Estimates for $n$-widths of sets of smooth functions on complex spheres

TL;DR

This paper provides sharp estimates for the n-widths of certain multiplier operators acting on function spaces on complex spheres, with applications to Sobolev and analytic function classes.

Contribution

It introduces new bounds for n-widths of multiplier operators on complex spheres, extending understanding of approximation properties of various function classes.

Findings

Derived order-sharp estimates for Kolmogorov n-widths.

Established bounds for linear, Gelfand, and Bernstein n-widths.

Applied results to Sobolev, differentiable, and analytic function classes.

Abstract

In this work we investigate $n$-widths of multiplier operators $\Lambda_*$ and $\Lambda$, defined for functions on the complex sphere $\Omega_d$ of $\mathbb{C}^d$, associated with sequences of multipliers of the type $\{\lambda_{m,n}^*\}_{m,n\in \mathbb{N}}$, $\lambda_{m,n}^*=\lambda(m+n)$ and $\{\lambda_{m,n}\}_{m,n\in \mathbb{N}}$, $\lambda_{m,n}=\lambda(\max\{m,n\})$, respectively, for a bounded function $\lambda$ defined on $[0,\infty)$. If the operators $\Lambda_{*}$ and $\Lambda$ are bounded from $L^p(\Omega_d)$ into $L^q(\Omega_d)$, $1\leq p,q\leq\infty$, and $U_p$ is the closed unit ball of $L^p(\Omega_d)$, we study lower and upper estimates for the $n$-widths of Kolmogorov, linear, of Gelfand and of Bernstein, of the sets $\Lambda_{*}U_p$ and $\Lambda U_p$ in $L^q(\Omega_d)$. As application we obtain, in particular, estimates for the Kolmogorov $n$-width of classes of Sobolev, of finitely differentiable, infinitely differentiable and analytic functions on the complex sphere, in $L^q(\Omega_d)$, which are order sharp in various important situations.

Equations

$$d_n(A,X):=\underset{X_n}{\inf }\underset{x\in A}{\sup }\underset{y\in X_n}{\inf }\Vert x-y{\Vert }_X,$$

$${\delta }_n(A,X):=\underset{P_n}{\inf }\underset{x\in A}{\sup }\Vert x-P_n(x){\Vert }_X,$$

$$d^n(A,X)=\underset{L^n}{\inf }\underset{x\in A\cap L^n}{\sup }\Vert x{\Vert }_X,$$

$$b_n(A,X)=\underset{X_{n+1}}{\sup }\sup \{\lambda :\lambda B_X\cap X_{n+1}\subseteq A\},$$

$$b_n(A,X)\le \min \{d_n(A,X),d^n(A,X)\}.$$

$$d_n(T)=d_n(T(B_X),Y),{\delta }_n(T)={\delta }_n(T(B_X),Y),d^n(T)=d^n(T(B_X),Y),b_n(T)=b_n(T(B_X),Y).$$

$$d^n(T^{\ast })=d_n(T).$$

$$f=\sum _{k=0}^{\infty }{Y}^{(k)},$$

$${\pi }_{k}f(x)={\int }_{S^{d-1}}f(y)Z_x^{(k)}(y)d_{{\tau }_d}(y),f\in L^2(S^{d-1}).$$

$$d_k=\dim {\mathcal{H}}_k=\left(\genfrac{}{}{0ex}{}{d+k-1}{k}\right)-\left(\genfrac{}{}{0ex}{}{d+k-3}{k-2}\right).$$

$$\sum _{j=1}^{d_k}\overline{Y_j^{(k)}(x)}{Y}_j^{(k)}(y)=Z_x^{(k)}(y)=\frac{d+2k-2}{\widetilde{{\omega }_d}(d-2)}{P}_k^{(d-2)/2}(⟨x,y⟩),x,y\in S^{d-1},$$

$${\tilde{Z}}^{(k)}(t)={\tilde{Z}}_d^{(k)}(t)=\frac{d+2k-2}{\widetilde{{\omega }_d}(d-2)}{P}_k^{(d-2)/2}(t),t\in [-1,1].$$

$$\overline{{\rm Y}}(z)=(x_1,y_1,\dots ,x_d,y_d),z_k=x_k+i{y}_k,k=1,\dots ,d.$$

$${\int }_{{\Omega }_d}fd{\sigma }_d={\int }_{S^{2d-1}}(f\circ {{\rm Y}}^{-1})d{\tau }_{2d}$$

$$\Vert f{\Vert }_p:={\left({\int }_{{\Omega }_d}|f(z){|}^{p}d{\sigma }_d(z)\right)}^{1/p}<\infty ,1\le p<\infty ,$$

$$\Vert f{\Vert }_{\infty }:=\underset{z\in {\Omega }_d}{\mathrm{esssup}}|f(z)|<\infty .$$

$$(f,g)={\int }_{{\Omega }_d}f(z)\overline{g(z)}d{\sigma }_d(z),f,g\in L^2({\Omega }_d).$$

$$p(z):=p(z,\overline{z})=\sum _{|\alpha |\le m,|\beta |\le n}{p}_{\alpha ,\beta }{z}^{\alpha }{\overline{z}}^{\beta },z\in {\mathbb{C}}^d,p_{\alpha ,\beta }\in \mathbb{C},\alpha ,\beta \in {\mathbb{N}}^d.$$

$${\Delta }_{(2d)}:=4\sum _{j=1}^d\frac{{\partial }^2}{\partial z_j\partial {\overline{z}}_j}$$

$$d_{m,n}=\dim {\mathcal{H}}_{m,n}=\left(\genfrac{}{}{0ex}{}{m+d-1}{m}\right)\left(\genfrac{}{}{0ex}{}{n+d-1}{n}\right)-\left(\genfrac{}{}{0ex}{}{m+d-2}{m-1}\right)\left(\genfrac{}{}{0ex}{}{n+d-2}{n-1}\right).$$

$$\frac{(m+n)(mn{)}^{d-2}}{(d-1)!(d-2)!}\le d_{m,n}\le \frac{(m+n)(mn{)}^{d-2}}{(d-1)!(d-2)!}+C(m+n)m^{d-2}{n}^{d-3},(m,n)\in {\mathbb{N}}^2.$$

$$f=\sum _{m,n\in \mathbb{N}}{Y}^{(m,n)},$$

$$R_{m,n}^{d-2}(z)=\{\begin{array}{cc}\frac{1}{P_n^{(d-2,m-n)}(1)}{z}^{m-n}{P}_n^{(d-2,m-n)}(2|z{|}^2-1),& \text{mbox}ifm\ge n,\\ \frac{1}{P_m^{(d-2,n-m)}(1)}{\overline{z}}^{n-m}{P}_m^{(d-2,n-m)}(2|z{|}^2-1),& \text{mbox}ifm<n,\end{array}$$

$$Z_w^{(m,n)}(z)=\sum _{k=1}^{d_{m,n}}\overline{Y_k(w)}{Y}_k(z)=\frac{d_{m,n}}{{\omega }_d}{R}_{m,n}^{d-2}(⟨z,w⟩),w,z\in {\Omega }_d,$$

$$Z_w^{(m,n)}(w)=\sum _{k=1}^{d_{m,n}}|Y_k(z){|}^2=\frac{d_{m,n}}{{\omega }_d}.$$

$${\tilde{Z}}^{(m,n)}(t)=\frac{d_{m,n}}{{\omega }_d}{R}_{m,n}^{d-2}(t),t\in \mathbb{D},$$

$$f\ast Z_e^{(m,n)}(w)=f\ast {\tilde{Z}}^{(m,n)}(w)$$

$$Z_x^{(k)}\circ {\rm Y}=\sum _{m+n=k}{Z}_w^{(m,n)},w={{\rm Y}}^{-1}(x).$$

$$\frac{2d+2k-2}{{\omega }_d(2d-2)}{P}_k^{(2d-2)/2}(⟨{\rm Y}(z),{\rm Y}(w)⟩)$$

$$f\sim \sum _{m,n\in \mathbb{N}}{\lambda }_{m,n}\phi \ast {\tilde{Z}}^{(m,n)},$$

Full text

**keywords: Complex sphere, Width, Multiplier, Smooth function.

MSC 2010 classification numbers: 41A46, 42B25, 42B15.

**

1 Introduction

In [2, 3, 10, 11, 12, 13, 14, 15, 16, 17, 18] techniques were developed to obtain estimates for $n$-widths of multiplier operators defined for functions on torus and on two-points homogeneous spaces $\mathbb{M}^{d}$: $\mathbb{S}^{d}$, $\mathbb{P}^{d}(\mathbb{R})$, $\mathbb{P}^{d}(\mathbb{C})$, $\mathbb{P}^{d}(\mathbb{H})$, $\mathbb{P}^{16}(Cay)$. In this work we continue the development of methods of estimating $n$-widths of multiplier operators.

The studies on asymptotic estimates for Kolmogorov $n$-widths of Sobolev classes on the circle were performed by several important mathematicians such as Rudin, Stechkin, Gluskin, Ismagilov, Maiorov, Makovoz and Scholz in [6, 20, 21, 29, 31, 32], they were initiated by Kolmogorov [9] in 1936 and completed by Kashin [7, 8] in 1977. Several techniques were applied in these different cases, among them we highlight a technique of discretization due to Maiorov and the Borsuk theorem. Observing the historical evolution of the study of $n$-widths, it is possible to note that it has been an usual practice to use different techniques in proofs of lower and upper bounds and in estimates for classes of finitely and infinitely differentiable functions (see, e.g. [28]). Among the mathematicians who worked with n-widths of sets of analytic functions, we highlight Babenko, Fisher, Micchelli, Taikov and Tichomirov, who obtained estimates for sets of analytic functions in Hardy spaces in [1, 5, 33, 34]. Among the different tools used, we can emphasize the properties of the class of Blaschke products of degree $m$ or less. In 1980, Melkman in [22] obtained estimates for a set of entire functions in $C[-T,T]$, using other technique. One of the goals of this work, is to give an unified treatment in the study of $n$-widths of sets of functions defined on complex spheres, determined by multiplier operators.

Let $\lambda:[0,\infty)\rightarrow\mathbb{C}$ be a bounded function and consider the sequences $\Lambda_{*}=(\lambda_{m,n}^{*})_{m,n\in\mathbb{N}}$, $\lambda_{m,n}^{*}=\lambda(m+n)$ and $\Lambda=(\lambda_{m,n})_{m,n\in\mathbb{N}}$, $\lambda_{m,n}=\lambda(\max\{m,n\})$. Let $\Omega_{d}$ be the unit sphere of $\mathbb{C}^{d}$ and for $m,n\in\mathbb{N}$, let $\mathcal{H}_{m,n}$ be the subspace of $L^{2}(\Omega_{d})$ of all complex spherical harmonics of bi-degree $(m,n)$. Given $f\in L^{2}(\Omega_{d})$, for each $m,n\in\mathbb{N}$, there is an unique $Y^{(m,n)}\in\mathcal{H}_{m,n}$ such that $f=\sum_{m,n}Y^{(m,n)}$ with convergence in the norm of $L^{2}(\Omega_{d})$ (see Section 2 for definitions and properties). For $f\in L^{2}(\Omega_{d})$ we define $\Lambda_{*}f=\sum_{m,n}\lambda_{m,n}^{*}Y^{(m,n)}$ and $\Lambda f=\sum_{m,n}\lambda_{m,n}Y^{(m,n)}$ where the convergence occurs in the norm of $L^{2}(\Omega_{d})$. Let $U_{p}$ be the closed unit ball of $L^{p}(\Omega_{d})$, $1\leq p\leq\infty$. In this paper we study asymptotic estimates for the $n$-widths of Kolmogorov, linear, of Gelfand and of Bernstein, of the sets $\Lambda_{*}U_{p}$ and $\Lambda U_{p}$ in $L^{q}(\Omega_{d})$, for $1\leq p,q\leq\infty$.

