Cauchy–Leray–Fantappiè formula for linearly convex domains

An important tool in analysis of functions of one complex variable is the Cauchy formula. However, in the case of several complex variables there is no unique and convenient formula of this sort. One can use the Szego projection $S$, but the kernel of the operator $S$ has usually no explicit expression. Another choice is the Cauchy–Leray–Fantappiè formula, which has rather explicit kernel for large classes of domains. In this paper we prove the boundedness properties of the Cauchy–Leray–Fantappiè integral for linearly convex domains, as an operator on $L^p$ and $BMO$.