Gowers norms for singular measures

Gowers introduced the notion of uniformity norm $\|f\|_{U^k(G)}$ of a bounded function $f:G\rightarrow\mathbb{R}$ on an abelian group $G$ in order to provide a Fourier-theoretic proof of Szemeredi's Theorem, that is, that a subset of the integers of positive upper density contains arbitrarily long arithmetic progressions. Since then, Gowers norms have found a number of other uses, both within and outside of Additive Combinatorics. The $U^k$ norm is defined in terms of an operator $\triangle^k : L^{\infty}(G)\mapsto L^{\infty} (G^{k+1})$. In this paper, we introduce an analogue of the object $\triangle^k f$ when $f$ is a singular measure on the torus $\mathbb{T}^d$, and similarly an object $\|\mu\|_{U^k}$. We provide criteria for $\triangle^k \mu$ to exist, which turns out to be equivalent to finiteness of $\||\mu|\|_{U^k}$, and show that when $\mu$ is absolutely continuous with density $f$, then the objects which we have introduced are reduced to the standard $\triangle^kf$ and $\|f\|_{U^k(\mathbb{T})}$. We further introduce a higher-order inner product between measures of finite $U^k$ norm and prove a Gowers-Cauchy-Schwarz inequality for this inner product.