Consider two Banach spaces X and Y. The norm of X will be denoted by $\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|$ or $\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|_{X}$ and the closed unit ball $\{x\in X:\|\hskip 0.72229ptx\hskip 0.72229pt\|\leq 1\}$ by $B_{X}$. Let A be a compact, convex and centrally symmetric subset of X. The Kolmogorov $n$-width of A in X is defined by

[TABLE]

where $X_{n}$ runs over all subspaces of X of dimension $n$. The linear $n$-width $\delta_{n}(A,X)$ is defined by

[TABLE]

where $P_{n}$ runs over all bounded linear operators $P_{n}:X\longrightarrow X$ whose range has dimension $n$. The Gelfand $n$-width of A in X is defined by

[TABLE]

where $L^{n}$ runs over all subspaces of X of codimension $n$. The Bernstein $n$-width of A in X is defined by

[TABLE]

where $X_{n+1}$ is any $(n+1)$-dimensional subspace of X. The following inequality is always valid:

[TABLE]

We define the $n$-widths of Kolmogorov, linear, Gelfand and Bernstein of a operator $T\in\mathcal{L}(X,Y)$, respectively, by

[TABLE]

Consider $T\in\mathcal{L}(X,Y)$ and let $T^{*}$ be its adjoint operator. If $T$ is compact or $Y$ is reflexive, then (see [28], p. 34)

[TABLE]

In Section 2 we make a brief study about real and complex spherical harmonics. In Section 3 we study estimates for $n$-widths of sets of functions on the complex sphere $\Omega_{d}$, associated with the multiplier operators of the type $\Lambda_{*}$. We prove that the $n$-widths of the sets $\Lambda_{*}U_{p}$ in $L^{q}(\Omega_{d})$ can be obtained from $n$-widths of sets of functions associated with multiplier operators on the real sphere $S^{2d-1}$. As application we obtain estimates for the Kolmogorov $n$-width of Sobolev classes and of finitely differentiable, infinitely differentiable and analytic functions on the complex sphere. In Section 4 and in the following sections we study $n$-widths of sets of functions on the complex sphere $\Omega_{d}$ associated with the multiplier operators of the type $\Lambda$. We note that in this case the results can not be transferred from results already obtained to the real sphere $S^{2d-1}$. In Section 4 we study estimates for Levy means of special norms on finite-dimensional Banach spaces which are associated with the multiplier sequences $(\lambda_{m,n})_{m,n\in\mathbb{N}}$.

In Section 5 we prove two theorems which provide us with lower and upper estimates for Kolmogorov $n$-widths of the sets $\Lambda U_{p}$ in $L^{q}(\Omega_{d})$, for general multiplier operators of the type $\Lambda$. The main tool used in these proofs are the results from Section 4.

In Section 6 we apply the results of Section 5 in the study of the Kolmogorov $n$-widths of the multiplier operators $\Lambda^{(1)}$ and $\Lambda^{(2)}$ associated with the functions $\lambda^{(1)}(t)=t^{-\gamma}(\ln t)^{-\xi}$ for $t>1$, $\lambda^{(1)}(t)=0$ for $0\leq t\leq 1$, and $\lambda^{(2)}(t)=e^{-\gamma t^{r}}$ for $t\geq 0$, where $\gamma,\xi,r\in\mathbb{R}$, $\gamma,r>0$ and $\xi\geq 0$, respectively. We have that the sets $\Lambda^{(1)}U_{p}$ and $\Lambda_{*}^{(1)}U_{p}$ are sets of finitely differentiable functions on $\Omega_{d}$, and $\Lambda^{(2)}U_{p}$ and $\Lambda_{*}^{(2)}U_{p}$ are sets of infinitely differentiable functions if $0<r<1$ and of analytic functions if $r\geq 1$ (see [24, 25]). We prove that estimates for the Kolmogorov $n$-widths of the sets $\Lambda^{(1)}U_{p}$ and $\Lambda^{(2)}U_{p}$ in $L^{q}(\Omega_{d})$ are sharp when $2\leq p,q<\infty$.

We note that the order of decrease of the $n$-widths $d_{n}(\Lambda^{(1)}U_{p},L^{q})$ and $d_{n}(\Lambda_{\ast}^{(1)}U_{p},L^{q})$ is the same, but it is different for the $n$-widths $d_{n}(\Lambda^{(2)}U_{p},L^{q})$ and $d_{n}(\Lambda_{\ast}^{(2)}U_{p},L^{q})$.

In this work there are several universal constants which enter into the estimates. These positive constants are mostly denoted by the letters $C,C_{1},C_{2},\ldots$. We did not carefully distinguish between the different constants, neither did we try to get good estimates for them. The same letter will be used to denote different universal constants in different parts of the paper. For ease of notation we will write $a_{n}\gg b_{n}$ for two sequences if $a_{n}\geq Cb_{n}$ for $n\in\mathbb{N}$, $a_{n}\ll b_{n}$ if $a_{n}\leq Cb_{n}$ for $n\in\mathbb{N}$, and $a_{n}\asymp b_{n}$ if $a_{n}\ll b_{n}$ and $a_{n}\gg b_{n}$. Also, we will write $(a)_{+}=a$ if $a>0$ and $(a)_{+}=0$ if $a\leq 0$.

2 Analysis on the spheres

We write $\mathbb{N}=\{0,1,2,\ldots\}$. For $x,y\in\mathbb{R}^{d}$ we denote by $\langle x,y\rangle$ the usual inner product $\langle x,y\rangle=x_{1}y_{1}+\cdots+x_{d}y_{d}$. Let

$S^{d-1}=\{x\in\mathbb{R}^{d}:\langle x,x\rangle=1\}$

be the unit sphere in $\mathbb{R}^{d}$. Let $\tau_{d}$ be the Lebesgue measure on $S^{d-1}$, consider the usual Banach spaces $L^{p}(S^{d-1})$, $1\leq p\leq\infty$ of $\tau_{d}$-measurable complex functions on $S^{d-1}$ and let $\widetilde{U_{p}}=\widetilde{U_{p}}(S^{d-1})=\{\phi\in L^{p}(S^{d-1}):\|\hskip 0.72229pt\phi\hskip 0.72229pt\|_{p}\leq 1\}$. We denote by $\mathcal{H}_{k}$ the subspace of $L^{2}(S^{d-1})$ of all spherical harmonics of degree $k$, that is, the space of the restrictions to $S^{d-1}$ of all polynomials $P(x_{1},\ldots,x_{d})$ which are homogeneous of degree $k$ and harmonic. Given $f\in L^{2}(S^{d-1})$, for each $k\in\mathbb{N}$, there exists an unique $Y^{(k)}\in\mathcal{H}_{k}$ such that

[TABLE]

where the series converges to $f$ in $L^{2}(S^{d-1})$. Let $\pi_{k}:L^{2}(S^{d-1})\longrightarrow\mathcal{H}_{k}$ be the harmonic projection operator, $\pi_{k}(f)=Y^{(k)}$. For each $x\in S^{d-1}$ and $k\in\mathbb{N}$, there exists an unique function $Z^{(k)}_{x}\in\mathcal{H}_{k}$, called the zonal harmonic of degree $k$ with pole $x$, such that

[TABLE]

We have that $\mathcal{H}_{k}\perp\mathcal{H}_{l}$, for $k\neq l$, with respect to the usual inner product in $L^{2}(S^{d-1})$ and for $k\geq 2$,

[TABLE]

Let $\{Y_{j}\}_{j=1}^{d_{k}}$ be an orthonormal basis of $\mathcal{H}_{k}$. The following addition formula is known:

[TABLE]

where $\widetilde{\omega_{d}}$ denotes the surface area of $S^{d-1}$ and $P^{(d-2)/2}_{k}$ denotes the Gegenbauer polynomial of degree $k$. Let us denote by $\widetilde{Z}^{(k)}_{d}$ the function

[TABLE]

Let $\overline{\Upsilon}=\overline{\Upsilon}_{d}:\mathbb{C}^{d}\rightarrow\mathbb{R}^{2d}$ be given by

[TABLE]

For $z,w\in\mathbb{C}^{d}$, we denote by $\langle z,w\rangle$ the usual inner product $\langle z,w\rangle=z_{1}\overline{w}_{1}+\cdots+z_{d}\overline{w}_{d}$. Let

$\Omega_{d}=\{z\in\mathbb{C}^{d}:\langle z,z\rangle=1\}$

be the unit sphere in $\mathbb{C}^{d}$. We denote by $\Upsilon$ or $\Upsilon_{d}$, the restriction of $\overline{\Upsilon}$ to $\Omega_{d}$. Since $\Upsilon(\Omega_{d})=S^{2d-1}$ we can define the measure $\sigma_{d}$ on the Borel $\sigma$-algebra $\mathcal{B}(\Omega_{d})$ of $\Omega_{d}$ by $\sigma_{d}(E)=\tau_{2d}(\Upsilon(E))$, $E\in\mathcal{B}(\Omega_{d})$. A measurable function $f:\Omega_{d}\longrightarrow\mathbb{C}$ is $\sigma_{d}$-integrable if

[TABLE]

is well defined. We denote by $L^{p}=L^{p}(\Omega_{d})$, $1\leq p\leq\infty$, the vector space of all measurable functions defined on $\Omega_{d}$ and with values in $\mathbb{C}$, satisfying

[TABLE]

[TABLE]

The space $L^{2}(\Omega_{d})$ is a Hilbert space with the inner product given by

[TABLE]

Let $U_{p}=U_{p}(\Omega_{d})=\{\phi\in L^{p}(\Omega_{d}):\|\hskip 0.72229pt\phi\hskip 0.72229pt\|_{p}\leq 1\}$, $\mathbb{D}=\{z\in\mathbb{C}:|z|\leq 1\}$ and let $\nu_{\alpha}$ be the Borel measure on $\mathbb{D}$ given by

$\nu_{\alpha}(E)=\displaystyle\int_{E}(1-|z|^{2})^{\alpha}d\mu_{1}(z),\hskip 28.45274ptE\in\mathcal{B}(\mathbb{D}).$

where $\alpha>-1$ and $\mu_{1}\circ\overline{\Upsilon}^{-1}_{1}$ is the Lebesgue measure on $\mathbb{R}^{2}$. For $f\in L^{1}(\Omega_{d})$ and $K\in L^{1}(\mathbb{D},\nu_{\alpha})$, the convolution function $f\ast K$ is defined by

$(f\ast K)(w)=\displaystyle\int_{\Omega_{d}}f(z)\overline{K(\langle z,w\rangle)}d\sigma_{d}(z),\hskip 42.67912ptw\in\Omega_{d}.$

Let $\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{N}^{d}$ and $z=(z_{1},\ldots,z_{d})\in\mathbb{C}^{d}$. We write $|\alpha|=\alpha_{1}+\cdots+\alpha_{d}$, $z^{\alpha}=z_{1}^{\alpha_{1}}\cdots z_{d}^{\alpha_{d}}$ and $\overline{z}=(\overline{z}_{1},\ldots,\overline{z}_{d})$. We denote by $\mathbb{P}(\mathbb{C}^{d})$ the vector space of all polynomials in the independent variables $z$ and $\overline{z}$ . If $p\in\mathbb{P}(\mathbb{C}^{d})$ then there are $m,n\in\mathbb{N}$ such that

[TABLE]

The vector subspace of $\mathbb{P}(\mathbb{C}^{d})$ formed by the polynomials that are homogeneous of degree $m$ in the variable $z$ and of degree $n$ in the variable $\overline{z}$, will be denoted by $\mathbb{P}_{m,n}(\mathbb{C}^{d})$. The subspace of $\mathbb{P}_{m,n}(\mathbb{C}^{d})$ of all polynomials that are in the kernel of the complex laplacian

[TABLE]

will be denoted by $\mathbb{H}_{m,n}(\mathbb{C}^{d})$.

