Nikolskii inequality for lacunary spherical polynomials

TL;DR

This paper improves the understanding of Nikolskii inequalities for lacunary spherical polynomials on spheres of dimension two or higher, showing that the inequality's order can be significantly better than classical bounds in many cases.

Contribution

It establishes sharper asymptotic bounds for Nikolskii inequalities for lacunary spherical polynomials on spheres of dimension at least two, extending classical results.

Findings

Improved asymptotic order of Nikolskii inequality for lacunary spherical polynomials.

Demonstrates significant improvements over classical bounds in many cases.

Contrasts the phenomenon with the case of the unit circle where no such improvement occurs.

Abstract

We prove that for $d\ge 2$, the asymptotic order of the usual Nikolskii inequality on $\mathbb{S}^d$ (also known as the reverse H\"{o}lder's inequality) can be significantly improved in many cases, for lacunary spherical polynomials of the form $f=\sum_{j=0}^m f_{n_j}$ with $f_{n_j}$ being a spherical harmonic of degree $n_j$ and $n_{j+1}-n_j\ge 3$. As is well known, for $d=1$, the Nikolskii inequality for trigonometric polynomials on the unit circle does not have such a phenomenon.

Equations

$$\|f\|_{p}:=\begin{cases}\displaystyle\Bigl{(}\int_{\SS^{d}}|f(x)|^{p}\,d\sigma(x)\Bigr{)}^{1/p},&0<p<\infty,\\ \operatorname*{ess\,sup}_{x\in\SS^{d}}|f(x)|,&p=\infty.\end{cases}$$

$${\mathrm{proj}}_{k}f(x)=\frac{k+\lambda }{\lambda }\frac{1}{{\omega }_d}{\int }_{{\text{SS}}^d}f(y)C_k^{\lambda }(x\cdot y)d\sigma (y),x\in {\text{SS}}^d,\lambda =\frac{d-1}{2},$$

$$\Vert f{\Vert }_q\le C_d{n}^{d(\frac{1}{p}-\frac{1}{q})}\Vert f{\Vert }_p,\forall f\in {\Pi }_n^d,0<p<q\le \infty .$$

$$\Vert f_n{\Vert }_q\le C{n}^{c(p,q)}\Vert f_n{\Vert }_p,\forall f_n\in {\mathcal{H}}_n^d,0<p<q\le \infty .$$

$$\Vert f_n{\Vert }_q\le C{n}^{\frac{d-1}{2}(\frac{1}{p}-\frac{1}{q})}\Vert f_n{\Vert }_p,\forall f_n\in {\mathcal{H}}_n^d,1\le p\le 2,p\le q\le p^{\prime }.$$

$$f(x)\sim \frac{a_0}{2}+\sum _{k=1}^{\infty }(a_k\cos n_{k}x+b_k\sin n_{k}x),\frac{n_{k+1}}{n_k}\ge \gamma >1,$$

$$\Vert f{\Vert }_{L_p(\mathbb{T})}\asymp \Vert f{\Vert }_{L_2(\mathbb{T})},0<p<\infty ,$$

$$f=\sum _{j=0}^m{f}_{n_j},f_{n_j}\in {\mathcal{H}}_{n_j}^d,j=0,1,\dots ,m,$$

$$\|f\|_{q}\leq C_{d}\bigl{(}n^{d-1-\ell_{0}}m\bigr{)}^{\frac{1}{p}-\frac{1}{q}}\|f\|_{p}\leq C_{d}n^{(d-\ell_{0})(\frac{1}{p}-\frac{1}{q})}\|f\|_{p},$$

$$\|f\|_{q}\leq C_{d}\bigl{(}n^{\frac{d-1}{2}}m\bigr{)}^{\frac{1}{p}-\frac{1}{q}}\|f\|_{p}\leq C_{d}n^{\frac{d+1}{2}\,(\frac{1}{p}-\frac{1}{q})}\|f\|_{p}.$$

$$\Vert f_n{\Vert }_{p^{\prime }}\le C{n}^{\frac{d+1}{2}(\frac{1}{p}-\frac{1}{p^{\prime }})}\Vert f_n{\Vert }_p,\forall f\in {\Pi }_{n,m,\ell }^{d}1\le p\le 2.$$

$${△}_h{a}_n=a_n-a_{n+h},{△}_h^{\ell +1}={△}_h{△}_h^{\ell },\ell =1,2,\dots .$$

$$R_n(\cos \theta ):=\frac{C_n^{\lambda }(\cos \theta )}{C_n^{\lambda }(1)},\theta \in [0,\pi ],$$

$${△}_h^{\ell }{R}_n(\cos \theta ):={△}_h^{\ell }{a}_n=\sum _{j=0}^{\ell }(-1{)}^j(\genfrac{}{}{0ex}{}{\ell }{j})R_{n+hj}(\cos \theta ),\ell =1,2,\dots ,n=0,1,\dots ,$$

$$\bigl{|}\triangle_{1}^{\ell}R_{n}(\cos\theta)\bigr{|}\leq C\theta^{\ell}(1+n\theta)^{-\frac{d-1}{2}},\quad\theta\in[0,\pi/2],\quad\ell\in{\mathbb{N}}.$$

$$\bigl{|}\triangle_{2}^{\ell}R_{n}(\cos\theta)\bigr{|}\leq C\theta^{\ell}(1+n\theta)^{-\frac{d-1}{2}},\quad\theta\in[0,\pi/2].$$

$${△}_2^{\ell }{R}_n(\cos \theta )=\sum _{j=0}^{\ell }(-1{)}^j(\genfrac{}{}{0ex}{}{\ell }{j})R_{n+2j}(\cos \theta ),$$

$$\bigl{|}\triangle_{2}^{\ell}R_{n}(\cos\theta)\bigr{|}\leq C\begin{cases}\theta^{\ell}(1+n\theta)^{-\frac{d-1}{2}},&\theta\in[0,\pi/2],\\ (\pi-\theta)^{\ell}(1+n(\pi-\theta))^{-\frac{d-1}{2}},&\theta\in[\pi/2,\pi].\end{cases}$$

$$P(x)=c_n\frac{1}{{\omega }_d}{\int }_{{\text{SS}}^d}P(y)R_n(x\cdot y)d\sigma (y),x\in {\text{SS}}^d,$$

$$c_n:=\frac{\Gamma (\frac{d+1}{2})}{2{\pi }^{(d+1)/2}}\frac{d+2n-1}{d+n-1}\frac{\Gamma (d+n)}{\Gamma (n+1)\Gamma (d)}\asymp n^{d-1},$$

$$P(x)$$

$$=c_n\frac{1}{{\omega }_d}{\int }_{{\text{SS}}^d}P(y){△}_2^{\ell }{R}_n(x\cdot y)d\sigma (y).$$

$$\bigl{|}\triangle_{2}^{\ell}R_{n}(\cos\theta)\bigr{|}\leq C\min\{n^{-\ell},n^{-\frac{d-1}{2}}\}.$$

$$Tg(x):=\frac{1}{{\omega }_d}{\int }_{{\text{SS}}^d}g(y)H(x\cdot y)d\sigma (y),x\in {\text{SS}}^d,g\in L^1({\text{SS}}^d),$$

$$H(\cos \theta )=\sum _{k=0}^m{c}_{n_k}\sum _{j=0}^{\ell }(-1{)}^j(\genfrac{}{}{0ex}{}{\ell }{j})R_{n_k+2j}(\cos \theta ).$$

$$Tg=\sum _{k=0}^m\sum _{j=0}^{\ell }(-1{)}^j(\genfrac{}{}{0ex}{}{\ell }{j})\frac{c_{n_k}}{c_{n_k+2j}}{\mathrm{proj}}_{n_k+2j}(g),$$

