# 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.

## Key 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.