The space of the restrictions of the polynomials in $\mathbb{P}_{m,n}(\mathbb{C}^{d})$ to the complex sphere $\Omega_{d}$ will be denoted by $\mathcal{P}_{m,n}=\mathcal{P}_{m,n}(\Omega_{d})$ and of the restrictions of the polynomials in $\mathbb{H}_{m,n}(\mathbb{C}^{d})$ to $\Omega_{d}$ by $\mathcal{H}_{m,n}=\mathcal{H}_{m,n}(\Omega_{d})$. We have that $\mathcal{H}_{m,n}\perp\mathcal{H}_{r,s}$, for $(m,n)\neq(r,s)$, with respect to the usual inner product in $L^{2}(\Omega_{d})$. The elements of $\mathcal{H}_{m,n}$ are called complex spherical harmonics of degree $m$ in $z$ and degree $n$ in $\overline{z}$. Let $m,n\in\mathbb{N}$ and $m+n=k$. Since $\mathcal{H}_{m,n}\subset\{Y\circ\Upsilon:Y\in\mathcal{H}_{k}\}$ and there is an orthonormal basis of $\mathcal{H}_{k}$ consisting of only real functions, then also there is an orthonormal basis of $\mathcal{H}_{m,n}$ consisting of only real functions. The dimension of $\mathcal{H}_{m,n}$ is given by

[TABLE]

We observe that there is an absolute constant $C>0$ such that

[TABLE]

It is known (see [30], Chapter 12) that if $f\in L^{2}(\Omega_{d})$ then there exists an unique decomposition

[TABLE]

where $Y^{(m,n)}\in\mathcal{H}_{m,n}$ for $m,n\in\mathbb{N}$ and the above series converges to $f$ in $L^{2}(\Omega_{d})$. The projection operator $\pi_{m,n}$ from $L^{2}(\Omega_{d})$ on $\mathcal{H}_{m,n}$ is defined by $\pi_{m,n}f=Y^{(m,n)}$. For every $w\in\Omega_{d}$ there exists an unique $Z_{w}^{(m,n)}\in\mathcal{H}_{m,n}$ that satisfies $(\pi_{m,n}f)(w)=(f,Z_{w}^{(m,n)})$, $f\in L^{2}(\Omega_{d})$. The function $Z_{w}^{(m,n)}$ is called the zonal harmonic of degree $(m,n)$ with pole $w$.

Next, we introduce some classes of disk polynomials. The disk polynomial of degree $m$ in $z$ and degree $n$ in $\overline{z}$ associated to the integer $d-2$ is the polynomial $R_{m,n}^{d-2}$ given by

[TABLE]

where $P_{k}^{(d-2,m-n)}$ is the usual Jacobi polynomial of degree $k$ associated to the pair of numbers $(d-2,m-n)$. Let $\{Y_{j}\}_{j=1}^{d_{m,n}}$ be an orthonormal basis of $\mathcal{H}_{m,n}$. The following addition formula is known:

[TABLE]

where $\omega_{d}=\sigma_{d}(\Omega_{d})=\tau_{2d}(S^{2d-1})$. In particular

[TABLE]

Let us denote by $\widetilde{Z}^{(m,n)}$ the function

[TABLE]

and let $e=e_{d}=(0,0,...,0,1)$, $f\in L^{2}(\Omega_{d})$. The convolution product $f\ast Z^{(m,n)}_{e}$ is given by

[TABLE]

It follow a relationship between the real zonal harmonics on $S^{2d-1}$ and the complex zonal harmonics on $\Omega_{d}$, as a consequence, we obtain an expression relating polynomials on the disk and Gegenbauer polynomials.

Proposition 2.1**.**

For any $k\in\mathbb{N}$ and $x\in S^{2d-1}$, we have that

[TABLE]

Corollary 2.2**.**

Given $z,w\in\Omega_{d}$,

[TABLE]

We denote $\hat{Z}^{(k)}_{d}=\sum_{m+n=k}\widetilde{Z}^{(m,n)}$ and we have from the above corollary that $\widetilde{Z}^{(k)}_{2d}(\langle\Upsilon(z),\Upsilon(w)\rangle)=\hat{Z}^{(k)}_{d}(\langle z,w\rangle)$, $z,w\in\Omega_{d}$. For a detailed study about real and complex spherical harmonics see [4, 23, 30].

3 Multiplier operators of type $\Lambda_{*}$

We denote by $\mathcal{H}$ the vector space generated by the union $\bigcup_{m,n}\mathcal{H}_{m,n}$. The elements of $\mathcal{H}$ are finite linear combinations of elements of the subspaces $\mathcal{H}_{m,n}$ and $\mathcal{H}$ is dense in $L^{p}(\Omega_{d})$ for $1\leq p<\infty$. Also we denote by $\widetilde{\mathcal{H}}$ the vector space generated by the union $\bigcup_{k}\mathcal{H}_{k}$ which is dense in $L^{p}(S^{2d-1})$ for $1\leq p<\infty$.

Let $\Lambda=\{\lambda_{m,n}\}_{m,n\in\mathbb{N}}$ be a sequence of complex numbers and $1\leq p,q\leq\infty$. If for any $\varphi\in L^{p}(\Omega_{d})$ there is a function $f=\Lambda\varphi\in L^{q}(\Omega_{d})$ with formal expansion in spherical harmonic

[TABLE]

such that $\|\hskip 0.72229pt\Lambda\hskip 0.72229pt\|_{p,q}=\sup\{\|\hskip 0.72229pt\Lambda\varphi\hskip 0.72229pt\|_{q}:\varphi\in U_{p}\}<\infty$, we say that the multiplier operator $\Lambda$ is bounded from $L^{p}$ into $L^{q}$ with norm $\|\hskip 0.72229pt\Lambda\hskip 0.72229pt\|_{p,q}$. Now let $\widetilde{\Lambda}=\{\lambda_{k}\}_{k\in\mathbb{N}}$ be a sequence of complex numbers and $1\leq p,q\leq\infty$. If for any $\varphi\in L^{p}(S^{2d-1})$ there is a function $f=\widetilde{\Lambda}\varphi\in L^{q}(S^{2d-1})$ with formal expansion in spherical harmonic

[TABLE]

such that $\|\hskip 0.72229pt\widetilde{\Lambda}\hskip 0.72229pt\|_{p,q}=\sup\{\|\hskip 0.72229pt\widetilde{\Lambda}\varphi\hskip 0.72229pt\|_{q}:\varphi\in\widetilde{U}_{p}\}<\infty$, we say that the multiplier operator $\widetilde{\Lambda}$ is bounded from $L^{p}$ into $L^{q}$ with norm $\|\hskip 0.72229pt\widetilde{\Lambda}\hskip 0.72229pt\|_{p,q}$.

Lemma 3.1**.**

Let $\lambda:[0,\infty)\longrightarrow\mathbb{C}$ be a bounded function, $\lambda_{m,n}^{\ast}=\lambda(m+n)$ and $\lambda_{k}=\lambda(k)$ for $m,n,k\in\mathbb{N}$. Consider the multiplier operators associated with the sequences $\Lambda_{\ast}=\{\lambda_{m,n}^{\ast}\}_{m,n\in\mathbb{N}}$ and $\widetilde{\Lambda}=\{\lambda_{k}\}_{k\in\mathbb{N}}$ defined on $\mathcal{H}$ and on $\widetilde{\mathcal{H}}$ respectively. Let $1\leq p,q\leq\infty$ and suppose that $\Lambda_{\ast}$ is bounded from $L^{p}$ to $L^{q}$. Then

[TABLE]

and

[TABLE]

Proof.

For $\varphi\in L^{2}(\Omega_{d})$ we get $\|\hskip 0.72229pt\Lambda_{\ast}\varphi\hskip 0.72229pt\|_{2}\leq\left(\sup_{m,n\in\mathbb{N}}|\lambda_{m,n}^{\ast}|\right)\|\hskip 0.72229pt\varphi\hskip 0.72229pt\|_{2}$, then $\Lambda_{\ast}$ is a bounded operator on $L^{2}(\Omega_{d})$. Thus

[TABLE]

where the above series converges in $L^{2}$. On the other hand, from Corollary 2.2, for $w\in\Omega_{d}$ we have

[TABLE]

Thus

[TABLE]

For $2\leq p\leq\infty$, we have that $L^{\infty}(\Omega_{d})\subseteq L^{p}(\Omega_{d})\subseteq L^{2}(\Omega_{d})$, then $\Lambda_{\ast}\varphi=(\widetilde{\Lambda}(\varphi\circ\Upsilon^{-1}))\circ\Upsilon$ for all $\varphi\in L^{p}(\Omega_{d})$ with $2\leq p\leq\infty$, in particular for $\varphi\in\mathcal{H}$. It follows for $1\leq p<\infty$ that

[TABLE]

Thus $\widetilde{\Lambda}$ is a bounded operator from $L^{p}(S^{2d-1})$ to $L^{q}(S^{2d-1})$.

Now consider $1\leq p\leq 2$. Given $\varphi\in L^{p}(\Omega_{d})$ and for each $n\in\mathbb{N}$, let $\varphi_{n}\in\mathcal{H}$ be such that $\varphi_{n}\rightarrow\varphi$ in $L^{p}(\Omega_{d})$. Then $\Lambda_{\ast}\varphi_{n}\longrightarrow\Lambda_{\ast}\varphi$ in $L^{q}(\Omega_{d})$, $\Lambda_{\ast}\varphi_{n}=(\widetilde{\Lambda}(\varphi_{n}\circ\Upsilon^{-1}))\circ\Upsilon$ and $\widetilde{\Lambda}(\varphi_{n}\circ\Upsilon^{-1})\longrightarrow\widetilde{\Lambda}(\varphi\circ\Upsilon^{-1})$ in $L^{q}(S^{2d-1})$. Therefore $\widetilde{\Lambda}(\varphi_{n}\circ\Upsilon^{-1})\circ\Upsilon\longrightarrow\widetilde{\Lambda}(\varphi\circ\Upsilon^{-1})\circ\Upsilon$ in $L^{q}(\Omega_{d})$, then $\Lambda_{\ast}\varphi=\widetilde{\Lambda}(\varphi\circ\Upsilon^{-1}))\circ\Upsilon$ q.t.p. . Thus we proved (3) and the proof of (4) follows easily from (3). ∎

Theorem 3.2**.**

Let $\Lambda_{\ast}$ and $\widetilde{\Lambda}$ as in Lemma 3.1. Then

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

By the Lemma 3.1

[TABLE]

where $X_{n}$ runs over all subspaces of $L^{q}(\Omega_{d})$ of dimension $n$ and $\widetilde{X}_{n}$ over all subspaces of $L^{q}(S^{2d-1})$ of dimension $n$;

[TABLE]

where $L^{n}$ runs over all subspaces of $L^{q}(\Omega_{d})$ of codimension $n$ and $\widetilde{L}^{n}$ over all subspaces of $L^{q}(S^{2d-1})$ of codimension $n$;

[TABLE]

where $P_{n}$ runs over all bounded linear operators $P_{n}:L^{q}(\Omega_{d})\longrightarrow L^{q}(\Omega_{d})$ whose range has dimension $n$ and $\widetilde{P}_{n}$ over all bounded linear operators $\widetilde{P}_{n}:L^{q}(S^{2d-1})\longrightarrow L^{q}(S^{2d-1})$ whose range has dimension $n$;

[TABLE]

where $X_{n+1}$ runs over all subspaces of $L^{q}(\Omega_{d})$ of dimension $n+1$ and $\widetilde{X}_{n+1}$ over all subspaces of $L^{q}(S^{2d-1})$ of dimension $n+1$. ∎

Remark 3.3**.**

For $\gamma\in\mathbb{R}$, we donote $\Lambda^{\gamma}=\{\lambda_{m,n}^{\gamma}\}_{m,n\in\mathbb{N}}$, with $\lambda_{m,n}^{\gamma}=\lambda^{\gamma}(m+n)$, where the function $\lambda^{\gamma}:[0,\infty)\longrightarrow\mathbb{R}$ is defined by $\lambda^{\gamma}(t)=(t(t+2d-2))^{-\gamma/2}$ if $t>0$ and $\lambda(0)=0$. The Sobolev space $\overline{W}_{p}^{\gamma}$ on $\Omega_{d}$ with $\gamma>0$, $1\leq p\leq\infty$, is defined as the set

[TABLE]

endowed with norm

[TABLE]

We define the Sobolev classe $W_{p}^{\gamma}$ as the set

[TABLE]

Now let $\widetilde{\Lambda}^{\gamma}=\{\lambda^{\gamma}_{k}\}_{k\in\mathbb{N}}$, $\lambda^{\gamma}_{k}=\lambda^{\gamma}(k)$. The Sobolev classe $\widetilde{W}_{p}^{\gamma}$ on $S^{2d-1}$ is given by

[TABLE]

Applying the Lemma 3.1 for the operator $\Lambda^{\gamma}$ we get

[TABLE]

and by Theorem 3.2 we have that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The estimates in the theorem below follow as a consequence of (5) and of estimates already obtained for the Kolmogorov $n$-width for the Sobolev classes $\widetilde{W}_{p}^{\gamma}$ on the real sphere $S^{2d-1}$, which can be found in [2, 3, 11, 12, 14, 16, 18].

Theorem 3.4**.**

For the Kolmogorov n-width of Sobolev classes on $\Omega_{d}$, we have in particular the following estimates:

• (a)

if $1\leq p=q\leq\infty$, $2\leq q\leq p<\infty$, $\gamma>0$,

$d_{n}(W_{p}^{\gamma}(\Omega_{d}),L^{q}(\Omega_{d}))\asymp n^{-\gamma/(2d-1)}$;

• (b)

if $1\leq p=q\leq\infty$, $2\leq q\leq p\leq\infty$, $\gamma>0$,

$n^{-\gamma/(2d-1)}(\ln n)^{-1/2}\ll d_{n}(W_{p}^{\gamma}(\Omega_{d}),L^{q}(\Omega_{d}))\ll n^{-\gamma/(2d-1)}$;

• (c)

if $2\leq q\leq p<\infty$, $\gamma/(2d-1)>1/2$,

$d_{n}(W_{p}^{\gamma}(\Omega_{d}),L^{q}(\Omega_{d}))\asymp n^{-\gamma/(2d-1)}$;

• (d)

if $2\leq q\leq p\leq\infty$, $\gamma/(2d-1)>1/2$,

$n^{-\gamma/(2d-1)}\ll d_{n}(W_{p}^{\gamma}(\Omega_{d}),L^{q}(\Omega_{d}))\ll n^{-\gamma/(2d-1)}(\ln n)^{1/2}$;

• (e)

if $1\leq p\leq q\leq 2$, $\gamma/(2d-1)>1/p-1/q$,

$d_{n}(W_{p}^{\gamma}(\Omega_{d}),L^{q}(\Omega_{d}))\asymp n^{-\gamma/(2d-1)+1/p-1/q}$;

• (f)

if $1\leq p\leq q\leq 2$, $\gamma>0$,

$d_{n}(W_{p}^{\gamma}(\Omega_{d}),L^{q}(\Omega_{d}))\asymp n^{-\gamma/(2d-1)}$;

• (g)

if $1\leq p\leq q\leq 2$, $\gamma>0$,

$n^{-\gamma/(2d-1)}(\ln n)^{-1/2}\ll d_{n}(W_{p}^{\gamma}(\Omega_{d}),L^{q}(\Omega_{d}))\ll n^{-\gamma/(2d-1)}$;

• (h)

if $1\leq p\leq 2\leq q<\infty$, $\gamma/(2d-1)>1/p$,

$d_{n}(W_{p}^{\gamma}(\Omega_{d}),L^{q}(\Omega_{d}))\asymp n^{-\gamma/(2d-1)+1/p-1/2}$;

• (i)

if $1\leq p\leq 2\leq q\leq\infty$, $\gamma/(2d-1)>1/p$,

$n^{-\gamma/(2d-1)+1/p-1/2}\ll d_{n}(W_{p}^{\gamma}(\Omega_{d}),L^{q}(\Omega_{d}))\ll n^{-\gamma/(2d-1)+1/p-1/2}(\ln n)^{1/2}$.

Other estimates for the Kolmogorov $n$-widths of the Sobolev classes $W_{p}^{\gamma}(\Omega_{d})$, as also to the $n$-widths of Gelfand, linear and Bernstein, can be obtained using (6)-(8).

Remark 3.5**.**

For $\gamma,\xi\in\mathbb{R}$, $\gamma>0$ and $\xi\geq 0$ , we denote $\Lambda_{\ast}^{(1)}=\{\lambda_{m,n}^{(1),\ast}\}_{m,n\in\mathbb{N}}$, with $\lambda_{m,n}^{(1),\ast}=\lambda^{(1)}(m+n)$, where the function $\lambda^{(1)}:[0,\infty)\longrightarrow\mathbb{R}$ is defined by $\lambda^{(1)}(t)=t^{-\gamma}(\ln t)^{-\xi}$ for $t>1$ and $\lambda^{(1)}(t)=0$ for $0\leq t\leq 1$. We have that $\Lambda_{\ast}^{(1)}U_{p}$ is a set of finitely differentiable functions on $\Omega_{d}$, in particular, $\Lambda_{\ast}^{(1)}U_{p}$ is a Sobolev-type class if $\xi=0$. Now for $\gamma,r\in\mathbb{R}$, with $\gamma,r>0$, we denote $\Lambda_{\ast}^{(2)}=\{\lambda_{m,n}^{(2),\ast}\}_{m,n\in\mathbb{N}}$ with $\lambda_{m,n}^{(2),\ast}=\lambda^{(2)}(m+n)$, where the function $\lambda^{(2)}:[0,\infty)\longrightarrow\mathbb{R}$ is defined by $\lambda^{(2)}(t)=e^{-\gamma t^{r}}$ for $t\geq 0$. We have that $\Lambda_{\ast}^{(2)}U_{p}$ is a set of infinitely differentiable $(0<r<1)$ or analytic $(r\geq 1)$ functions on $\Omega_{d}$.

The results of the next three theorems follow as a consequence of the Theorem 3.2 and of results proved in [18].

Theorem 3.6**.**

If $1\leq p\leq\infty$, $2\leq q\leq\infty$ and $\gamma/(2d-1)>1/p$, we have

$d_{n}(\Lambda_{\ast}^{(1)}U_{p},L^{q})\ll n^{-\gamma/(2d-1)+(1/p-1/2)_{+}}(\ln n)^{-\xi}\left\{\begin{matrix}q^{1/2},&\mbox{if }q<\infty,\\ (\ln n)^{1/2},&\mbox{if }q=\infty,\end{matrix}\right.$**

and for $\gamma/(2d-1)>1/p-1/q$,

$d_{n}(\Lambda_{\ast}^{(1)}U_{p},L^{q})\gg n^{-\gamma/(2d-1)}(\ln n)^{-\xi}\vartheta_{n}$,

where

$\vartheta_{n}=\left\{\begin{matrix}1,&\mbox{if }1\leq p\leq 2,1<q\leq 2,\\ 1,&\mbox{if }2\leq p<\infty,2\leq q\leq\infty,\\ 1,&\mbox{if }1\leq p\leq 2\leq q\leq\infty,\\ (\ln n)^{-1/2},&\mbox{if }1\leq p\leq 2,q=1,\\ (\ln n)^{-1/2},&\mbox{if }p=\infty,2\leq q\leq\infty.\end{matrix}\right.$**

In particular, if $\gamma/(2d-1)>1/p,2\leq p,q<\infty$,

[TABLE]

Theorem 3.7**.**

For $\gamma,r\in\mathbb{R}$ such that $\gamma>0$, $0<r\leq 1$ and for all $k\in\mathbb{N}$ we have that

[TABLE]

where

[TABLE]

and $\vartheta_{k}$ is given in the previous theorem.

Theorem 3.8**.**

For $0<r\leq 1$, $1\leq p\leq\infty$, $2\leq q\leq\infty$ and for all $k\in\mathbb{N}$, we have that

[TABLE]

where $\mathcal{R}_{\ast}$ is the constant given in the previous theorem.

Other estimates for the Kolmogorov $n$-width for the operator $\Lambda_{\ast}^{(2)}$ when $r\in\mathbb{R}$, $r>1$, can also be obtained from results in [18].

4 Estimates for Levy means

In this paper, from this point forward, we will consider only the multiplier operators $\Lambda$ associated with the norm $|(m,n)|=\max\{m,n\}$.

Given $l,N,m,n,M_{1},M_{2}\in\mathbb{N}$, with $M_{1}<M_{2}$, we consider the following notations:

$A_{l}=\{(m,n)\in\mathbb{N}^{2}:|(m,n)|\leq l\},\hskip 11.38092ptA_{-1}=\emptyset\hskip 11.38092pt,a_{l}=\#(A_{l}\setminus A_{l-1})$

[TABLE]

[TABLE]

In this section and in the following sections we will consider $\mathcal{H}_{m,n}$ and $\mathcal{T}_{M_{1},M_{2}}$ as real vector subspaces of the real vector spaces $L^{p}(\Omega_{d},\mathbb{R})$, $1\leq p\leq\infty$.

Proposition 4.1**.**

There are positive constants $C_{1},C_{2},C_{3}$ such that

[TABLE]

[TABLE]

In particular, $\dim\mathcal{H}_{l}\asymp l^{2d-2}$ and $\dim\mathcal{T}_{N}\asymp N^{2d-1}$.

Proof.

The proofs of (9) and (10) are easily obtained using (2) and integration to estimate finite sums. ∎

Let $E=(\mathbb{R}^{n},\|\cdot\|)$ be a $n$-dimensional real Banach space with unit ball $B_{E}=\{x\in\mathbb{R}^{n}:\|\hskip 0.72229ptx\hskip 0.72229pt\|\leq 1\}$, and let $|||x|||=(\sum_{k=1}^{n}|x_{k}|^{2})^{1/2}$ and $\langle x,y\rangle=\sum_{k=1}^{n}x_{k}y_{k}$ be the euclidean norm and the inner product on $\mathbb{R}^{n}$. Let $S^{n-1}=\{x\in\mathbb{R}^{n}:|||x|||=1\}$ be the euclidean unit sphere in $\mathbb{R}^{n}$. The Levy mean of the norm $\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|$ on $\mathbb{R}^{n}$ is defined by

[TABLE]

where $\mu$ denotes the normalized Lebesgue measure on $S^{n-1}$.

Remark 4.2**.**

For $M_{1}+1\leq l\leq M_{2}$ let $A_{l}\setminus A_{l-1}=\{(m_{j}^{l},n_{j}^{l}):1\leq j\leq a_{l}\}$ such that $|(m_{j}^{l},n_{j}^{l})|\leq|(m_{j+1}^{l},n_{j+1}^{l})|$ for $1\leq j\leq a_{l}-1$ and let $\{Y_{1}^{(m_{j}^{l},n_{j}^{l})},\ldots,Y_{d_{m_{j}^{l},n_{j}^{l}}}^{(m_{j}^{l},n_{j}^{l})}\}$ be an orthonormal basis of $\mathcal{H}_{m_{j}^{l},n_{j}^{l}}$ consisting of only real functions. We denote $Y_{i}^{l,j}=Y_{i}^{(m_{j}^{l},n_{j}^{l})}$, $d_{l,j}=d_{m_{j}^{l},n_{j}^{l}}$ and $\mathcal{H}_{l,j}=\mathcal{H}_{m_{j}^{l},n_{j}^{l}}$. We consider the orthonormal basis

$\{\xi_{k}\}_{k=1}^{s}=\displaystyle\bigcup_{l=M_{1}+1}^{M_{2}}\displaystyle\bigcup_{j=1}^{a_{l}}\{Y_{1}^{l,j},\ldots,Y_{d_{l,j}}^{l,j}\}$**

of $\mathcal{T}_{M_{1},M_{2}}$ endowed with the order $Y_{1}^{M_{1}+1,1},\ldots,Y_{d_{M_{1}+1,1}}^{M_{1}+1,1},\ldots,Y_{1}^{M_{1}+1,a_{M_{1}+1}},\ldots,Y_{d_{M_{1}+1,a_{M_{1}+1}}}^{M_{1}+1,a_{M_{1}+1}},\\ \ldots,Y_{1}^{M_{2},1},\ldots,Y_{d_{M_{2},1}}^{M_{2},1},\ldots,Y_{1}^{M_{2},a_{M_{2}}},\ldots,Y_{d_{M_{2},a_{M_{2}}}}^{M_{2},a_{M_{2}}}$. Let $J:\mathbb{R}^{s}\longrightarrow\mathcal{T}_{M_{1},M_{2}}$ be the coordinate isomorphism which assigns to $\alpha=(\alpha_{1},\ldots,\alpha_{s})\in\mathbb{R}^{s}$ the function

$J(\alpha)=\displaystyle\sum_{k=1}^{s}\alpha_{k}\xi_{k}=\displaystyle\sum_{l=M_{1}+1}^{M_{2}}\displaystyle\sum_{j=1}^{a_{l}}\displaystyle\sum_{i=1}^{d_{l,j}}\alpha_{i}^{l,j}Y_{i}^{l,j}$**

*where $(\alpha_{1},\ldots,\alpha_{s})=(\alpha_{1}^{M_{1}+1,1},\ldots,\alpha_{d_{M_{1}+1,1}}^{M_{1}+1,1},\ldots,\alpha_{1}^{M_{1}+1,a_{M_{1}+1}},\ldots,\alpha_{d_{M_{1}+1,a_{M_{1}+1}}}^{M_{1}+1,a_{M_{1}+1}},\ldots,$

$\alpha_{1}^{M_{2},1},\ldots,\alpha_{d_{M_{2},1}}^{M_{2},1},\ldots,\alpha_{1}^{M_{2},a_{M_{2}}},\ldots,\alpha_{d_{M_{2},a_{M_{2}}}}^{M_{2},a_{M_{2}}})$.

Consider a function $\lambda:[0,\infty)\longrightarrow\mathbb{R}$, such that $\lambda(t)\neq 0$ for $t\geq 0$ and let $\Lambda=\{\lambda_{m,n}\}_{m,n\in\mathbb{N}}$ be the sequence of multipliers defined by $\lambda_{m,n}=\lambda(|(m,n)|)$. For $M_{1}+1\leq l\leq M_{2}$ and $1\leq j\leq a_{l}$, we write $\lambda_{j}^{l}=\lambda_{m_{j}^{l},n_{j}^{l}}=\lambda(|(m_{j}^{l},n_{j}^{l})|)$. Let $\Lambda_{s}=\{\widetilde{\lambda}_{k}\}_{k=1}^{s}$ be the numerical sequence*

[TABLE]

Consider the multiplier operator $\Lambda_{s}$ on $\mathcal{T}_{M_{1},M_{2}}$ defined by

$\Lambda_{s}\left(\displaystyle\sum_{k=1}^{s}\alpha_{k}\xi_{k}\right)=\displaystyle\sum_{k=1}^{s}\widetilde{\lambda}_{k}\alpha_{k}\xi_{k}$.

We also denote by $\Lambda_{s}$ the multiplier operator defined on $\mathbb{R}^{s}$ by

[TABLE]

For $\xi\in\mathcal{T}_{M_{1},M_{2}}$ and $1\leq p\leq\infty$, we define

$\|\hskip 0.72229pt\xi\hskip 0.72229pt\|_{\Lambda_{s},p}=\|\hskip 0.72229pt\Lambda_{s}\xi\hskip 0.72229pt\|_{p}$,

and for $\alpha\in\mathbb{R}^{s}$ we define

$\|\hskip 0.72229pt\alpha\hskip 0.72229pt\|_{(\Lambda_{s},p)}=\|\hskip 0.72229ptJ(\alpha)\hskip 0.72229pt\|_{\Lambda_{s},p}$.

The application $\mathcal{T}_{M_{1},M_{2}}\ni\xi\longmapsto\|\hskip 0.72229pt\xi\hskip 0.72229pt\|_{\Lambda_{s},p}$ is a norm on $\mathcal{T}_{M_{1},M_{2}}$ and the application $\mathbb{R}^{s}\ni\alpha\longmapsto\|\hskip 0.72229pt\alpha\hskip 0.72229pt\|_{(\Lambda_{s},p)}$ is a norm on $\mathbb{R}^{s}$. We denote

$B_{\Lambda_{s},p}^{s}=B_{\Lambda,p}^{s}=\{\xi\in\mathcal{T}_{M_{1},M_{2}}:\|\hskip 0.72229pt\xi\hskip 0.72229pt\|_{\Lambda_{s},p}\leq 1\},$**

$B_{(\Lambda_{s},p)}^{n}=B_{(\Lambda,p)}^{s}=\{\alpha\in\mathbb{C}^{s}:\|\hskip 0.72229pt\alpha\hskip 0.72229pt\|_{(\Lambda_{s},p)}\leq 1\}.$**

If $\Lambda_{s}$ is the identity operator I, we will write $\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|_{I,p}=\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|_{p}$, $\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|_{(I,p)}=\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|_{(p)}$, $B_{I,p}^{s}=B_{p}^{s}$ and $B_{(I,p)}^{s}=B_{(p)}^{s}$.

Theorem 4.3**.**

Let $\lambda:[0,\infty)\rightarrow\mathbb{R}$ be a positive and monotonic function, $s=\dim\mathcal{T}_{M_{1},M_{2}}$ and consider the orthonormal system $\{\xi_{k}\}^{s}_{k=1}$ of $\mathcal{T}_{M_{1},M_{2}}$ and the multiplier operator $\Lambda_{s}$ on $\mathcal{T}_{M_{1},M_{2}}$ as in Remark 4.2. If $\lambda$ is non-increasing, then there is an absolute constant $C>0$ such that:

• (a)

If $2\leq p<\infty$, we have

[TABLE]

• (b)

If $p=\infty$, we have

[TABLE]

• (c)

If $1\leq p\leq 2$, we have

[TABLE]

• (d)

If $p=2$, we have

[TABLE]

If $\lambda$ is non-decreasing, then we obtain the estimates in $(a)$, $(b)$, $(c)$ and $(d)$ permuting $\lambda(l)$ for $\lambda(l-1)$.

Proof.

Suppose $\lambda$ a non-increasing function. For a continuous function $f$ on $S^{n-1}$ consider the function $\widetilde{f}$ defined on $\mathbb{R}^{n}\backslash\{0\}$ by $\widetilde{f}(x)=|||x|||^{2}f\left(x/|||x|||\right)$. It is known that

[TABLE]

where $d\gamma(x)=e^{-\pi|||x|||^{2}}dx$ denotes the Gaussian measure on $\mathbb{R}^{n}$. Let $\{r_{k}\}_{k=1}^{\infty}$ be the sequence of Rademacher’s functions given by $r_{k}(\theta)=\textrm{sign}\hskip 2.84544pt\sin(2^{k}\pi\theta)$, $\theta\in[0,1]$, $k\in\mathbb{N}$ and let

[TABLE]

It follows by Lemma 2.1 in Kwapień [19], p. 585, that if $h:\mathbb{R}^{n}\longrightarrow\mathbb{R}$ is a continuous function satisfying

$h(x_{1},\ldots,x_{n})e^{-\sum_{k=1}^{n}|x_{k}|}\rightarrow 0$, uniformly when $\sum_{k=1}^{n}|x_{k}|\rightarrow\infty$, then

[TABLE]

Now, we consider $f(x)=\|\hskip 0.72229ptx\hskip 0.72229pt\|_{(\Lambda_{s},p)}^{2}$, $x\in S^{s-1}$ and $h(x)=\widetilde{f}(x)=f(x)$, $x\in\mathbb{R}^{s}$ applying (11) and (12), we obtain

[TABLE]

Denoting $\widetilde{\xi}_{(i-1)m+k}(z)=m^{-1/2}\xi_{i}(z)$, $z\in\Omega_{d}$ and $\widetilde{\widetilde{\lambda}}_{(i-1)m+k}=\widetilde{\lambda}_{i}$, for $i=1,\ldots,s;$ $k=1,\ldots,m$ and $m=1,2,\ldots$

we obtain

[TABLE]

Hence, by (13) and (14),

[TABLE]

Using the addition formula we get

[TABLE]

and analogously,

[TABLE]

Therefore, from Jensen’s inequality, Khintchine’s inequality (see [27], p. 41) and (15), we obtain for $2\leq p\leq\infty$,

[TABLE]

where $C_{1}$ is obtained from the fact that $\gamma(p)\asymp p^{1/2}$, and hence we get the upper estimate in $(a)$. On the other hand, for $p=1$, it follows from Khintchine’s inequality, Jensen’s inequality and (15)

[TABLE]

Since the Levy mean is an increasing function of p, it follows that

[TABLE]

thus we obtain the lower estimate in $(c)$.

Now, we will obtain the inequalities in $(d)$. For $x\in\mathbb{R}^{s}$, we have

[TABLE]

and

[TABLE]

Thus

[TABLE]

and consequently

[TABLE]

Analogously we get

[TABLE]

Therefore we obtain $(d)$. Since the Levy mean $M(\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|_{(\Lambda_{s},p)})$ is an increasing function of $p$ for $1\leq p\leq\infty$, the lower estimates in $(a)$,$(b)$ and the upper estimate in $(c)$ follow from $(d)$.

Finally, we will find the upper estimate in $(b)$. Any polynomial $t_{s}\in\mathcal{T}_{M_{1},M_{2}}$, can be expressed as $t_{s}=D_{M_{1},M_{2}}\ast t_{s}$, where $D_{M_{1},M_{2}}=\sum_{(m,n)\in A_{M_{2}}\setminus A_{M_{1}}}Z_{e}^{(m,n)}$ and from Young’s inequality, we get

[TABLE]

and since $D_{M_{1},M_{2}}=D_{M_{1},M_{2}}\ast D_{M_{1},M_{2}}$, then

[TABLE]

and thus $\|\hskip 0.72229ptt_{s}\hskip 0.72229pt\|_{\infty}\leq s\|\hskip 0.72229ptt_{s}\hskip 0.72229pt\|_{1}/\omega_{d}$. Furthermore, if I denotes the identity operator, we have

[TABLE]

Applying the Riesz-Thorin Interpolation Theorem to the above inequalities , we obtain

[TABLE]

Similarly, we can show that

[TABLE]

To obtain the upper estimate in $(b)$, we apply (16) and the upper estimate in $(a)$ with $p=\ln s$, and we get

[TABLE]

If $\lambda$ is a non-decreasing function, the proof is analogous. ∎

5 Estimates for $n$-widths of general multiplier operators

Consider a $n$-dimensional Banach space $E=(\mathbb{R}^{n},\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|)$. The dual norm of $\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|$ is defined by $\|\hskip 0.72229ptx\hskip 0.72229pt\|^{0}=\sup\{|\langle x,y\rangle|:y\in B_{E}\}$, where $\langle x,y\rangle$ denotes the usual inner product of the elements $x,y\in\mathbb{R}^{n}$. The dual space $(\mathbb{R}^{n},\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|^{0})$ of $E$ will be denoted by $E^{0}$.

Theorem 5.1**.**

([26]) There exists an absolute constant $C>0$ such that, for every $0<\rho<1$ and $n\in\mathbb{N}$, there exists a subspace $F_{k}\subseteq\mathbb{R}^{n}$, with $\dim F_{k}=k>\rho n$ and

[TABLE]

Theorem 5.2**.**

Let $1\leq q\leq p\leq 2$, $0<\rho<1$, $s=\dim\mathcal{T}_{N}$, $\mathcal{T}_{N}=\bigoplus_{l=0}^{N}\mathcal{H}_{l}$, $d_{l}=\dim\mathcal{H}_{l}$ and let $\lambda:[0,\infty)\rightarrow\mathbb{R}$ be a positive and non-increasing function with $\lambda(t)\neq 0$ for $t\geq 0$ and $\Lambda=\{\lambda_{m,n}\}_{m,n\in\mathbb{N}}$, $\lambda_{m,n}=\lambda(|(m,n)|)$. Then there is an absolute constant $C>0$ such that

[TABLE]

where

$\vartheta_{q,s}=\left\{\begin{matrix}(1-1/q)^{1/2},&\mbox{ }q>1,\\ (\ln s)^{-1/2},&\mbox{ }q=1,\end{matrix}\right.$**

and $[\rho s-1]$ denotes the integer part of the number $\rho s-1$.

Proof.

Let $x,y\in\mathbb{R}^{s}=J^{-1}\mathcal{T}_{N}$. The Hölder’s inequality implies that

[TABLE]

where $1/q+1/q^{\prime}=1$ and $\Lambda^{-1}=\{\lambda_{m,n}^{-1}\}_{m,n\in\mathbb{N}}$. Taking $0<\rho<1$, the Theorem 5.1 guarantees the existence of a subspace $F_{k}\subseteq\mathbb{R}^{s}$, $\dim F_{k}=k>\rho s$, such that

[TABLE]

for all $x\in F_{k}$. Thus, for $\epsilon=(C^{\prime})^{-1}(1-\rho)^{1/2}\left(M\left(\|\hskip 0.72229pt\cdot\hskip 0.72229pt\|_{(\Lambda_{s}^{-1},q^{\prime})}\right)\right)^{-1}$, we have that $\epsilon B_{(\Lambda_{s},q)}^{s}\cap F_{k}\subseteq B_{(2)}^{s}$.

From Theorem 4.3, since $2\leq q^{\prime}\leq\infty$, it follows that

[TABLE]

and since $1/q+1/q^{\prime}=1$,

[TABLE]

We have that $\Lambda_{s}=\Lambda\lvert_{\mathcal{T}_{N}}$ and since $B_{2}^{s}\subset U_{2}$, then $\Lambda_{s}(B_{2}^{s})\subset\Lambda(U_{2})$, therefore

[TABLE]

Taking $X_{k}=J(F_{k})$ we obtain $\epsilon B_{q}^{s}\cap X_{k}\subseteq\Lambda_{s}B_{2}^{s}$. Thus by the definition of Bernstein $n$-width

$b_{[\rho s-1]}(\Lambda_{s}B_{2}^{s},L^{q})\geq b_{k-1}(\Lambda_{s}B_{2}^{s},L^{q})\geq\epsilon$

and consequently

[TABLE]

thus concluding the proof of the theorem. ∎

The next result follows applying Theorem 5.2, (1) and basic properties of $n$-widths.

Corollary 5.3**.**

In the conditions of Theorem 5.2, we have

$d_{[\rho s-1]}(\Lambda U_{p},L^{p})\geq C^{\prime}(1-\rho)^{1/2}s^{1/2}\left(\displaystyle\sum_{l=1}^{N}(\lambda(l))^{-2}d_{l}\right)^{-1/2}\kappa_{s}$,

where

$\kappa_{s}=\left\{\begin{matrix}1,&\mbox{ }1\leq p\leq 2,1<q\leq 2,\\ 1,&\mbox{ }2\leq p<\infty,2\leq q\leq\infty,\\ 1,&\mbox{ }1\leq p\leq 2\leq q\leq\infty,\\ (\ln s)^{-1/2},&\mbox{ }1\leq p\leq 2,q=1,\\ (\ln s)^{-1/2},&\mbox{ }p=\infty,2\leq q\leq\infty.\end{matrix}\right.$**

Theorem 5.4**.**

Let $\lambda:(0,\infty)\longrightarrow\mathbb{R}$ be a positive and non-increasing function and let $\Lambda=\{\lambda_{m,n}\}_{m,n\in\mathbb{N}}$, $\lambda_{m,n}=\lambda(|(m,n)|)$. Suppose $1\leq p\leq 2\leq q\leq\infty$ and that the multiplier operator $\Lambda$ is bounded from $L^{1}$ to $L^{2}$. Let $\{N_{k}\}_{k=0}^{\infty}$ and $\{m_{k}\}_{k=0}^{M}$ be sequences of natural numbers such that $N_{k}<N_{k+1}$, $N_{0}=0$ and $\sum_{k=0}^{M}m_{k}\leq\beta$. Then there exists an absolute constant $C>0$ such that

$d_{\beta}(\Lambda U_{p},L^{q})\leq C\left(\displaystyle\sum_{k=1}^{M}\lambda(N_{k})\varrho_{m_{k}}+\displaystyle\sum_{k=M+1}^{\infty}\lambda(N_{k})\left(\theta_{N_{k},N_{k+1}}\right)^{1/p-1/q}\right)$,

where

[TABLE]

Proof.

Let $\mathcal{T}_{M_{1},M_{2}}=\bigoplus_{l=M_{1}+1}^{M_{2}}\mathcal{H}_{l}$, $s=\dim\mathcal{T}_{M_{1},M_{2}}=\sum_{l=M_{1}+1}^{M_{2}}d_{l}$, $B_{p}^{s}=U_{p}\cap\mathcal{T}_{M_{1},M_{2}}$, $B_{(p)}^{s}=J^{-1}B_{p}^{s}$, $\lambda(0)=0$ and $0<\rho<1$. The Theorem 5.1 guarantees the existence of a subspace $F_{k}$ of $\mathbb{R}^{s}$, with $\dim F_{k}=k>\rho s$, such that, for all $\alpha\in F_{k}$,

[TABLE]

If $m=s-k$, then $(1-\rho)^{-1/2}<\left(s/m\right)^{1/2}$. Applying the Theorem 4.3 for $\Lambda_{s}=Id$, we get

[TABLE]

Therefore, by (1)

[TABLE]

Let $B_{p}^{N_{k},N_{k+1}}=U_{p}\cap\mathcal{T}_{N_{k},N_{k+1}}$ and for $f\in\Lambda U_{p}\subseteq L^{2}(\Omega_{d})$, let $S_{N}(f)=\sum_{(m,n)\in A_{N}}f\ast\widetilde{Z}^{(m,n)}$, $\phi_{N_{k},N_{k+1}}(f):=S_{N_{k+1}}(f)-S_{N_{k}}(f)$, $k\geq 1$, $\phi_{N_{0},N_{1}}(f)=S_{N_{1}}(f)$. Since $S_{N_{k}}(f)\longrightarrow f$ in $L^{2}(\Omega_{d})$, then

[TABLE]

Then it is easy to see that

[TABLE]

But, for each $\varphi\in U_{p}$, $\phi_{N_{k},N_{k+1}}(\Lambda\varphi)=\Lambda(\phi_{N_{k},N_{k+1}}(\varphi))$, therefore

[TABLE]

Now, given $\varphi\in U_{p}$,

[TABLE]

and by Young’s inequality

[TABLE]

Using properties of the zonal harmonic functions, we have

[TABLE]

and thus by (21), for $p=1$ we obtain $\|\hskip 0.72229pt\phi_{N_{k},N_{k+1}}(\varphi)\hskip 0.72229pt\|_{2}\leq\omega_{d}^{-1/2}\theta_{N_{k},N_{k+1}}^{1/2}\|\hskip 0.72229pt\varphi\hskip 0.72229pt\|_{1}$. Furthermore, for $p=2$, and $\varphi\in U_{2}$ we get $\|\hskip 0.72229pt\phi_{N_{k},N_{k+1}}(\varphi)\hskip 0.72229pt\|_{2}\leq\|\hskip 0.72229pt\varphi\hskip 0.72229pt\|_{2}$. Applying the Riesz-Thorin Interpolation theorem to the last two inequalities, it follows for $1\leq p\leq 2$ that

[TABLE]

Hence, from (20) and (19),

[TABLE]

By (17) for $2\leq q\leq\infty$, we have for $\varphi\in B_{2}^{N_{k},N_{k+1}}$ that

[TABLE]

and therefore $B_{2}^{N_{k},N_{k+1}}\subseteq\omega_{d}^{1/q-1/2}\theta_{N_{k},N_{k+1}}^{1/2-1/q}B_{q}^{N_{k},N_{k+1}}$. Consequently by (22)

[TABLE]

Finally, using (18), (23) and properties of $n$-widths, we get

[TABLE]

∎

Remark 5.5**.**

We will improve the estimate obtained in the previous theorem, specifying the sequences $N_{k}$ and $m_{k}$. We define $N_{1}=N\in\mathbb{N}$ and

[TABLE]

Let $\theta_{N_{k},N_{k+1}}$ be as in Theorem 5.4. For $\epsilon>0$ we define

[TABLE]

Hence,

[TABLE]

where $C_{\epsilon}>0$, depends only on $\epsilon$. Applying the previous theorem for

[TABLE]

and writing $d_{\beta}=d_{\beta}(\Lambda U_{p},L^{q})$, we have

[TABLE]

Definition 5.6**.**

Let $N_{k},M$ and $\theta_{N_{k},N_{k+1}}$ be as in the previous remark. We say that $\Lambda=\{\lambda_{m,n}\}_{m,n\in\mathbb{N}}\in K_{\epsilon,p}$, $\epsilon>0$, $1\leq p\leq 2$, if $\lambda(k+1)\leq\lambda(k)$, $N_{k+1}>N_{k}$, for all $k\in\mathbb{N}$ and if for all $N\in\mathbb{N}$, there exists a constant $C_{\epsilon,p}$, depending only on $d$, $\epsilon$ and $p$, such that

[TABLE]

Corollary 5.7**.**

Let $\Lambda=\{\lambda_{m,n}\}_{m,n\in\mathbb{N}}$ and $\theta_{N_{k},N_{k+1}}$ be as in the previous theorem, and let $\epsilon>0$, $N_{k}$, $M$, $\{m_{k}\}_{k=0}^{M}$ and $\beta$ be as in the previos remark. Let $1\leq p\leq 2\leq q\leq\infty$ and suppose that $\Lambda\in K_{\epsilon,p}$, for a fixed $\epsilon>0$. Then there exists a constant $C_{\epsilon,p}>0$, such that

[TABLE]

Remark 5.8**.**

We note that the Theorems 5.2 and 5.4, and the Corollaries 5.3 and 5.7 hold if we consider a function $\lambda$ such that $t\rightarrow|\lambda(t)|$ is a positive and non-increasing fuction. Simply we change $\lambda(t)$ by $|\lambda(t)|$.

6 Applications

Let $\lambda^{(1)},\lambda^{(2)}:[0,\infty)\rightarrow\mathbb{R}$ be defined by $\lambda^{(1)}(t)=t^{-\gamma}(\ln t)^{-\xi}$ for $t>1$, $\lambda^{(1)}(t)=0$ for $0\leq t\leq 1$, and $\lambda^{(2)}(t)=e^{-\gamma t^{r}}$, $t\geq 0$, where $\gamma,\xi,r\in\mathbb{R}$, $\gamma,r>0$, $\xi\geq 0$. In this section we consider the multiplier operators associated with the sequences $\Lambda^{(1)}=(\lambda_{m,n}^{(1)})_{m,n\in\mathbb{N}}$ and $\Lambda^{(2)}=(\lambda_{m,n}^{(2)})_{m,n\in\mathbb{N}}$ where $\lambda_{m,n}^{(1)}=\lambda^{(1)}(|(m,n)|)$, $\lambda_{m,n}^{(2)}=\lambda^{(2)}(|(m,n)|)$, $|(m,n)|=\max\{m,n\}$.

Remark 6.1**.**

We will prove that, if $\gamma>(2d-1)/2$, then the multiplier operator $\Lambda^{(1)}$ is bounded from $L^{1}$ to $L^{2}$. Given $\varphi\in U_{1}$, for each $k\in\mathbb{N}^{*}$, let

[TABLE]

We obtain that

[TABLE]

and thus since $d_{l}\asymp l^{2d-2}$, we obtain

[TABLE]

Therefore

$\|\hskip 0.72229pt\Lambda^{(1)}\varphi\hskip 0.72229pt\|_{2}=\displaystyle\sup_{k}\|\hskip 0.72229pt\Lambda^{(1)}\varphi_{k}\hskip 0.72229pt\|_{2}\leq(C_{3})^{1/2}$,

and thus $\Lambda^{(1)}$ is bounded from $L^{1}$ to $L^{2}$.

Theorem 6.2**.**

For $1\leq p\leq\infty$, $2\leq q\leq\infty$, $\gamma>(2d-1)/p$ if $1\leq p\leq 2$ and $\gamma>(2d-1)/2$ if $p\geq 2$, and for all $m\in\mathbb{N}$, we have

[TABLE]

Proof.

Suppose $1\leq p\leq 2\leq q\leq\infty$, fix $\delta>0$ and let $\lambda_{1},\lambda_{2}:(0,\infty)\rightarrow\mathbb{R}$ be defined by $\lambda_{1}(t)=t^{-\gamma}$ and $\lambda_{2}(t)=t^{-\gamma-\delta}$. Let $a>1$ and let $b,b_{1},b_{2}\in\mathbb{R}$ such that $e\lambda(b)=\lambda(a)$, $e\lambda_{1}(b_{1})=\lambda_{1}(a)$ and $e\lambda_{2}(b_{2})=\lambda_{2}(a)$. We have that $b,b_{1},b_{2}>a$ and that $b_{1}=e^{1/\gamma}a$, $b_{2}=e^{1/(r+\delta)}a$ and $b_{1}=b\left(\ln b/\ln a\right)^{\xi/\gamma}$. Since $b>a$, $b_{1}>b$ and $b>b_{2}$, we obtain $e^{1/(\gamma+\delta)}a<b<e^{1/\gamma}a$.

Taking $a=N_{k}$, we get $e^{1/(\gamma+\delta)}N_{k}<b<e^{1/\gamma}N_{k}$. But $N_{k}<b\leq N_{k+1}<b+1$ and therefore

[TABLE]

Considering that $d_{l}=\dim\mathcal{H}_{l}\asymp l^{2d-2}$, integrating the function $x^{2d-2}$, we obtain

[TABLE]

Furthermore using (27), we get

[TABLE]

Thus, by (29) and (28), it follows that

[TABLE]

and using (27) we obtain

[TABLE]

Let $M=[\epsilon^{-1}\ln\theta_{N_{1},N_{2}}]$ be as in Remark 5.5. We have that $M\asymp\epsilon^{-1}\ln N\asymp\epsilon^{-1}\ln s$. By (30) and (31),

[TABLE]

Since $M\asymp\epsilon^{-1}\ln N$, then there exists a positive constant $C$ such that

[TABLE]

and since $1\leq p\leq 2\leq q\leq\infty$ and $\gamma>(2d-1)(1/p-1/q)$, we have that $1-(1/p-1/q)(2d-1)/\gamma>0$ and thus

[TABLE]

Therefore we obtain by (32) that

[TABLE]

So, taking $0<\epsilon<C(1-(2d-1)(1/p-1/q)/\gamma)/(2d-1)(1/p-1/q)$, we get

[TABLE]

Next we will prove that $\Lambda^{(1)}\in K_{\epsilon,p}$ for some $\epsilon>0$. Remembering that $M\asymp\epsilon^{-1}\ln N$, by (30) and (31), it follows that

[TABLE]

Since $\gamma/(2d-1)>1/p$, we have $t=-(1-\epsilon/2-(2d-1)/\gamma p)<0$ if $0<\epsilon/2<1-(2d-1)/\gamma p$ and consequently

[TABLE]

By (31) there exists a constant $C_{1}$, such that $C_{1}e^{(2d-1)/(\gamma+\delta)}N^{2d-1}\leq\theta_{N_{1},N_{2}}$ and then

[TABLE]

that is, $\Lambda^{(1)}\in K_{\epsilon,p}$. Now, by Remark 6.1 we have that $\Lambda^{(1)}$ is bounded from $L^{1}(\Omega_{d})$ to $L^{2}(\Omega_{d})$, and hence from Corollary 5.7 and (33), we obtain

[TABLE]

For $1\leq k\leq M$, using (30), (31), the definition of $M$, and the fact that $s\asymp N^{2d-1}$, we obtain

[TABLE]

and thus $\ln(\theta_{N_{k},N_{k+1}})\ll\ln(s^{1+C/\gamma\epsilon})=(1+C/\gamma\epsilon)\ln s\ll\ln s$. Furthermore, by (30) and (31), it follows that $\theta_{N_{1},N_{2}}^{1/p-1/2}\ll N^{(2d-1)(1/p-1/2)}$ and therefore

[TABLE]

Now, since $s\asymp N^{2d-1}$, then $N^{-\gamma}\asymp s^{-\gamma/(2d-1)}$, $N^{(2d-1)(1/p-1/2)}\asymp s^{(1/p-1/2)}$ and $(\ln N)^{-\xi}\asymp(\ln N^{d})^{-\xi}\asymp(\ln s)^{-\xi}$. Consequently

[TABLE]

From Remark 5.5, $\beta\asymp s+\sum_{j=1}^{M}e^{-\epsilon j}\theta_{N_{1},N_{2}}$ and keeping in mind that $s\asymp N^{2d-1}$ and $M\asymp\epsilon^{-1}\ln N$, by (30) and (31) we get

[TABLE]

that is, there exists a constant $C_{2}\in\mathbb{N}$ such that $\beta\leq C_{2}N^{2d-1}$. Given $m\in\mathbb{N}$, let $N\in\mathbb{N}$ such that $C_{2}N^{2d-1}\leq m\leq C_{2}(N+1)^{2d-1}$. It follows by (34) that

$d_{m}(\Lambda^{(1)}U_{p},L^{q})\ll m^{-\gamma/(2d-1)+(1/p-1/2)}(\ln m)^{-\xi}\left\{\begin{matrix}q^{1/2},&\mbox{ }2\leq q<\infty,\\ (\ln m)^{1/2},&\mbox{}q=\infty,\end{matrix}\right.$

for $1\leq p\leq 2$. The case $2\leq p\leq\infty$, follows because $\Lambda(U_{p})\subset\Lambda(U_{2})$.∎

Theorem 6.3**.**

For $\gamma>(2d-1)/2$ and for all $m\in\mathbb{N}$, we have

[TABLE]

where

$\kappa_{m}=\left\{\begin{matrix}1,&\mbox{ }1\leq p\leq 2,1<q\leq 2,\\ 1,&\mbox{ }2\leq p<\infty,2\leq q\leq\infty,\\ 1,&\mbox{ }1\leq p\leq 2\leq q\leq\infty,\\ (\ln m)^{-1/2},&\mbox{ }1\leq p\leq 2,q=1,\\ (\ln m)^{-1/2},&\mbox{ }p=\infty,2\leq q\leq\infty.\end{matrix}\right.$**

Proof.

