Scalar and vector Muckenhoupt weights

We inspect the relationship between the Ap,q condition for families of norms on vector valued functions and the Ap condition for scalar weights. In particular we will show if we are considering a norm-valued function ρ(·) such that, uniformly in all nonzero vectors x, ρ(·)(x) p ∈ Ap and ρ∗(·)(x)q ∈ Aq then the following hold: If p = q = 2, and functions take values in R2 then ρ ∈ A2,2. If p = q = 2 and func-tions take values in Rd, d ≥ 6, ρ need not be an A2,2 weight. If ρ satisfies the relatively weak A0,0 condition in addition to the scalar conditions mentioned above, then ρ ∈ Ap,q. The famous Hunt–Muckenhoupt–Wheeden (Coifman–Fefferman) Theo-rem states that the Hilbert Transform T is a bounded operator in the weighted space Lp(w) (1 < p < ∞) if and only if the weight satisfies the so-called Muckenhoupt Ap condition: