{m}=0}^{\infty}\frac{1}{(n-1)\cdots (n-m-1)} \int _{0}^{ \infty }p_{n-m+1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times (n+k_{1}-m+1) \int _{0}^{\infty }p_{n+k_{1}-m+2,k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}\biggl(\frac{u_{2}}{1+u_{1}} \biggr)\cdots \\ &\quad{}\times \Biggl(n-1+\sum_{\ell =1}^{m-1}k_{\ell} \Biggr) \int _{0}^{ \infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \\ &\quad{}\times \Phi \biggl( \frac{ \vert f(u_{1},u_{2},\ldots ,u_{m}) \vert }{\lambda} \biggr)\,\mathrm {d}\biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr)\leq 1 \Biggr\} \\ &=\inf_{\lambda >0} \Biggl\{ \lambda : \sum_{k_{1}=0}^{\infty}\cdots \sum _{k_{m}=0}^{\infty} \frac{n+k_{1}-m+1}{n-1} \frac{n+k_{1}+k_{2}-m+2}{n-2}\cdots \\ &\quad{}\times \frac{n+\sum_{\ell =1}^{m-1}k_{\ell}-1}{n-m+1} \int _{0}^{ \infty }p_{n-m+1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \int _{0}^{\infty }p_{n+k_{1}-m+2,k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr) \\ &\quad{}\times \,\mathrm {d}\biggl(\frac{u_{2}}{1+u_{1}} \biggr)\cdots \int _{0}^{ \infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \\ &\quad{}\times \Phi \biggl( \frac{ \vert f(u_{1},u_{2},\ldots ,u_{m}) \vert }{\lambda} \biggr)\,\mathrm {d}\biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr)\leq 1 \Biggr\} \\ &=\inf_{\lambda >0} \biggl\{ \lambda : \int _{0}^{\infty} \,\mathrm {d}u_{1} \int _{0}^{ \infty} \,\mathrm {d}u_{2}\cdots \int _{0}^{\infty }\Phi \biggl( \frac{ \vert f(u_{1},u_{2},\ldots ,u_{m}) \vert }{\lambda} \biggr)\,\mathrm {d}u_{m}\leq 1 \biggr\} \\ &= \Vert f \Vert _{(\Phi )}. \end{aligned}$$

By the double Inequality (1), we complete the proof of Lemma 1. □

Lemma 2

Let $f\in L_{\Phi}^{*}(\mathbb{R}^{2}_{0})$, $\Psi \in \Delta _{2}$, and $n>2$. Then

$$ \bigl\Vert V_{n,2}(f)-f \bigr\Vert _{\Phi}\le \frac{C}{n} \Biggl( \Vert f \Vert _{\Phi}+\sum _{i=1}^{2} \bigl\Vert \varphi _{i}^{2}D_{i}^{2}f \bigr\Vert _{\Phi} \Biggr). $$

Proof

Let

$$ z=\frac{x_{2}}{1+x_{1}} \quad \text{and}\quad g_{u_{1}}(t)=f \bigl(u_{1},(1+u_{1})t \bigr) $$

for $0\le t<\infty$. Utilizing the decomposition formula

$$\begin{aligned} V_{n,2}(f,\boldsymbol{x})&=\sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times \sum_{k_{2}=0}^{\infty }p_{n+k_{1},k_{2}} \biggl( \frac{x_{2}}{1+x_{1}} \biggr) (n+k_{1}-1) \\ &\quad{}\times \int _{0}^{\infty }p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr) f \biggl(u_{1},(1+u_{1}) \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}\biggl(\frac{u_{2}}{1+u_{1}} \biggr) \end{aligned}$$

yields

$$\begin{aligned} V_{n,2}(f,\boldsymbol{x})-f(\boldsymbol{x}) &=\sum _{k_{1}=0}^{ \infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \bigl(V_{n+k_{1},1} (g_{u_{1}},z) \\ &\quad{}-g_{u_{1}}(z) \bigr)\,\mathrm {d}u_{1}+V_{n,1}^{*} \bigl(h(\cdot ),x_{1} \bigr) -h(x_{1}), \end{aligned}$$

(5)

where

$$ h(u_{1})=h(u_{1},\boldsymbol{x})\triangleq f \biggl(u_{1}, \frac{(1+u_{1})x_{2}}{1+x_{1}} \biggr),\quad 0\leq u_{1}< \infty , $$

the notation ≜ means “define”, and

$$ V_{n,1}^{*}(g,x)=\sum_{i=0}^{\infty }p_{n,i}(x) (n-2) \int _{0}^{ \infty }p_{n-1,i}(t)g(t)\,\mathrm {d}t. $$

Now we start out to estimate

$$ J_{1}=\sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{ \infty }p_{n-1,k_{1}}(u_{1}) \bigl(V_{n+k_{1},1} (g_{u_{1}},z) -g_{u_{1}}(z) \bigr) \,\mathrm {d}u_{1}. $$

From [17], we obtain

$$ \bigl\vert V_{n,1}(f,x)-f(x) \bigr\vert \leq \frac {C}{n}\bigl[ \bigl\vert f(x) \bigr\vert + \bigl\vert \varphi ^{2}(x)f''(x) \bigr\vert \bigr]. $$

(6)

From the Inequality (6), Jensen’s inequality, and the convexity of $\Phi (t)$, it follows

$$\begin{aligned} & \int _{0}^{\infty} \int _{0}^{\infty}\Phi \biggl(\frac{1}{\lambda} \vert J_{1} \vert \biggr)\,\mathrm {d}x_{1}\,\mathrm {d}x_{2} \\ &\quad = \int _{0}^{\infty} \int _{0}^{\infty}\Phi \Biggl(\frac{1}{\lambda} \Biggl\vert \sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \\ &\quad \quad{}\times \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \bigl(V_{n+k_{1},1} (g_{u_{1}},z)-g_{u_{1}}(z) \bigr) \,\mathrm {d}u_{1} \Biggr\vert \Biggr)\,\mathrm {d}x_{1}\,\mathrm {d}x_{2} \\ &\quad \le \int _{0}^{\infty} \int _{0}^{\infty}\sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \\ &\quad \quad{}\times \Phi \biggl(\frac{1}{\lambda} \bigl\vert V_{n+k_{1},1} (g_{u_{1}},z)-g_{u_{1}}(z) \bigr\vert \biggr)\,\mathrm {d}u_{1}\,\mathrm {d}x_{1}\,\mathrm {d}x_{2} \\ &\quad \leq \int _{0}^{\infty} \int _{0}^{\infty}\sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \\ &\quad \quad{}\times \Phi \biggl(\frac{C}{\lambda (n+k_{1})} \bigl( \bigl\vert g_{u_{1}}(z) \bigr\vert + \bigl\vert \varphi ^{2}(z)g''_{u_{1}}(z) \bigr\vert \bigr) \biggr)\,\mathrm {d}u_{1}\,\mathrm {d}x_{1} \,\mathrm {d}x_{2} \\ &\quad =\sum_{k_{1}=0}^{\infty} \int _{0}^{\infty }p_{n,k_{1}}(x_{1}) (1+x_{1}) \,\mathrm {d}x_{1}(n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad \quad{}\times \int _{0}^{\infty}\Phi \biggl( \frac{C}{\lambda (n+k_{1})} \bigl( \bigl\vert g_{u_{1}}(z) \bigr\vert + \bigl\vert \varphi ^{2}(z)g''_{u_{1}}(z) \bigr\vert \bigr) \biggr)\,\mathrm {d}z \\ &\quad =\sum_{k_{1}=0}^{\infty} \frac{n+k_{1}-1}{n-1} \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad \quad{}\times \int _{0}^{\infty}\Phi \biggl( \frac{C}{\lambda (n+k_{1})} \bigl( \bigl\vert g_{u_{1}}(z) \bigr\vert + \bigl\vert \varphi ^{2}(z)g''_{u_{1}}(z) \bigr\vert \bigr) \biggr)\,\mathrm {d}z \\ &\quad \leq \sum_{k_{1}=0}^{\infty} \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad \quad{}\times \int _{0}^{\infty}\Phi \biggl( \frac{C(n+k_{1}-1)}{\lambda (n-1)(n+k_{1})} \bigl( \bigl\vert g_{u_{1}}(z) \bigr\vert + \bigl\vert \varphi ^{2}(z)g''_{u_{1}}(z) \bigr\vert \bigr) \biggr)\,\mathrm {d}z \\ &\quad \leq \sum_{k_{1}=0}^{\infty} \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \int _{0}^{\infty}\Phi \biggl(\frac{C}{n\lambda} \bigl( \bigl\vert g_{u_{1}}(z) \bigr\vert + \bigl\vert \varphi ^{2}(z)g''_{u_{1}}(z) \bigr\vert \bigr) \biggr)\,\mathrm {d}z \\ &\quad = \int _{0}^{\infty} \,\mathrm {d}u_{1} \int _{0}^{\infty}\Phi \biggl( \frac{C}{n\lambda} \bigl( \bigl\vert g_{u_{1}}(z) \bigr\vert + \bigl\vert \varphi ^{2}(z)g''_{u_{1}}(z) \bigr\vert \bigr) \biggr)\,\mathrm {d}z \\ &\quad \leq \frac{1}{2} \int _{0}^{\infty} \,\mathrm {d}u_{1} \int _{0}^{\infty}\Phi \biggl(\frac{C}{n\lambda} \bigl\vert g_{u_{1}}(z) \bigr\vert \biggr)\,\mathrm {d}z \\ &\quad \quad{}+\frac{1}{2} \int _{0}^{\infty} \,\mathrm {d}u_{1} \int _{0}^{\infty}\Phi \biggl(\frac{C}{n\lambda} \bigl\vert \varphi ^{2}(z)g''_{u_{1}}(z) \bigr\vert \biggr)\,\mathrm {d}z. \end{aligned}$$

