# Sets of $p$-restriction and $p$-spectral synthesis

## TL;DR

This paper explores conditions under which sets in Euclidean space fail to have the p-restriction property and extends spectral synthesis concepts to p-restriction sets, demonstrating the sphere's spectral synthesis for certain p-values.

## Contribution

It provides new sufficient conditions for the failure of p-restriction and extends spectral synthesis to L^p spaces for sets with p-restriction, including the unit sphere.

## Key findings

- Identifies conditions for failure of p-restriction property.
- Extends spectral synthesis to L^p spaces for p > 1.
- Shows the unit sphere is p-spectral in certain dimensions.

## Abstract

In this paper we investigate the restriction problem. More precisely, we give sufficient conditions for the failure of a set $E$ in $\mathbb{R}^n$ to have the $p$-restriction property. We also extend the concept of spectral synthesis to $L^p(\mathbb{R}^n)$ for sets of $p$-restriction when $p > 1$. We use our results to show that there are $p$-values for which the unit sphere is a set of $p$-spectral synthesis in $\mathbb{R}^n$ when $n \geq 3$.