$$\Vert H{\Vert }_{\infty }\le C\sum _{k=0}^m{n}_k^{d-1-{\ell }_0}\le Cm{n}^{d-1-{\ell }_0},$$

$$\Vert Tg{\Vert }_{\infty }\le Cm{n}^{d-1-{\ell }_0}\Vert g{\Vert }_1,\forall g\in L^1({\text{SS}}^d).$$

$$\Vert Tg{\Vert }_2\le C\Vert g{\Vert }_2,\forall g\in L^2({\text{SS}}^d).$$

$$\Vert Tg{\Vert }_{p^{\prime }}\le C(m{n}^{d-1-{\ell }_0}{)}^{\frac{1}{p}-\frac{1}{p^{\prime }}}\Vert g{\Vert }_p,\forall g\in L^p({\text{SS}}^d).$$

$$\Vert f{\Vert }_{p^{\prime }}\le C(m{n}^{d-1-{\ell }_0}{)}^{\frac{1}{p}-\frac{1}{p^{\prime }}}\Vert f{\Vert }_p,1\le p\le 2.$$

Full text

F. D. was supported by NSERC Canada under the grant RGPIN 04702 Dai. D. G. was supported by the Russian Science Foundation under grant 18-11-00199. S. T. was partially supported by MTM 2017-87409-P, 2017 SGR 358, and by the CERCA Programme of the Generalitat de Catalunya.

1. Introduction

Let $\SS^{d}=\{x\in\mathbb{R}^{d+1}\colon|x|=1\}$ denote the unit sphere of ${\mathbb{R}}^{d+1}$ equipped with the usual surface Lebesgue measure $d\sigma(x)$, and $\omega_{d}$ the surface area of the sphere $\SS^{d}$; that is, $\omega_{d}:=\sigma(\SS^{d})=2\pi^{\frac{d+1}{2}}/\Gamma(\frac{d+1}{2})$. Here, $|\mathop{\cdot}|$ denotes the Euclidean norm of ${\mathbb{R}}^{d+1}$. Given $0<p\leq\infty$, we denote by $L^{p}(\SS^{d})$ the usual Lebesgue $L^{p}$-space defined with respect to the measure $d\sigma(x)$ on $\SS^{d}$, and $\|\mathop{\cdot}\|_{p}=\|\mathop{\cdot}\|_{L^{p}(\SS^{d})}$ the quasi-norm of $L^{p}(\SS^{d})$; that is,

[TABLE]

In what follows $c$, $C$ will denote positive constants whose value may change with each occurrence. The notation $A\asymp B$ means that $c^{-1}A\leq B\leq cA.$

Let $\Pi_{n}^{d}$ denote the space of all spherical polynomials of degree at most $n$ on $\SS^{d}$ (i.e., restrictions on $\SS^{d}$ of polynomials in $d+1$ variables of total degree at most $n$), and $\mathcal{H}_{n}^{d}$ the space of all spherical harmonics of degree $n$ on $\SS^{d}$. As is well known (see, e.g., [2, Chap. 1]), both $\mathcal{H}_{n}^{d}$ and $\Pi_{n}^{d}$ are finite dimensional spaces with $\dim\mathcal{H}_{n}^{d}\asymp n^{d-1}$ and $\dim\Pi_{n}^{d}\asymp n^{d}$.

The spaces $\mathcal{H}_{k}^{d}$ are mutually orthogonal with respect to the inner product of $L^{2}(\SS^{d})$, and the orthogonal projection $\operatorname{proj}_{k}$ of $L^{2}(\SS^{d})$ onto the space $\mathcal{H}_{k}^{d}$ can be expressed as a spherical convolution:

[TABLE]

where the $C_{k}^{\lambda}$ denote the Gegenbauer polynomials as defined in [1, Sec. 10.9].