On the other hand, by definition, we can deduce

$$ \varphi ^{2}(t)g''_{u_{1}}(t)=t(1+t) (1+u_{1})^{2}D_{2}^{2}f \bigl(u_{1},(1+u_{1})t\bigr)= \bigl(\varphi _{2}^{2}D_{2}^{2}f \bigr) \bigl(u_{1},(1+u_{1})t \bigr) $$

and

$$ \begin{aligned} & \int _{0}^{\infty} \int _{0}^{\infty}\Phi \biggl( \frac{1}{\lambda} \vert J_{1} \vert \biggr)\,\mathrm {d}x_{1}\,\mathrm {d}x_{2} \\ &\quad \leq \frac{1}{2} \int _{0}^{\infty}\frac{\,\mathrm {d}u_{1}}{1+u_{1}} \int _{0}^{ \infty }\Phi \biggl(\frac{C}{n\lambda} \biggl\vert f \biggl(u_{1}, \frac{(1+u_{1})x_{2}}{1+x_{1}} \biggr) \biggr\vert \biggr) \,\mathrm {d}\biggl( \frac{(1+u_{1})x_{2}}{1+x_{1}} \biggr) \\ &\quad \quad{}+\frac{1}{2} \int _{0}^{\infty}\frac{\,\mathrm {d}u_{1}}{1+u_{1}} \int _{0}^{ \infty }\Phi \biggl(\frac{C}{n\lambda} \biggl\vert \bigl(\varphi _{2}^{2}D_{2}^{2}f \bigr) \biggl(u_{1},\frac{(1+u_{1})x_{2}}{1+x_{1}} \biggr) \biggr\vert \biggr)\,\mathrm {d}\biggl(\frac{(1+u_{1})x_{2}}{1+x_{1}} \biggr) \\ &\quad \leq \frac{1}{2} \int _{0}^{\infty} \,\mathrm {d}u_{1} \int _{0}^{\infty }\Phi \biggl(\frac{C}{n\lambda} \bigl\vert f(u_{1},u_{2}) \bigr\vert \biggr)\,\mathrm {d}u_{2} \\ &\quad \quad{}+\frac{1}{2} \int _{0}^{\infty} \,\mathrm {d}u_{1} \int _{0}^{\infty}\Phi \biggl(\frac {C}{n\lambda} \bigl\vert \bigl(\varphi _{2}^{2}D_{2}^{2}f \bigr) (u_{1},u_{2}) \bigr\vert \biggr)\,\mathrm {d}u_{2}. \end{aligned} $$

(7)

To estimate the second term $J_{2}=V_{n,1}^{*} (h(\cdot ),x_{1} ) -h(x_{1})$, we use a similar method as estimating (6) and acquire

$$ \bigl\vert V_{n,1}^{*}(f,x)-f(x) \bigr\vert \leq \frac {C}{n}\bigl[ \bigl\vert f(x) \bigr\vert + \bigl\vert \varphi ^{2}(x)f''(x) \bigr\vert \bigr]. $$

(8)

By the Inequality (8) and the convexity of $\Phi (t)$, we arrive at

$$\begin{aligned} \int _{\mathbb{R}^{2}_{0}}\Phi \biggl(\frac {1}{\lambda} \vert J_{2} \vert \biggr)\,\mathrm {d}\boldsymbol{x} &= \int _{0}^{\infty} \int _{0}^{\infty}\Phi \biggl(\frac {1}{\lambda} \bigl\vert V_{n,1}^{*}\bigl(h(\cdot ),x_{1} \bigr)-h(x_{1}) \bigr\vert \biggr)\,\mathrm {d}x_{1}\,\mathrm {d}x_{2} \\ &\le \int _{0}^{\infty} \int _{0}^{\infty}\Phi ( \frac {C}{n\lambda}\bigl( \bigl\vert h(x_{1}) \bigr\vert + \bigl\vert \varphi ^{2}(x_{1})h''(x_{1}) \bigr\vert \bigr) \,\mathrm {d}x_{1}\,\mathrm {d}x_{2} \\ &\le \frac{1}{2} \int _{0}^{\infty} \int _{0}^{\infty}\Phi \biggl( \frac {C}{n\lambda} \bigl\vert h(x_{1}) \bigr\vert \biggr)\,\mathrm {d}x_{1} \,\mathrm {d}x_{2} \\ &\quad{}+\frac{1}{2} \int _{0}^{\infty} \int _{0}^{\infty}\Phi \biggl( \frac {C}{n\lambda} \bigl\vert \varphi ^{2}(x_{1})h''(x_{1}) \bigr\vert \biggr)\,\mathrm {d}x_{1} \,\mathrm {d}x_{2}. \end{aligned}$$

When denoting $\varphi _{12}(\boldsymbol{x})=\varphi _{21}(\boldsymbol{x}) \triangleq \sqrt{x_{1}x_{2}}$, $D_{12}^{2}=\frac{\partial ^{2}}{\partial x_{1}\partial x_{2}}$, and $D_{21}^{2}=\frac{\partial ^{2}}{\partial x_{2}\partial x_{1}}$, we can write

$$\begin{aligned} \bigl\vert \varphi ^{2}(u)h''(u) \bigr\vert &= \biggl\vert u(1+u) \biggl[D_{1}^{2}f+ \frac{x_{2}}{1+x_{1}}D_{12}^{2}f+\frac{x_{2}}{1+x_{1}}D_{21}^{2}f \\ &\quad{}+\frac{x_{2}^{2}}{(1+x_{1})^{2}}D_{22}^{2}f \biggr] \biggl(u,(1+u) \frac{x_{2}}{1+x_{1}} \biggr) \biggr\vert \\ &= \biggl\vert \biggl(\frac{1+x_{1}}{1+x_{1}+x_{2}}\varphi _{1}^{2}D_{1}^{2}f+ \varphi _{12}^{2}D_{12}^{2}f+ \varphi _{21}^{2}D_{21}^{2}f \\ &\quad{}+\frac {u}{1+u}\frac{x_{2}}{1+x_{1}+x_{2}}\varphi _{2}^{2}D_{2}^{2}f \biggr) \biggl(u,(1+u)\frac{x_{2}}{1+x_{1}} \biggr) \biggr\vert . \end{aligned}$$

By virtue of the facts that $\varphi _{12}(\boldsymbol{x})$ is not bigger than $\varphi _{1}(\boldsymbol{x})$ or $\varphi _{2}(\boldsymbol{x})$ and that

$$ \bigl\vert D_{12}^{2} f(\boldsymbol{x}) \bigr\vert \le \sup \bigl\{ \bigl\vert D_{1}^{2} f( \boldsymbol{x}) \bigr\vert , \bigl\vert D_{2}^{2} f( \boldsymbol{x}) \bigr\vert \bigr\} $$

in [6, Lemma 2.1], we obtain

$$ \int _{\mathbb{R}^{2}_{0}}\Phi \biggl(\frac{C}{\lambda n}\bigl(\varphi ^{2} \bigl\vert h'' \bigr\vert ,x_{1}\bigr) \biggr)\,\mathrm {d}\boldsymbol{x} \le \int _{\mathbb{R}^{2}_{0}}\Phi \Biggl( \frac {C}{\lambda n}\sum _{i=1}^{2} \bigl\vert \bigl(\varphi _{i}^{2}D_{i}^{2}f\bigr) ( \boldsymbol{x}) \bigr\vert \Biggr)\,\mathrm {d}\boldsymbol{x} $$

and

$$\begin{aligned} \int _{\mathbb{R}^{2}_{0}}\Phi \biggl(\frac {1}{\lambda} \vert J_{2} \vert \biggr)\,\mathrm {d}\boldsymbol{x} &\le \frac{1}{2} \int _{\mathbb{R}^{2}_{0}} \Phi \biggl(\frac {C}{\lambda n} \bigl\vert f( \boldsymbol{x}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{x} \\ &\quad{}+\frac{1}{2} \int _{\mathbb{R}^{2}_{0}}\Phi \Biggl( \frac {C}{\lambda n}\sum _{i=1}^{2} \bigl\vert \bigl(\varphi _{i}^{2}D_{i}^{2}f\bigr) ( \boldsymbol{x}) \bigr\vert \Biggr)\,\mathrm {d}\boldsymbol{x}. \end{aligned}$$

Combining the above inequality with (5) and (7) and paying attention to computation formulas of norm and the double Inequality (1) yield

$$ \bigl\Vert V_{n,2}(f)-f \bigr\Vert _{\Phi }\le \Vert J_{1} \Vert _{\Phi}+ \Vert J_{2} \Vert _{\Phi }\le \frac{C}{n} \Biggl( \Vert f \Vert _{\Phi}+\sum_{i=1}^{2} \bigl\Vert \varphi _{i}^{2}D_{i}^{2}f \bigr\Vert _{\Phi} \Biggr). $$

The proof of Lemma 2 is complete. □

Proof of Theorem 1

Our proof is based on induction on the dimension m and on a decomposition for the Baskakov–Durrmeyer operator.

For $m\geq 1$, the proof of Theorem 1 follows from combining Lemmas 1 and 2 with the estimates

$$ \bigl\Vert V_{n,m}(f)-f \bigr\Vert _{\Phi }\le C \textstyle\begin{cases} \Vert f \Vert _{\Phi}, & f\in L_{\Phi}^{*}(\mathbb{R}^{m}_{0}), \\ \frac{1}{n}\sum_{i=1}^{m} \Vert \varphi _{i}^{2}D_{i}^{2}f \Vert _{\Phi}+ \Vert f \Vert _{ \Phi}, & f\in W_{\varphi}^{2,\Phi}(\mathbb{R}^{m}_{0}). \end{cases} $$

(9)

The first estimate in (9) can be derived from Lemma 1. By Lemma 2, the second estimate in (9) is valid for $m=1,2$. If the second estimate in (9) is valid for $m=r\geq 2$, that is

$$ \bigl\Vert V_{n,r}(f)-f \bigr\Vert _{\Phi}\leq \frac{C}{n}\sum_{i=1}^{r} \bigl\Vert \varphi _{i}^{2}D_{i}^{2}f \bigr\Vert _{\Phi}+ \Vert f \Vert _{\Phi}, $$

(10)

then we have to further verify its validity for $m=r+1$.

Let

$$\begin{aligned}& \boldsymbol{x}^{*} =(x_{2},x_{3},\ldots ,x_{r+1}), \qquad \boldsymbol{x}=\bigl(x_{1}, \boldsymbol{x}^{*}\bigr)\in \mathbb{R}^{r+1}_{0}, \\& \boldsymbol{k}^{*} =(k_{2},k_{3},\ldots k_{r+1}),\qquad \boldsymbol{k}=\bigl(k_{1}, \boldsymbol{x}^{*}\bigr)\in \mathbb{N}^{r+1}_{0}, \\& \boldsymbol{z} =\frac{\boldsymbol{x}^{*}}{1+x_{1}}= \biggl( \frac{x_{2}}{1+x_{1}},\ldots , \frac{x_{r+1}}{1+x_{1}} \biggr)=(z_{1}, \ldots ,z_{r}), \\& \boldsymbol{t}^{*} =\frac{\boldsymbol{u}^{*}}{1+u_{1}}= \biggl( \frac{u_{2}}{1+u_{1}}, \ldots ,\frac{u_{r+1}}{1+u_{1}} \biggr)=(t_{1}, \ldots ,t_{r}). \end{aligned}$$

We claim that the decomposition formula

$$\begin{aligned} V_{n,r+1}(f,\boldsymbol{x})&=\sum _{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-1) \int _{0}^{\infty }p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times \sum_{\boldsymbol{k}^{*}=0}^{\infty }p_{n+k_{1}, \boldsymbol{k}^{*}} \biggl(\frac{\boldsymbol{x}^{*}}{1+x_{1}} \biggr) (n-2) \cdots (n-r-1) \\ &\quad{}\times \int _{\mathbb{R}^{r}_{0}}p_{n+k_{1},\boldsymbol{k}^{*}} \biggl(\frac{\boldsymbol{u}^{*}}{1+u_{1}} \biggr) f\bigl(u_{1}, \boldsymbol{u}^{*}\bigr)\,\mathrm {d}\boldsymbol{u}^{*} \\ &=\sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \sum_{\boldsymbol{k}^{*}=0}^{\infty }p_{n+k_{1}, \boldsymbol{k}^{*}} \biggl(\frac{\boldsymbol{x}^{*}}{1+x_{1}} \biggr) \\ &\quad{}\times \frac{(n-3)\cdots (n-r-1)}{(n-2+k_{1})\cdots (n+k_{1}-r)}(n+k_{1}-1) (n-2+k_{1}) \cdots (n+k_{1}-r) \\ &\quad{}\times \int _{\mathbb{R}^{r}_{0}}p_{n+k_{1},\boldsymbol{k}^{*}} \biggl(\frac{\boldsymbol{u}^{*}}{1+u_{1}} \biggr) f \biggl(u_{1},(1+u_{1}) \frac{\boldsymbol{u}^{*}}{1+u_{1}} \biggr)\,\mathrm {d}\biggl( \frac{\boldsymbol{u}^{*}}{1+u_{1}} \biggr) \\ &\leq \sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{ \infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \sum_{\boldsymbol{k}^{*}=0}^{ \infty }p_{n+k_{1},\boldsymbol{k}^{*}}( \boldsymbol{z}) (n+k_{1}-1)\cdots \\ &\quad{}\times (n+k_{1}-r) \int _{\mathbb{R}^{r}_{0}}p_{n+k_{1}, \boldsymbol{k}^{*}}\bigl(\boldsymbol{t}^{*} \bigr)f \bigl(u_{1},(1+u_{1}) \boldsymbol{t}^{*} \bigr) \,\mathrm {d}\boldsymbol{t}^{*} \\ &=\sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1})V_{n+k_{1},r} \bigl(g_{u_{1}}(\cdot ),\boldsymbol{z} \bigr)\,\mathrm {d}u_{1} \end{aligned}$$

is valid, where $g_{u_{1}}(t)=f (u_{1},(1+u_{1})t )$ for $0\leq t<\infty$. From the above formula, it follows that

$$ \begin{aligned} V_{n,r+1}(f, \boldsymbol{x})-f(\boldsymbol{x})&\leq \sum_{k_{1}=0}^{ \infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \\ &\quad{}\times \bigl[V_{n+k_{1},r}\bigl(g_{u_{1}}(\cdot ), \boldsymbol{z}\bigr)-g_{u_{1}}( \boldsymbol{z})\bigr]\,\mathrm {d}u_{1} \\ &\quad{}+V_{n,1}^{*}\bigl(h(\cdot ),x_{1} \bigr)-h(x_{1}) \\ &\triangleq T_{1}+T_{2}, \end{aligned} $$

(11)

where

$$ h(u_{1})\triangleq h(x_{1},\boldsymbol{x})\triangleq f \biggl(u_{1},(1+u_{1}) \frac{\boldsymbol{x}^{*}}{1+x_{1}} \biggr), \quad 0\leq u_{1}< \infty . $$

