On the problem of existence in principal value of a Calderón–Zygmund operator on a space of non‐homogeneous type

For classical Calder\'{o}n-Zygmund operators (CZOs) acting in $\mathbb{R}^d$, Calder\'on and Zygmund showed that if the operator is bounded in $L^2(m_d)$, where $m_d$ is the $d$-dimensional Lebesgue measure, then the principal value integral exists $m_d$-almost everywhere. Tolsa proved an analogous result for the Cauchy transform where the Lebesgue measure is replaced by any (non-atomic, locally finite, Borel) measure. On the other hand, there are several examples showing that one cannot expect Tolsa's theorem to extend to a class of, say, smooth homogeneous kernels sharing the same degree of homogeneity as the Cauchy transform. For instance, Nazarov and the first author found a finite measure $\mu$ in $\mathbb{C}$ for which the operator associated to the Huovinen kernel $K(z)=\frac{z^k}{|z|^{k+1}}$, with $k\geq 3$ odd, is bounded in $L^2(\mu)$, but the operator fails to exist in the sense of principal value $\mu$-almost everywhere. For a wide class of $s$-dimensional CZOs (with $s\in (0,d)$) acting in $\mathbb{R}^d$, Mattila and Verdera showed that the underlying measure $\mu$ having zero $s$-density ($\lim_{r\to 0}\frac{\mu(B(x,r))}{r^s}=0$ $\mu$-almost every $x\in \mathbb{R}^d$) is a sufficient condition for the $L^2(\mu)$ boundedness of the CZO to imply the existence of the principal value integral. Building upon the basic scheme introduced by Mattila-Verdera, we introduce more general sufficient conditions on a measure $\mu$, given in terms of the transportation cost to a certain collection of ``symmetric'' measures associated to the kernel. The conditions we introduce are necessary and sufficient for the Riesz and Huovinen transforms to exist in principal value.