We have that $d_{l}\asymp l^{2d-2}$ and $s=\dim\mathcal{T}_{N}\asymp N^{2d-1}$, and thus

[TABLE]

Applying the Corollary 5.7 with $\rho=1/2$, we obtain

[TABLE]

Finally, taking $m=[s/3]\leq[(s-2)/2]$ we have that $m\asymp s$ and thus

[TABLE]

∎

Remark 6.4**.**

We will prove that the multiplier operator $\Lambda^{(2)}$ is bounded from $L^{p}$ to $L^{q}$, for $1\leq p,q\leq\infty$. We will show that this result is true for $p=1$ and $q=\infty$ and the other cases will follow immediately from inequalities between norms. Given $\varphi\in U_{1}$, we have that

[TABLE]

and thus

[TABLE]

where $a_{l}(\omega)=\sum_{(m,n)\in A_{l}\setminus A_{l-1}}\lambda_{m,n}\varphi\ast\widetilde{Z}^{(m,n)}(\omega)$. Let

[TABLE]

We have that $D_{l}=\overline{D_{l}}\ast\overline{D_{l}}$ and $a_{l}=\varphi\ast D_{l}$. It follows by Young’s inequality that

[TABLE]

and hence

[TABLE]

Moreover there is a constant $C_{2}$ such that $e^{-\gamma l^{r}}l^{2d}\leq C_{2}$, for all $l\in\mathbb{N}$ and therefore

[TABLE]

Thus

[TABLE]

so $\Lambda^{(2)}$ is bounded from $L^{1}(\Omega_{d})$ to $L^{\infty}(\Omega_{d})$ .

Theorem 6.5**.**

Consider the sequences $\phi_{k}=\dim\mathcal{T}_{k}$, $\psi_{k}=\phi_{k}-\phi_{k}^{1-r/(2d-1)}-1$ and $\kappa_{k}$ as in Theorem 6.3. Then

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

Proof.

For $\theta,\eta,r>0$ and $\eta\geq r-1$, we have

[TABLE]

Since there is a positive constant $C_{1}$ such that $(N+1)^{r}\leq N^{r}+C_{1}N^{-1}$, then

[TABLE]

Let us fix $N\in\mathbb{N}$ and let $s=\dim\mathcal{T}_{N}$, $D=2/d!(d-1)!$. By the Proposition 4.1 we have

[TABLE]

From the above estimates, we obtain $N^{r}\leq D^{-r/(2d-1)}s^{r/(2d-1)}$ and $N^{r}\geq D^{-r/(2d-1)}s^{r/(2d-1)}-C_{4}N^{r-1}$ for a positive constant $C_{4}$ and for $r>0$. Therefore, for $\mathcal{R}=\gamma D^{-r/(2d-1)}$ and the fact that $s\asymp N^{2d-1}$, we get

[TABLE]

for $r>0$. The above estimates also hold if we put $N+1$ in the place of $N$. If $\rho=1-s^{-r/(2d-1)}$, using (40), we get

[TABLE]

and hence it follows by Corollary 5.7 and (41) that

[TABLE]

The above estimate says that, for $r>0$

[TABLE]

but, since $\phi_{N}=\dim\mathcal{T}_{N}\asymp N^{2d-1}$, we have that $\kappa_{\phi_{N}}\asymp\kappa_{N}$ and therefore we get (36). Furthermore, if $r>2d-1$, then $[\psi_{N}]=[\phi_{N}-\phi_{N}^{1-{r/(2d-1)}}-1]=\phi_{N}-1$ and thus (38) follows from (36).

Consider $0<r\leq 2d-1$. Then

[TABLE]

and there exists a positive constant $C_{r}$ such that $0<S_{s}\leq C_{r}$ for all $s\in\mathbb{N}$. Therefore

[TABLE]

thus

[TABLE]

and from (36) we get

[TABLE]

Now let $0<r\leq 1$. Using the Mean Value Theorem we get $(N+1)^{r}-N^{r}=r(N+c)^{r-1}<rN^{r-1}\leq r$, and then $1<e^{-\gamma N^{r}}/e^{-\gamma(N+1)^{r}}\leq e^{\gamma r}$. From (36) and (41) it follows that

[TABLE]

and using (42) we obtain

[TABLE]

If $k\in\mathbb{N}$ and $[\psi_{N}]<k\leq[\psi_{N+1}]$, then from (44)

[TABLE]

and thus we proved (39).

Finally, if $0<r\leq 2d-1$, we get $e^{\mathcal{R}l^{r/(2d-1)}}\asymp e^{\mathcal{R}(l+1)^{r/(2d-1)}}$, $l\in\mathbb{N}$ and thus it follows from (43) that

[TABLE]

that is, we proved (37). ∎

Theorem 6.6**.**

Let $\phi_{k}$ be as in Theorem 6.5. Then for $0<r\leq 1$, $1\leq p\leq\infty$, $2\leq q\leq\infty$ and for all $k\in\mathbb{N}$, we have

[TABLE]

and for all $r>1$ and all $k\in\mathbb{N}$,

[TABLE]

where $\mathcal{R}$ is the constant given in Theorem 6.5.

Proof.

Suppose $0<r\leq 1$ and let $\lambda(x)=e^{-\gamma x^{r}}$. For $k\in\mathbb{N}$, let $x_{k}\in\mathbb{R}$, $e\lambda(x_{k})=\lambda(N_{k})$. Then $x_{k}=(N_{k}^{r}+1/\gamma)^{1/r}$ and $N_{k}<x_{k}\leq N_{k+1}<x_{k}+1$. Therefore

[TABLE]

We have that

[TABLE]

and since $0<r\leq 1$, then $(N_{k}^{r}+1/\gamma)^{1-1/r}\leq 1$ and therefore $N_{k+1}^{r}\leq N_{k}^{r}+C_{1}$. Thus

[TABLE]

and using that $N_{1}=N$, we show that

[TABLE]

By (47), $N_{k+1}\asymp(N_{k}^{r}+1/\gamma)^{1/r}=N_{k}\left(1+N_{k}^{-r}/\gamma\right)^{1/r}$, and then

[TABLE]

Thus $N_{k+1}-N_{k}\asymp N_{k}^{1-r}$. On the other hand

[TABLE]

consequently, from (48), (49) and for $\epsilon>0$ and $k$ sufficiently large, we obtain

[TABLE]

Hence, for any $\delta>0$, there exists $C_{2}>0$ such that

[TABLE]

and taking $\delta^{\prime}=\delta(2d-r-1)/r$, we get

[TABLE]

For $1\leq p\leq 2$,

[TABLE]

where $\epsilon$, $p$ e $\delta$ are chosen satisfying $\epsilon+\delta^{\prime}/p<1$. Thus

[TABLE]

and then $\Lambda^{(2)}\in K_{2\epsilon,p}$. Therefore it follows from Corollary 5.7 that

[TABLE]

Furthermore, by (49) and (50) we have that

[TABLE]

where $\delta^{\prime\prime}=\delta^{\prime}(1/p-1/q)$, $\delta^{\prime\prime}<1$. We have that $M\asymp\epsilon^{-1}\ln\theta_{N_{1},N_{2}}$ and $\theta_{N_{1},N_{2}}\asymp N^{2d-r-1}$, and hence

[TABLE]

if $0<\epsilon<C_{5}(1-\delta^{\prime\prime})/(1/p-1/q)$. Then it follows by (49) and (51) that

[TABLE]

By (48) and (49), we get $\theta_{N_{k},N_{k+1}}\asymp N_{k}^{2d-r-1}=\left(N_{k}^{r}\right)^{(2d-r-1)/r}\leq(N^{r}+C_{1}(k-1))^{(2d-r-1)/r}$ and considering that $M\leq\epsilon^{-1}\ln\theta_{N_{1},N_{2}}$, $\theta_{N_{1},N_{2}}\asymp N^{2d-r-1}$ and $N^{2d-1}\asymp s$ we get $\ln\theta_{N_{k},N_{k+1}}\asymp\ln s$. Therefore by (52)

[TABLE]

From Remark 5.5, (24) and (49), we get

[TABLE]

and since $N^{2d-1}\asymp s$, it follows that $\beta\leq s+C_{9}s^{1-r/(2d-1)}$ and then

[TABLE]

Let $\mathcal{T}_{N}=s+C_{9}s^{1-r/(2d-1)}$. By (41) we have $-\gamma N^{r}\leq-\mathcal{R}s^{r/(2d-1)}+C_{10}s^{(r-1)/(2d-1)}$ and hence $C_{10}s^{(r-1)/(2d-1)}<C_{11}$. Thus

[TABLE]

Taking $N$ sufficiently large, we have $|C_{9}\phi_{N}^{-r/(2d-1)}|<1$ and hence we obtain

[TABLE]

where $0\leq S_{N}\leq C_{r}$ and therefore $-\gamma N^{r}+\mathcal{R}\tau_{N}^{r/(2d-1)}\leq C_{12}$. Let $l\in\mathbb{N}$, $[\tau_{N}]\leq l\leq[\tau_{N+1}]$. We have $1<e^{-\gamma N^{r}}/e^{-\gamma(N+1)^{r}}\leq e^{\gamma r}$, so, using (53), we obtain

[TABLE]

proving (45) for $1\leq p\leq 2\leq q\leq\infty$. The case $2\leq p\leq\infty$ follows because $\Lambda U_{p}\subset\Lambda U_{2}$.

Now for $r>1$ we will apply the Theorem 5.4. We have $-\gamma(k+1)^{r}+\gamma k^{r}\leq-\gamma rk^{r-1}$ for $k\geq 1$ and therefore $e\lambda(k+1)\leq e^{-\gamma rk^{r-1}+1}\lambda(k)$. Let $a\in\mathbb{N}$ such that $e^{-\gamma rk^{r-1}+1}\leq 1$, for all $k\geq a$. Consider $N\geq a$, $N_{0}=0$, $N_{1}=N$, $N_{k+1}=N+k$, $M=0$, $\beta=m_{0}=n=\phi_{N}$. Applying Theorem 5.4 for $1\leq p\leq 2\leq q\leq\infty$ we obtain

[TABLE]

Since $\lambda(N+k)\leq e^{-k}\lambda(N)$, for $1\leq p\leq 2\leq q\leq\infty$ we obtain

[TABLE]

Now for $2\leq p,q\leq\infty$, we obtain

[TABLE]

and this proves (46). ∎