On a theorem of Muchenhoupt-Wheeden in generalized Morrey spaces

Abstract

In this paper we find a condition on a function w which ensures the equivalence of norms of the Riesz potential and the fractional maximal function in generalized Morrey spaces M_p,w(Rn).

Recall that I_α * μ and M_αμ denote the Riesz potential and the fractional maximal function associated with a non-negative measure μ on Rn, respectively. That is,
I_α * μ(x) = ∫_Rn dμ(y) / |x - y|^n-α, 0 < α < n,
and
M_αμ(x) = sup_r>0 r^α-n μ(B(x,r)), 0 ≤ α < n,
where B(x,r) denotes the open ball centered at x of radius r. If dμ(x) = f(x) dx, then I_α * μ and M_αμ will be denoted by I_αf and M_αf, respectively.

Recall that, for 0 < α < n,
M_αf(x) ≲ I_α * μ(x)
for any x ∈ Rn.

By A ≲ B we mean that A ≤ cB with some positive constant c independent of appropriate quantities. If A ≲ B and B ≲ A, we write A ≈ B and say that A and B are equivalent.

Let us denote by L_{p, loc⁺}(Rn) the set of all non-negative functions from L_p^loc(Rn). The well-known Morrey spaces M_p,α, introduced by C. Morrey in 1938 in connection with the study of partial differential equations, were widely investigated during the last decades, including the study of classical operators of harmonic analysis - maximal, singular and potential operators - in generalizations of these spaces (the so-called Morrey-type spaces).