By the inequality

$$\begin{aligned} &\int _{\mathbb{R}^{r}_{0}}\Phi \biggl(\frac{1}{\lambda} \bigl\vert V_{n,r}(f, \boldsymbol{x})-f(\boldsymbol{x}) \bigr\vert \biggr) \,\mathrm {d}\boldsymbol{x} \\ &\quad \leq \int _{\mathbb{R}^{r}_{0}}\Phi \Biggl(\frac{C}{n\lambda}\sum _{i=1}^{r} \bigl\vert \bigl( \varphi _{i}^{2}D_{i}^{2}f\bigr) ( \boldsymbol{x}) \bigr\vert \Biggr)\,\mathrm {d}\boldsymbol{x} + \int _{\mathbb{R}^{r}_{0}}\Phi \biggl(\frac{C}{n\lambda} \bigl\vert f( \boldsymbol{x}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{x}, \end{aligned}$$

which can be obtained from (10) and Jensen’s inequality, we arrive at

$$\begin{aligned} \int _{\mathbb{R}^{r+1}_{0}}\Phi \biggl(\frac{1}{\lambda} \vert T_{1} \vert \biggr)\,\mathrm {d}\boldsymbol{x} &= \int _{\mathbb{R}^{r+1}_{0}}\Phi \Biggl( \frac{1}{\lambda} \Biggl\vert \sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \\ &\quad{}\times \bigl[V_{n+k_{1},r} \bigl(g_{u_{1}}(\cdot ), \boldsymbol{z} \bigr)-g_{u_{1}}(\boldsymbol{z}) \bigr]\,\mathrm {d}u_{1} \Biggr\vert \Biggr)\,\mathrm {d}\boldsymbol{x} \\ &\leq \int _{\mathbb{R}^{r+1}_{0}}\sum_{k_{1}=0}^{\infty }p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \\ &\quad{}\times \Phi \biggl(\frac{1}{\lambda} \bigl\vert V_{n+k_{1},r} \bigl(g_{u_{1}}( \cdot ),\boldsymbol{z} \bigr) -g_{u_{1}}( \boldsymbol{z}) \bigr\vert \biggr)\,\mathrm {d}u_{1} \,\mathrm {d}\boldsymbol{x} \\ &=\sum_{k_{1}=0}^{\infty} \int _{0}^{\infty }p_{n,k_{1}}(x_{1}) (1+x_{1}) \,\mathrm {d}x_{1}(n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times \int _{\mathbb{R}^{r}_{0}}\Phi \biggl(\frac{1}{\lambda} \bigl\vert V_{n+k_{1},r} \bigl(g_{u_{1}}(\cdot ),\boldsymbol{z} \bigr) -g_{u_{1}}( \boldsymbol{z}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{z} \\ &=\sum_{k_{1}=0}^{\infty} \frac{n+k_{1}-1}{n-1} \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times \int _{\mathbb{R}^{r}_{0}}\Phi \biggl(\frac{1}{\lambda} \bigl\vert V_{n+k_{1},r} \bigl(g_{u_{1}}(\cdot ),\boldsymbol{z} \bigr) -g_{u_{1}}( \boldsymbol{z}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{z} \\ &\leq \sum_{k_{1}=0}^{\infty} \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times \int _{\mathbb{R}^{r}_{0}}\Phi \biggl( \frac{n+k_{1}-1}{\lambda (n-1)} \bigl\vert V_{n+k_{1},r} \bigl(g_{u_{1}}(\cdot ), \boldsymbol{z} \bigr) -g_{u_{1}}(\boldsymbol{z}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{z} \\ &\leq \sum_{k_{1}=0}^{\infty} \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad{}\times \Biggl[ \int _{\mathbb{R}^{r}_{0}}\Phi \Biggl( \frac{C}{n\lambda} \sum _{i=1}^{r} \bigl\vert \bigl(\varphi _{i}^{2}D_{i}^{2}g_{u_{1}} \bigr) ( \boldsymbol{z}) \bigr\vert \Biggr)\,\mathrm {d}\boldsymbol{z} \\ &\quad {}+ \int _{\mathbb{R}^{r}_{0}} \Phi \biggl(\frac{C}{n\lambda} \bigl\vert g_{u_{1}}(\boldsymbol{z}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{z} \Biggr]. \end{aligned}$$

On the other hand, by definition, we can deduce

$$ \begin{aligned} \varphi _{i}^{2}( \boldsymbol{x})D_{i}^{2}g_{u_{1}}( \boldsymbol{x}) &=x_{i}\bigl(1+ \vert \boldsymbol{x} \vert \bigr) (1+u_{1})^{2}D_{i+1}^{2}f \bigl(u_{1},(1+u_{1}) \boldsymbol{x} \bigr) \\ &= \bigl(\varphi _{i+1}^{2}D_{i+1}^{2}f \bigr) \bigl(u_{1},(1+u_{1}) \boldsymbol{x} \bigr). \end{aligned} $$

As a result, we obtain

$$\begin{aligned} &\int _{\mathbb{R}^{r+1}_{0}}\Phi \biggl(\frac {1}{\lambda} \vert T_{1} \vert \biggr)\,\mathrm {d}\boldsymbol{x} \\ &\quad \le \int _{0}^{\infty}\sum_{k_{1}=0}^{ \infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \Biggl[ \int _{\mathbb{R}^{r}_{0}} \Phi \Biggl(\frac{C}{n\lambda}\sum _{i=1}^{r+1} \\ &\quad \quad {}\times \bigl\vert \bigl(\varphi _{i}^{2}D_{i}^{2}f \bigr) \bigl(u_{1},(1+u_{1}) \boldsymbol{z} \bigr) \bigr\vert \Biggr)\,\mathrm {d}\boldsymbol{z} + \int _{ \mathbb{R}^{r}_{0}}\Phi \biggl(\frac{C}{n\lambda} \bigl\vert f \bigl(u_{1},(1+u_{1}) \boldsymbol{z} \bigr) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{z} \Biggr] \\ &\quad \leq \int _{0}^{\infty} \,\mathrm {d}u_{1} \int _{\mathbb{R}^{r}_{0}}\Phi \Biggl(\frac{C}{n\lambda} \sum _{i=1}^{r+1} \bigl\vert \bigl(\varphi _{i}^{2}D_{i}^{2}f\bigr) \bigl(u_{1},(1+u_{1})\boldsymbol{z} \bigr) \bigr\vert \Biggr)\,\mathrm {d}\bigl((1+u_{1}) \boldsymbol{z}\bigr) \\ &\quad \quad{}+ \int _{0}^{\infty} \,\mathrm {d}u_{1} \int _{\mathbb{R}^{r}_{0}} \Phi \biggl(\frac {C}{n\lambda} \bigl\vert f \bigl(u_{1},(1+u_{1})\boldsymbol{z} \bigr) \bigr\vert \biggr)\,\mathrm {d}\bigl((1+u_{1})\boldsymbol{z}\bigr) \\ &\quad = \int _{\mathbb{R}^{r+1}_{0}}\Phi \Biggl(\frac{C}{n\lambda}\sum _{i=1}^{r+1} \bigl\vert \bigl( \varphi _{i}^{2}D_{i}^{2}f\bigr) ( \boldsymbol{u}) \bigr\vert \Biggr)\,\mathrm {d}\boldsymbol{u}+ \int _{\mathbb{R}^{r+1}_{0}}\Phi \biggl( \frac{C}{n\lambda} \bigl\vert f( \boldsymbol{u}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{u}. \end{aligned}$$

(12)

By the Inequality (8) and the convexity of $\Phi (t)$, we acquire

$$\begin{aligned} \int _{\mathbb{R}^{r+1}_{0}}\Phi \biggl(\frac {1}{\lambda} \vert T_{2} \vert \biggr)\,\mathrm {d}\boldsymbol{x} &= \int _{\mathbb{R}^{r+1}_{0}}\Phi \biggl( \frac {1}{\lambda} \bigl\vert V_{n,1}^{*} \bigl(h(\cdot ),x_{1} \bigr)-h(x_{1}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{x} \\ &\le \int _{\mathbb{R}^{r+1}_{0}}\Phi \biggl(\frac {C}{n\lambda} \bigl( \bigl\vert h(x_{1}) \bigr\vert +\varphi ^{2}(x_{1}) \bigl\vert h''(x_{1}) \bigr\vert \bigr) \biggr)\,\mathrm {d}\boldsymbol{x}. \end{aligned}$$

Denoting $\varphi _{ij}(\boldsymbol{x})=\sqrt{x_{i}x_{j}}$ for $1\le i< j\leq r+1$ and $D_{ij}^{2}=\frac{\partial ^{2}}{\partial x_{i}\partial x_{j}}$, we have

$$\begin{aligned} \varphi ^{2}(u)h''(u) &=u(1+u) \Biggl[D_{1}^{2}f+\sum _{i=2}^{r+1} \frac{x_{i}}{1+x_{1}}D_{1i}^{2}f+ \sum_{i=2}^{r+1} \frac{x_{i}}{1+x_{1}}D_{i1}^{2}f \\ &\quad{}+\sum_{i=2}^{r+1}\sum _{j=2}^{r+1} \frac{x_{i}x_{j}}{(1+x_{1})^{2}}D_{ij}^{2}f \Biggr] \biggl(u, \frac{(1+u)\boldsymbol{x}^{*}}{1+x_{1}} \biggr) \\ &= \Biggl(\frac{1+x_{1}}{1+ \vert \boldsymbol{x} \vert }\varphi _{1}^{2}D_{1}^{2}f+ \sum _{i=2}^{r+1}\varphi _{1i}^{2}D_{1i}^{2}f+ \sum_{i=2}^{r+1} \varphi _{i1}^{2}D_{i1}^{2}f \\ &\quad{}+\sum_{i=2}^{r+1} \frac {u}{1+u} \frac{x_{i}}{1+ \vert \boldsymbol{x} \vert }\varphi _{i}^{2}D_{i}^{2}f +\sum_{i,j=2,i \neq j}^{r+1}\frac {u}{1+u} \varphi _{ij}^{2}D_{ij}^{2}f \Biggr) \biggl(u, \frac{(1+u)\boldsymbol{x}^{*}}{1+x_{1}} \biggr). \end{aligned}$$

Recalling that $\varphi _{ij}(\boldsymbol{x})$ is not bigger than $\varphi _{i}(\boldsymbol{x})$ or $\varphi _{j}(\boldsymbol{x})$, and using the fact

$$ \bigl\vert D_{ij}^{2}f(\boldsymbol{x}) \bigr\vert \le \sup_{1\le i\le r+1} \bigl\vert D_{i}^{2}f( \boldsymbol{x}) \bigr\vert $$

in [6, Lemma 2.1], we obtain

$$ \begin{aligned} \int _{\mathbb{R}^{r+1}_{0}}\Phi \biggl(\frac {1}{\lambda} \vert T_{2} \vert \biggr)\,\mathrm {d}\boldsymbol{x} &\le \frac{1}{2} \int _{\mathbb{R}^{r+1}_{0}} \Phi \biggl(\frac {C}{n\lambda} \bigl\vert f( \boldsymbol{x}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{x} \\ &\quad{}+ \int _{\mathbb{R}^{r+1}_{0}}\Phi \Biggl(\frac {C}{n\lambda} \sum _{i=1}^{r+1} \bigl\vert \bigl(\varphi _{i}^{2}D_{i}^{2}f\bigr) ( \boldsymbol{x}) \bigr\vert \Biggr)\,\mathrm {d}\boldsymbol{x}. \end{aligned} $$

(13)

Combining the Inequalities (11), (12), and (13) and paying attention to computation of norm and the Inequality (1), we obtain the second estimate of (9) for any $m\geq 2$.

For $g\in W_{\varphi}^{2,\Phi}(\mathbb{R}^{m}_{0})$, using (2), (9), and Lemma 1 gives

$$\begin{aligned} \bigl\Vert V_{n,m}(f)-f \bigr\Vert _{\Phi}&\le \bigl\Vert V_{n,m}(f)-V_{n,m}(g) \bigr\Vert _{\Phi}+ \bigl\Vert V_{n,m}(g)-g \bigr\Vert _{\Phi}+ \Vert f-g \Vert _{\Phi} \\ &\le C \Vert f-g \Vert _{\Phi}+ \frac{C}{n} \Biggl( \Vert g \Vert _{\Phi}+\sum _{i=1}^{m} \bigl\Vert \varphi _{i}^{2}D_{i}^{2}g \bigr\Vert _{\Phi}+\frac{1}{n}\sum_{i=1}^{m} \bigl\Vert D_{i}^{2}g \bigr\Vert _{\Phi} \Biggr) \\ &\le C \Biggl( \Vert f-g \Vert _{\Phi}+\frac{1}{n}\sum _{i=1}^{m} \bigl\Vert \varphi _{i}^{2}D_{i}^{2}g \bigr\Vert _{\Phi}+\frac{1}{n^{2}}\sum_{i=1}^{m} \bigl\Vert D_{i}^{2}g \bigr\Vert _{\Phi} \Biggr)+ \frac{C}{n} \Vert f \Vert _{\Phi} \\ &\le C \biggl[\omega _{2,\varphi} \biggl(f,\frac{1}{n^{1/2}} \biggr)_{ \Phi}+\frac{1}{n} \Vert f \Vert _{\Phi} \biggr]. \end{aligned}$$

The proof of Theorem 1 is complete. □

4 Proof of inverse theorem

In order to prove the inverse theorem, we need several lemmas.

Lemma 3

Let $f\in W_{\varphi}^{2,\Phi}(\mathbb{R}^{m}_{0})$ and $n>m$ for $n,m\in \mathbb{N}$. Then

$$ \bigl\Vert \varphi _{i}^{2}D_{i}^{2}V_{n,m}(f) \bigr\Vert _{(\Phi )} \le \bigl\Vert \varphi _{i}^{2}D_{i}^{2}f \bigr\Vert _{(\Phi )},\quad i=1,2,\ldots ,m. $$

Proof

By straight computation, we have

$$\begin{aligned} \bigl\vert \varphi ^{2}(x)V_{n,1}''(f,x) \bigr\vert &= \Biggl\vert \varphi ^{2}(x)\sum _{k=0}^{ \infty}(n-1)p''_{n,k}(x) \int _{0}^{\infty}p_{n,k}(t)f(t)\,\mathrm {d}t \Biggr\vert \\ &= \Biggl\vert \varphi ^{2}(x)\sum _{i=0}^{2}\sum _{k=i}^{\infty}(n-1) \frac{(n+k-1)!}{k!(n-1)!} \binom{2}{i} \bigl(D^{i}x^{k} \bigr) \\ &\quad{}\times \bigl(D^{2-i}(1+x)^{-n-k} \bigr) \int _{0}^{\infty}p_{n,k}(t)f(t) \,\mathrm {d}t \Biggr\vert \\ &= \Biggl\vert \sum_{k=0}^{\infty}(n-1) \frac{(n+k+1)(k+1)}{(k+2)(n+k)}p_{n,k+1}(x) \int _{0}^{\infty}p_{n,k+1}(t)\varphi ^{2}(t)f''(t)\,\mathrm {d}t \Biggr\vert \\ &\leq \sum_{k=0}^{\infty}(n-1)p_{n,k+1}(x) \int _{0}^{\infty}p_{n,k+1}(t) \varphi ^{2}(t) \bigl\vert f''(t) \bigr\vert \,\mathrm {d}t, \end{aligned}$$

where $\frac{(n+k+1)(k+1)}{(k+2)(n+k)}\leq 1$ for $k\ge 0$ and $n\in \mathbb{N}$. Using Jensen’s inequality, we derive

$$ \int _{0}^{\infty}\Phi \biggl(\frac{1}{\lambda} \bigl\vert \varphi ^{2}(x)V_{n,1}''(f,x) \bigr\vert \biggr)\,\mathrm {d}x\leq \int _{0}^{\infty}\Phi \biggl( \frac{1}{\lambda} \varphi ^{2}(t) \bigl\vert f''(t) \bigr\vert \biggr)\,\mathrm {d}t. $$

(14)

