Let $\mu$ be a finite compactly supported Borel measure on the real line. On the space $L^2(\mu)$ consider the integral operators $$ (K_a f)(x)=\int k_a(x, y)f(y)d\mu(y) $$ with $$ k_a(x, y)=\frac{a\cdot(x-y)}{|x-y|^2+a^2}, \qquad a>0. $$ From the representation $$ k_a(x, y)=\int_{-\infty}^\infty e^{\frac{itx}a}e^{-\frac{ity}a}\ te^{-|t|}\ dt $$ it follows that the trace class norms of the operators $K_a$ are uniformly bounded. This is a consequence of the fact that the integral operators with kernels $e^{\frac{itx}a}e^{-\frac{ity}a}$ are rank-one operators whose norm is equal to the full mass of $\mu$.
Question. Describe the class of measures $\mu$ for which the trace class norms of the operators $K_a$ tend to 0 as $a\searrow 0$.
Of special interest is the question if this is possible for measures containing a continuous component.
