Capacities associated with scalar signed riesz kernels, and analytic capacity

Analytic capacity is associated with the Cauchy kernel 1/z and the space L∞. One has likewise capacities associated with the real and imaginary parts of the Cauchy kernel and L∞. Striking results of Tolsa and a simple remark show that these three capacities are comparable. We present an extension of this fact to Rn, n ≥ 3, involving the vector-valued Riesz kernel of homogeneity -1 and n - 1 of its components.