The classical Nikolskii inequality for spherical polynomials reads as follows (see, e.g., [6]):

[TABLE]

In [3], continuing Sogge’s investigations [7], we obtained the sharp asymptotic order of the following Nikolskii inequality for spherical harmonics $f_{n}$ of degree $n$, that is,

[TABLE]

In many cases, these sharp estimates turn out to be remarkably better than the corresponding estimate for spherical polynomials (1.2). In particular, we have that

[TABLE]

Furthermore, this estimate is sharp.

These results, in particular, shows that there are no exponents $0<p<q\leq\infty$ such that the equivalence $\|f_{n}\|_{q}\asymp\|f_{n}\|_{p}$ holds for any $f_{n}\in\mathcal{H}_{n}^{d}$. This in turn implies that no analogue of the following Zygmund theorem for lacunary trigonometric series [9] for spherical polynomials: For any trigonometric series of the form

[TABLE]

one has

[TABLE]

*where the equivalent constants depend only on $p$ and $\gamma$. *

In this paper, we prove that for $d\geq 2$, the asymptotic order of the Nikolskii inequality can be significantly improved when restricted on a wide class of “lacunary” spherical polynomials, although the order is sharp on the whole space of spherical polynomials. To be precise, given positive integers $\ell$, $m$ with $m\leq n/\ell$, we denote by $\Pi_{n,m,\ell}^{d}$ the class of all spherical polynomials $f$ that can be represented in the form

[TABLE]

for some sequence of nonnegative integers $\{n_{j}\}_{j=0}^{m}\subset[0,n]$ such that $n_{j}-n_{j-1}\geq 2\ell+1$ for all $j=1,\dots,m$. Given $f\in\Pi_{n,m,\ell}^{d}$, we denote by $N_{f}$ the largest integer $j$ for which $\operatorname{proj}_{j}f\neq 0$. Note that $\Pi_{n,m,\ell}^{d}$ is not a linear space.

Our main goal in this paper is to show

Theorem 1**.**

Let $(p,q)$ be a pair of exponents satisfying either of the following two conditions: (i) $0<p\leq 1$ and $p\leq q$ or (ii) $1\leq p\leq 2$ and $p\leq q\leq p^{\prime}$. If $f\in\Pi_{n,m,\ell}^{d}$, and $d\geq 2$, then we have

[TABLE]

where $\ell_{0}=\min\{\ell,\frac{d-1}{2}\}$. In particular, if $\ell\geq\frac{d-1}{2}$, then

[TABLE]

It is worth mentioning that the asymptotic order of the Nikolskii exponent for “lacunary” spherical polynomials lies between the classical exponent provided by (1.2) and the one for the spherical harmonics given by (1.3). In particular, we have that for $\ell\geq\frac{d-1}{2}$,

[TABLE]

It is also clear that inequality (1.7) generalizes (1.4).

Note that no improvement can be achieved in the order of the Nikolskii inequality for similar “lacunary” trigonometric polynomials on the unit circle, cf. (1.5).

The Nikolskii type inequalities are closely related to the Remez type inequalities in a very general setting, as was shown in [8, pp. 601-602]. Moreover, these inequalities play a crucial role in establishing a Sobolev-type embedding result for the Besov spaces: B^{r}_{q}\big{(}L_{p}(\mathbb{S}^{d})\big{)}\hookrightarrow L^{q}(\mathbb{S}^{d}), (see [5, Cor. 4] and [4, Sec.8]). As a result, Theorem 1 can be applied to improve the Remez type inequalities as well as the limiting smoothness parameter r=\frac{d+1}{2}\bigl{(}\frac{1}{p}-\frac{1}{q}\bigr{)}_{+} in place of r=d\bigl{(}\frac{1}{p}-\frac{1}{q}\bigr{)}_{+} for “lacunary” spherical polynomials, or “lacunary” spherical functions $f\in\bigcup_{n}\Pi_{n,m,\ell}^{d}$ with $\ell\geq\frac{d-1}{2}$.