Let

$$ g_{u^{*}}(t)=f \Biggl(u_{1},u_{2},\ldots ,u_{m-1}, \Biggl(1+\sum_{k=1}^{m-1}u_{k} \Biggr)t \Biggr), \quad 0\leq t< \infty $$

and $z=\frac{x_{m}}{1+\sum_{\ell =1}^{m-1}x_{\ell}}$. Then, by the Inequality (4) and for $m>1$, we have

$$\begin{aligned} V_{n,m}(f,\boldsymbol{x})&=\sum_{k_{1}=0}^{\infty}p_{n,k_{1}}(x_{1}) (n-1) \int _{0}^{\infty }p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1}\sum_{k_{2}=0}^{\infty}p_{n+k_{1},k_{2}} \biggl(\frac{x_{2}}{1+x_{1}} \biggr) \\ &\quad{}\times (n-2) \int _{0}^{\infty }p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad{}\times \sum_{k_{m}=0}^{\infty}p_{n+\sum _{\ell =1}^{m-1}k_{ \ell},k_{m}} \biggl(\frac{x_{m}}{1+\sum_{\ell =1}^{m-1}x_{\ell}} \biggr) (n-m) \\ &\quad{}\times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr)f(u_{1}, \ldots ,u_{m}) \,\mathrm {d}u_{m} \\ &=\sum_{k_{1}=0}^{\infty}p_{n,k_{1}}(x_{1}) (n-1) \int _{0}^{\infty }p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1}\sum_{k_{2}=0}^{\infty}p_{n+k_{1},k_{2}} \biggl( \frac{x_{2}}{1+x_{1}} \biggr) (n-2) \\ &\quad{}\times \int _{0}^{\infty }p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \sum _{k_{m}=0}^{\infty} p_{n+ \sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{x_{m}}{1+\sum_{\ell =1}^{m-1}x_{\ell}} \biggr) \\ &\quad{}\times (n-m) \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-1}k_{ \ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \Biggl(1+\sum_{k=1}^{m-1}u_{k} \Biggr) \\ &\quad{}\times f \Biggl(u_{1},\ldots ,u_{m-1}, \Biggl(1+ \sum_{k=1}^{m-1}u_{k} \Biggr) \frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \Biggr) \,\mathrm {d}\biggl(\frac{u_{m}}{1+\sum_{\ell =1}^{m-1}u_{\ell}} \biggr) \\ &=\sum_{k_{1}=0}^{\infty}p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1}\sum_{k_{2}=0}^{\infty}p_{n+k_{1},k_{2}} \biggl( \frac{x_{2}}{1+x_{1}} \biggr) (n-3) \\ &\quad{}\times \int _{0}^{\infty }p_{n+k_{1}-1,k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad{}\times \sum_{k_{m-1}=0}^{\infty}p_{n+\sum _{\ell =1}^{m-2}k_{ \ell},k_{m-1}} \biggl(\frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr) (n-m) \\ &\quad{}\times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-2}k_{\ell}-1,k_{m-1}} \biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr) \,\mathrm {d}u_{m-1} \\ &\quad{}\times \sum_{k_{m}=0}^{\infty}p_{n+\sum _{\ell =1}^{m-1}k_{ \ell},k_{m}}(z) \Biggl(n-1+\sum_{\ell =1}^{m-1}k_{\ell} \Biggr) \\ &\quad{}\times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}}(t)f \Biggl(u_{1},\ldots ,u_{m-1}, \Biggl(1+\sum _{k=1}^{m-1}u_{k} \Biggr)t \Biggr) \,\mathrm {d}t \\ &=\sum_{k_{1}=0}^{\infty}p_{n,k_{1}}(x_{1}) (n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \sum_{k_{2}=0}^{\infty}p_{n+k_{1},k_{2}} \biggl( \frac{x_{2}}{1+x_{1}} \biggr) \\ &\quad{}\times (n-3) \int _{0}^{\infty }p_{n+k_{1}-1,k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad{}\times \sum_{k_{m-1}=0}^{\infty} p_{n+\sum _{\ell =1}^{m-2}k_{ \ell},k_{m-1}} \biggl(\frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr) (n-m) \\ &\quad{}\times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-2}k_{\ell}-1,k_{m-1}} \biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr) V_{n+ \sum _{\ell =1}^{m-1}k_{\ell},1}(g_{u^{*}},z)\,\mathrm {d}u_{m-1}. \end{aligned}$$

(15)

Using the Inequalities (14), (15), and Jensen’s inequality, we see that

$$\begin{aligned} & \int _{\mathbb{R}^{m}_{0}} \Phi \biggl(\frac {1}{\lambda} \bigl\vert \varphi _{m}^{2}(\boldsymbol{x})D_{m}^{2}V_{n,m}(f, \boldsymbol{x}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{x} \\ &\quad \leq \int _{0}^{\infty}\sum_{k_{1}=0}^{\infty}p_{n,k_{1}}(x_{1}) \,\mathrm {d}x_{1}(n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad \quad{}\times \int _{0}^{ \infty}\sum_{k_{2}=0}^{\infty} p_{n+k_{1},k_{2}} \biggl( \frac{x_{2}}{1+x_{1}} \biggr)\,\mathrm {d}x_{2}(n-2+k_{1}) \int _{0}^{\infty }p_{n+k_{1}-1,k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{m-1}=0}^{\infty}p_{n+\sum _{ \ell =1}^{m-2}k_{\ell},k_{m-1}} \biggl( \frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr)\,\mathrm {d}x_{m-1} \\ &\quad \quad{}\times \Biggl(n-2+\sum_{\ell =1}^{m-2}k_{\ell} \Biggr) \int _{0}^{ \infty }p_{n+\sum _{\ell =1}^{m-2}k_{\ell}-1,k_{m-1}} \biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr)\,\mathrm {d}u_{m-1} \\ &\quad \quad{}\times \int _{0}^{\infty}\Phi \biggl( \frac{(n-3)(n-4)\cdots (n-m) (1+\sum_{k=1}^{m-1}x_{k} )^{2}}{\lambda (n-2+k_{1})(n-2+k_{1}+k_{2})\cdots (n-2+\sum_{\ell =1}^{m-2}k_{\ell} )} \\ &\quad \quad{}\times \varphi _{m}^{2}(z) \bigl\vert V''_{n+\sum _{\ell =1}^{m-1}k_{\ell},1}(g_{u^{*}},z) \bigr\vert \biggr) \Biggl(1+\sum_{k=1}^{m-1}x_{k} \Biggr)\,\mathrm {d}z \\ &\quad \leq \int _{0}^{\infty}\sum_{k_{1}=0}^{\infty}p_{n-1,k_{1}}(x_{1}) \,\mathrm {d}x_{1}(n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \int _{0}^{ \infty}\sum_{k_{2}=0}^{\infty}(n-2 \\ &\quad \quad{}+k_{1})p_{n+k_{1}-1,k_{2}} \biggl(\frac{x_{2}}{1+x_{1}} \biggr) \,\mathrm {d}\biggl(\frac{x_{2}}{1+x_{1}} \biggr) \int _{0}^{\infty }p_{n+k_{1}-1,k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{m-1}=0}^{\infty}p_{n+\sum _{ \ell =1}^{m-2}k_{\ell}-1,k_{m-1}} \biggl( \frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr) \,\mathrm {d}\biggl( \frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr) \\ &\quad \quad{}\times \Biggl(n-2+\sum_{\ell =1}^{m-2}k_{\ell} \Biggr) \int _{0}^{ \infty }p_{n+\sum _{\ell =1}^{m-2}k_{\ell}-1,k_{m-1}} \biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr) \\ &\quad \quad{}\times \frac{n+\sum_{\ell =1}^{m-2}k_{\ell}-1}{n-1}\,\mathrm {d}u_{m-1} \int _{0}^{\infty}\sum_{k_{m}=0}^{\infty}p_{n+\sum _{\ell =1}^{m-1}k_{ \ell},k_{m}+1}(z) \,\mathrm {d}z \\ &\quad \quad{}\times \int _{0}^{\infty}\Phi \biggl( \frac{(n-3)(n-4)\cdots (n-m)}{\lambda (n-2+k_{1})(n-2+k_{1}+k_{2})\cdots (n-2+\sum_{\ell =1}^{m-2}k_{\ell} )} \\ &\quad \quad{}\times \varphi ^{2}(t) \bigl\vert g''_{u^{*}}(t) \bigr\vert \biggr) \Biggl(n-1+\sum_{ \ell =1}^{m-1}k_{\ell} \Biggr) p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}+1}(t) \,\mathrm {d}t \\ &\quad = \int _{0}^{\infty}\sum_{k_{1}=0}^{\infty}p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \int _{0}^{\infty}\sum_{k_{2}=0}^{\infty} p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{m-1}=0}^{\infty} \frac{n+\sum_{\ell =1}^{m-2}k_{\ell}-1}{n+\sum_{\ell =1}^{m-1}k_{\ell}} p_{n+\sum _{\ell =1}^{m-2}k_{\ell},k_{m-1}} \biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr) \,\mathrm {d}u_{m-1} \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{m}=0}^{\infty}p_{n+\sum _{ \ell =1}^{m-1}k_{\ell},k_{m}+1} \biggl( \frac{u_{m}}{1+\sum_{k=1}^{m-1}u_{k}} \biggr) \\ &\quad \quad{}\times \Phi \biggl( \frac{(n-3)(n-4)\cdots (n-m)}{\lambda (n-2+k_{1})(n-2+k_{1}+k_{2})\cdots (n-2+\sum_{\ell =1}^{m-2}k_{\ell} )} \\ &\quad \quad{}\times \varphi ^{2}_{m}(u_{1},u_{2}, \ldots ,u_{m}) \biggl\vert \frac{\partial ^{2}}{\partial u_{m}^{2}}f(u_{1},u_{2}, \ldots ,u_{m}) \biggr\vert \biggr) \,\mathrm {d}u_{m} \\ &\quad \leq \int _{\mathbb{R}^{m}_{0}} \Phi \biggl(\frac {1}{\lambda} \varphi _{m}^{2}(\boldsymbol{u}) \bigl\vert D_{m}^{2}f(\boldsymbol{u}) \bigr\vert \biggr) \,\mathrm {d}\boldsymbol{u}. \end{aligned}$$

