The dyadic Riesz vector I

We derive a dyadic model operator for the Riesz vector. We show linear lower $L^p$ bounds for $1<p<\mathrm{\infty }$ between this model operator and the Riesz vector, when applied to functions with values in Banach spaces. By a lower bound we mean that the boundedness of the Riesz vector implies the boundedness of the dydic Riesz vector.