2. Proof of Theorem 1

We start with some useful definitions. Given $h\in{\mathbb{N}}$, and a sequence $\{a_{n}\}_{n=0}^{\infty}$ of real numbers, define (see, for instance, [3])

[TABLE]

Next, let

[TABLE]

denote the normalized Gegenbauer polynomial, and for a step $h\in{\mathbb{N}}$, define

[TABLE]

with $a_{n}:=R_{n}(\cos\theta)$. Here and throughout, the difference operator in $\triangle_{h}^{\ell}R_{n}(\cos\theta)$ is always acting on the integer $n$. In the case when the step $h=1$, we have the following estimate ([2, Lemma B.5.1]):

[TABLE]

On the other hand, however, the $\ell$-th order difference $\triangle_{1}^{\ell}R_{n}(\cos\theta)$ with step $h=1$ does not provide a desirable upper estimate when $\theta$ is close to $\pi$, and as will be seen in our later proof, estimate (2.1) itself will not be enough for our purpose.

To overcome this difficulty, instead of the difference with step $1$, we consider the $\ell$-th order difference $\triangle_{2}^{\ell}R_{n}(\cos\theta)$ with step $h=2$. Since $\triangle_{2}^{\ell}a_{n}=\sum_{j=0}^{\ell}\binom{\ell}{j}\triangle_{1}^{\ell}a_{n+j},$ on one hand, (2.1) implies that

[TABLE]

On the other hand, however, since

[TABLE]

and since $R_{n+2j}(-z)=(-1)^{n}R_{n+2j}(z)$ ([1, Sec. 10.9]), we have $\triangle_{2}^{\ell}R_{n}(\cos(\pi-\theta))=(-1)^{n}\triangle_{2}^{\ell}R_{n}(\cos\theta).$ It follows that

[TABLE]

By (1.1), we obtain that for every $P\in\mathcal{H}_{n}^{d}$,

[TABLE]

where

[TABLE]

and $x\mathop{\cdot}y$ denotes the dot product of $x,y\in{\mathbb{R}}^{d}$. Since $R_{j}(x\,\cdot\,)\in\mathcal{H}_{j}^{d}$ for any fixed $x\in\SS^{d}$, it follows by the orthogonality of spherical harmonics that for any $P\in\mathcal{H}_{n}^{d}$, and any $\ell\in{\mathbb{N}}$,

[TABLE]

By (2.2), for any positive integer $\ell$,

[TABLE]

Let $f\in\Pi_{n,m,\ell}^{d}$ be given in (1.6) with $n_{m}=n$. Define the operator

[TABLE]

where

[TABLE]

Clearly,

[TABLE]

and hence $Tf=f$. Since, by (2.3),

[TABLE]

it follows that

[TABLE]

On the other hand, by Plancherel’s formula, we have

[TABLE]

Thus, applying the Riesz-Thorin interpolation theorem, we deduce that for $1\leq p\leq 2$,

[TABLE]

Taking $Tf=f$ we arrive at

[TABLE]

Further, log-convexity of ${L^{p}}$ norms, namely $\|f\|_{q}\leq\|f\|_{p}^{\theta}\|f\|_{p^{\prime}}^{1-\theta}$ with $\frac{\theta}{p}+\frac{1-\theta}{p^{\prime}}=\frac{1}{q}$ and $0\leq\theta\leq 1$, implies

[TABLE]

where $1\leq p\leq 2$ and $p\leq q\leq p^{\prime}.$

To complete the proof we have to show that this inequality is valid for $0<p<1$ and $p\leq q$. Let first $q=\infty$. Using (2.4) with $p=1$, we have

[TABLE]

This yields that

[TABLE]

If $q<\infty$, we write

[TABLE]

Applying (2.5) implies

[TABLE]

completing the proof.