Hence, from the computation formula of the form, it follows that

$$ \bigl\Vert \varphi ^{2}_{m}D_{m}^{2}V_{n,m}(f) \bigr\Vert _{(\Phi )}\leq \bigl\Vert \varphi ^{2}_{m}D_{m}^{2}(f) \bigr\Vert _{(\Phi )}. $$

Similarly, we can prove the same results for $i=1, 2, \ldots , m-1$. The proof of Lemma 3 is thus complete. □

Lemma 4

Let $f\in L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$ and $n>m$ for $n,m\in \mathbb{N}$. Then

$$ \bigl\Vert \varphi _{i}^{2}D_{i}^{2}V_{n,m}(f) \bigr\Vert _{(\Phi )}\le 4n \Vert f \Vert _{(\Phi )}, \quad i=1,2,\ldots ,m. $$

Proof

By straight calculation, for $m=1$, we have

$$\begin{aligned} \bigl\vert \varphi ^{2}(x)V_{n,1}''(f,x) \bigr\vert &= \Biggl\vert \varphi ^{2}(x)\sum _{k=0}^{ \infty}(n-1)p''_{n,k}(x) \int _{0}^{\infty}p_{n,k}(t)f(t)\,\mathrm {d}t \Biggr\vert \\ &= \Biggl\vert \sum_{i=0}^{2}\sum _{k=i}^{\infty}\varphi ^{2}(x) (n-1) \frac{(n+k-1)!}{k!(n-1)!}\binom{2}{i} \\ &\quad{}\times \bigl(D^{i}x^{k} \bigr) \bigl(D^{2-i}(1+x)^{-n-k} \bigr) \int _{0}^{\infty}p_{n,k}(t)f(t)\,\mathrm {d}t \Biggr\vert \\ &= \Biggl\vert \frac{n-1}{\varphi ^{2}(x)}\sum_{k=0}^{\infty}p_{n,k}(x) \bigl[(nx-k)^{2}-k(2x+1) \\ &\quad{}+nx^{2} \bigr] \int _{0}^{\infty}p_{n,k}(t)f(t)\,\mathrm {d}t \Biggr\vert \\ &\leq \frac{n-1}{\varphi ^{2}(x)}\sum_{k=0}^{\infty}p_{n,k}(x) \bigl[(nx-k)^{2}+k(2x+3) \\ &\quad{}+nx^{2} \bigr] \int _{0}^{\infty}p_{n,k}(t) \bigl\vert f(t) \bigr\vert \,\mathrm {d}t \\ &\triangleq 4n(n-1)\sum_{k=0}^{\infty}\beta _{n,k}(x) \int _{0}^{ \infty}p_{n,k}(t) \bigl\vert f(t) \bigr\vert \,\mathrm {d}t, \end{aligned}$$

(16)

where

$$ \beta _{n,k}(x)= \frac{p_{n,k}(x) [(nx-k)^{2}+k(2x+3)+nx^{2} ]}{4n\varphi ^{2}(x)}. $$

Moreover, we can verify that

$$ \sum_{k=0}^{\infty}\beta _{n,k}(x)=1 \quad \text{and}\quad \int _{0}^{ \infty}\beta _{n,k}(x)\,\mathrm {d}x=\frac{1}{2} \biggl(\frac{1}{n}+ \frac{1}{n-1} \biggr) $$

for $n\geq 2$. By the Inequalities (4) and (16), for $m>1$, we have

$$\begin{aligned} & \int _{\mathbb{R}^{m}_{0}} \Phi \biggl(\frac {1}{\lambda} \bigl\vert \varphi _{m}^{2}(\boldsymbol{x})D_{m}^{2}V_{n,m}(f, \boldsymbol{x}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{x} \\ &\quad = \int _{0}^{\infty}\sum_{k_{1}=0}^{\infty}p_{n,k_{1}}(x_{1}) \,\mathrm {d}x_{1}(n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{2}=0}^{\infty} p_{n+k_{1},k_{2}} \biggl(\frac{x_{2}}{1+x_{1}} \biggr)\,\mathrm {d}x_{2}(n-2+k_{1}) \\ &\quad \quad{}\times \int _{0}^{\infty }p_{n+k_{1}-1,k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2} \cdots \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{m-1}=0}^{\infty}p_{n+\sum _{ \ell =1}^{m-2}k_{\ell},k_{m-1}} \biggl( \frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr)\,\mathrm {d}x_{m-1} \\ &\quad \quad{}\times \Biggl(n-2+\sum_{\ell =1}^{m-2}k_{\ell} \Biggr) \int _{0}^{ \infty }p_{n+\sum _{\ell =1}^{m-2}k_{\ell}-1,k_{m-1}} \biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr)\,\mathrm {d}u_{m-1} \\ &\quad \quad{}\times \int _{0}^{\infty}\Phi \biggl( \frac{(n-3)(n-4)\cdots (n-m) (1+\sum_{k=1}^{m-1}x_{k} )^{2}}{ \lambda (n-2+k_{1})(n-2+k_{1}+k_{2})\cdots (n-2+\sum_{\ell =1}^{m-2}k_{\ell} )} \\ &\quad \quad{}\times \varphi ^{2}(z) \bigl\vert V''_{n+\sum _{\ell =1}^{m-1}k_{\ell},1}(g_{u^{*}},z) \bigr\vert \biggr) \Biggl(1+\sum_{k=1}^{m-1}x_{k} \Biggr)\,\mathrm {d}z \\ &\quad \leq \int _{0}^{\infty}\sum_{k_{1}=0}^{\infty}p_{n-1,k_{1}}(x_{1}) \,\mathrm {d}x_{1}(n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{2}=0}^{\infty}p_{n+k_{1}-1,k_{2}} \biggl(\frac{x_{2}}{1+x_{1}} \biggr) (n-2+k_{1})\,\mathrm {d}\biggl( \frac{x_{2}}{1+x_{1}} \biggr) \\ &\quad \quad{}\times \int _{0}^{\infty }p_{n+k_{1}-1,k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \int _{0}^{\infty} \Biggl(n-2+ \sum _{\ell =1}^{m-2}k_{\ell} \Biggr) \\ &\quad \quad{}\times \sum_{k_{m-1}=0}^{\infty}p_{n+\sum _{\ell =1}^{m-2}k_{ \ell}-1,k_{m-1}} \biggl(\frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr) \,\mathrm {d}\biggl(\frac{x_{m-1}}{1+\sum_{k=1}^{m-2}x_{k}} \biggr) \\ &\quad \quad{}\times \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-2}k_{\ell}-1,k_{m-1}} \biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr) \frac{n+\sum_{\ell =1}^{m-2}k_{\ell}-1}{n-1}\,\mathrm {d}u_{m-1} \\ &\quad \quad{}\times \int _{0}^{\infty}\Phi \Biggl( \frac{4(n-3)(n-4)\cdots (n-m)(n+\sum_{\ell =1}^{m-1}k_{\ell}) (n-1+\sum_{\ell =1}^{m-1}k_{\ell} )}{ \lambda (n-2+k_{1})(n-2+k_{1}+k_{2})\cdots (n-2+\sum_{\ell =1}^{m-2}k_{\ell} )} \\ &\quad \quad{}\times \sum_{k_{m}=0}^{\infty}\beta _{n+\sum _{\ell =1}^{m-1}k_{ \ell},k_{m}}(z) \int _{0}^{\infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}}(t) \bigl\vert g_{u^{*}}(t) \bigr\vert \,\mathrm {d}t \Biggr)\,\mathrm {d}z \\ &\quad \leq \int _{0}^{\infty}\sum_{k_{1}=0}^{\infty}p_{n-1,k_{1}}(x_{1}) \,\mathrm {d}x_{1}(n-2) \int _{0}^{\infty }p_{n-1,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \int _{0}^{ \infty}\sum_{k_{2}=0}^{\infty}(n-2 \\ &\quad \quad{}+k_{1})p_{n+k_{1}-1,k_{2}} \biggl(\frac{x_{2}}{1+x_{1}} \biggr) \,\mathrm {d}\biggl(\frac{x_{2}}{1+x_{1}} \biggr) \int _{0}^{\infty }p_{n+k_{1}-1,k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{m}=0}^{\infty} \frac{n+\sum_{\ell =1}^{m-2}k_{\ell}-1}{n-1}\beta _{n+\sum _{\ell =1}^{m-1}k_{ \ell},k_{m}}(z)\,\mathrm {d}z \\ &\quad \quad{}\times \Biggl(n-1+\sum_{\ell =1}^{m-1}k_{\ell} \Biggr) \int _{0}^{ \infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}}(t) \\ &\quad \quad{}\times \Phi \biggl( \frac{4(n-3)(n-4)\cdots (n-m)(n+\sum_{\ell =1}^{m-1}k_{\ell})}{\lambda (n-2+k_{1})(n-2+k_{1}+k_{2})\cdots (n-2+\sum_{\ell =1}^{m-2}k_{\ell} )} \bigl\vert g_{u^{*}}(t) \bigr\vert \biggr)\,\mathrm {d}t \\ &\quad = \int _{0}^{\infty}\sum_{k_{1}=0}^{\infty}p_{n,k_{1}}(u_{1}) \,\mathrm {d}u_{1} \int _{0}^{\infty}\sum_{k_{2}=0}^{\infty} p_{n+k_{1},k_{2}} \biggl( \frac{u_{2}}{1+u_{1}} \biggr)\,\mathrm {d}u_{2}\cdots \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{m-1}=0}^{\infty}p_{n+\sum _{ \ell =1}^{m-2}k_{\ell},k_{m-1}} \biggl( \frac{u_{m-1}}{1+\sum_{\ell =1}^{m-2}u_{\ell}} \biggr) \frac{n+\sum_{\ell =1}^{m-2}k_{\ell}-1}{n+\sum_{\ell =1}^{m-1}k_{\ell}} \,\mathrm {d}u_{m-1} \\ &\quad \quad{}\times \int _{0}^{\infty}\sum_{k_{m}=0}^{\infty} \beta _{n+ \sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{x_{m}}{1+\sum_{k=1}^{m-1}x_{k}} \biggr) \,\mathrm {d}\biggl( \frac{x_{m}}{1+\sum_{k=1}^{m-1}x_{k}} \biggr) \\ &\quad \quad{}\times \Biggl(n-1+\sum_{\ell =1}^{m-1}k_{\ell} \Biggr) \int _{0}^{ \infty }p_{n+\sum _{\ell =1}^{m-1}k_{\ell},k_{m}} \biggl( \frac{u_{m}}{1+\sum_{k=1}^{m-1}u_{k}} \biggr) \\ &\quad \quad{}\times \Phi \biggl( \frac{4(n-3)(n-4)\cdots (n-m) (n+\sum_{\ell =1}^{m-1}k_{\ell} )}{\lambda (n-2+k_{1})\cdots (n-2+\sum_{\ell =1}^{m-2}k_{\ell} )} \bigl\vert f(u_{1},\ldots ,u_{m}) \bigr\vert \biggr)\,\mathrm {d}u_{m} \\ &\quad \leq \int _{0}^{\infty} \,\mathrm {d}u_{1} \int _{0}^{\infty} \,\mathrm {d}u_{2}\cdots \int _{0}^{\infty}\Phi \biggl(\frac{4n}{\lambda} \bigl\vert f(u_{1},\ldots ,u_{m}) \bigr\vert \biggr) \,\mathrm {d}u_{m} \\ &\quad = \int _{\mathbb{R}^{m}_{0}}\Phi \biggl(\frac{4n}{\lambda} \bigl\vert f( \boldsymbol{u}) \bigr\vert \biggr)\,\mathrm {d}\boldsymbol{u}. \end{aligned}$$

