this space below.

For now we fix a particular choice of norms on the scale of Besov space defined in terms of Littlewood-Paley theory. Let $W_0:=1$ and for $n\in {{\mathbb N}}_+$ we define $$W_n(z):=\sum_{k=2^{n-1}}^{2^{n+1}} \min\left(\frac{k-2^{n-1}}{2^n-2^{n-1}},\frac{2^{n+1}-k}{2^{n+1}-2^{n}}\right) (z^k+z^{-k}).$$ The polynomials $W_n$ are characterized by the property that their Fourier coefficients $(\hat{W}_n(k))_{k\in {{\mathbb Z}}}$ is linearly interpolating between $\hat{W}_n(2^{n-1})=\hat{W}_n(2^{n+1})=0$, $\hat{W}_n(2^n)=1$ and $\hat{W}_n(k)=\hat{W}_n(-k)$. In particular, $\sum_{n=0}^\infty \hat{W}_n(k)=1$ for any $k$. For a function $f$ on $S^1$, we define $$\Phi_n(f):=W_n*f.$$ A well known result from Littlewood-Paley theory guarantees that for any function $f$ on $S^1$, $$\|f\|_{L^p(S^1)}\sim \|(\Phi_n f)_{n\in {{\mathbb N}}}\|_{L^p(S^1\times {{\mathbb N}})}.$$

We define $$\|f\|_{B^{1/p}_{p,p}(S^1)}:=\|(2^{n/p}\Phi_n f)_{n\in {{\mathbb N}}}\|_{L^p(S^1\times {{\mathbb N}})}.$$

\[pellerhem\] Let $f$ be a function on $S^1$ extending holomorphically to the unit disc with $f(0)=0$. Then $H_{\overline{f}}\in \mathcal{L}^p(L^2(S^1))$ if and only if $f\in B^{1/p}_{p,p}(S^1)$. Moreover, for any $p_0>1$ there is a constant $C>0$ such that $$C^{-1}\|f\|_{B^{1/p}_{p,p}(S^1)}\leq \|H_{\overline{f}}\|_{\mathcal{L}^p(L^2(S^1))}\leq C\|f\|_{B^{1/p}_{p,p}(S^1)}, \quad \forall p\in [1,p_0].$$

The reader can note that the statement in [@peller Chapter 6.2, Theorem 2.1] does not give a uniform constant, but existence of a uniform constant follows from the fact that the proof is by interpolation. We shall use Peller’s theorem to compute and estimate Dixmier traces. To do so, it will be 