Hence, from the computation formula of the form, it follows that

$$ \bigl\Vert \varphi ^{2}_{m}D_{m}^{2}V_{n,m}(f) \bigr\Vert _{(\Phi )}\leq 4n \Vert f \Vert _{(\Phi )}. $$

Similarly, we can prove the same results for $i=1, 2, \ldots , m-1$. The proof of Lemma 4 is thus complete. □

Proof of Theorem 2

Let

$$ v_{n}=\frac{1}{n} \bigl\Vert \varphi ^{2}_{i}D_{i}^{2}V_{n,m}(f) \bigr\Vert _{(\Phi )}, \quad i=1,2,\ldots ,m $$

and

$$ \tau _{k}=4 \bigl\Vert V_{k,m}(f)-f \bigr\Vert _{(\Phi )}. $$

It is obvious that $v_{1}=0$. From Lemmas 3 and 4, it follows that

$$\begin{aligned} v_{n}&\leq \frac{1}{n} \bigl\Vert \varphi ^{2}_{i}D_{i}^{2}V_{n,m} \bigl(V_{k,m}(f) \bigr) \bigr\Vert _{(\Phi )}+ \frac{1}{n} \bigl\Vert \varphi ^{2}_{i}D_{i}^{2}V_{n,m} \bigl(V_{k,m}(f)-f \bigr) \bigr\Vert _{(\Phi )} \\ &\leq \frac{1}{n} \bigl\Vert \varphi ^{2}_{i}D_{i}^{2}V_{k,m}(f) \bigr\Vert _{(\Phi )}+4 \bigl\Vert V_{k,m}(f)-f \bigr\Vert _{(\Phi )} \\ &=\frac{k}{n}v_{k}+\tau _{k}. \end{aligned}$$

By [28, Lemma 2.1], we acquire $v_{n}\leq \frac{C}{n}\sum_{k=1}^{n}\tau _{k}$. Therefore,

$$ \bigl\Vert \varphi ^{2}_{i}D_{i}^{2}V_{k,m}(f) \bigr\Vert _{(\Phi )}\leq C\sum_{k=1}^{n} \bigl\Vert V_{k,m}(f)-f \bigr\Vert _{(\Phi )}. $$

Using the double Inequality (1), we obtain

$$ \bigl\Vert \varphi ^{2}_{i}D_{i}^{2}V_{k,m}(f) \bigr\Vert _{\Phi}\leq C\sum_{k=1}^{n} \bigl\Vert V_{k,m}(f)-f \bigr\Vert _{\Phi}. $$

For $n\geq 2$, there exists $s\in \mathbb{N}$ such that $\frac{n}{2}\leq s\leq n$ and

$$ \bigl\Vert V_{s,m}(f)-f \bigr\Vert _{\Phi}\leq \bigl\Vert V_{k,m}(f)-f \bigr\Vert _{\Phi},\quad \frac{n}{2} \leq k\leq n. $$

Accordingly, we have

$$ \bigl\Vert V_{s,m}(f)-f \bigr\Vert _{\Phi}\leq \frac{2}{n}\sum_{\frac{n}{2}\leq k\leq n} \bigl\Vert V_{k,m}(f)-f \bigr\Vert _{\Phi}\leq \frac{2}{n} \sum_{k=1}^{n} \bigl\Vert V_{k,m}(f)-f \bigr\Vert _{\Phi}. $$

Hence, by the definition of the K–functional, we deduce

$$\begin{aligned} \tilde{K}_{2,\varphi} \biggl(f,\frac{1}{n} \biggr)&\leq \bigl\Vert V_{s,m}(f)-f \bigr\Vert _{\Phi}+ \frac{1}{n} \max_{1\leq i\leq m} \bigl\Vert \varphi ^{2}_{i}D_{i}^{2}V_{s,m}(f) \bigr\Vert _{\Phi} \\ &\leq \frac{C}{n}\sum_{k=1}^{n} \bigl\Vert V_{k,m}(f)-f \bigr\Vert _{\Phi}. \end{aligned}$$

Finally, using (3), we finish the proof of Theorem 2. □

5 Conclusions

In this paper, using the equivalent theorem between the modified K-functional and modulus of smoothness, employing a decomposition technique, and considering some properties of multivariate Baskakov–Durrmeyer operators in the form of Lemmas 1, 2, 3, and 4, we obtained a direct theorem and weak type inverse theorem in the Orlicz spaces $f\in L_{\Phi}^{*}(\mathbb{R}^{m}_{0})$.