A characterization of two weight norm inequalities for fractional and Poisson integrals

For $1>p⩽q>\mathrm{\infty }$ and $w(x)$ , $v(x)$ nonnegative functions on ${\mathbf{R}}^n$ , we show that the weighted inequality \[ ${\left(\int |Tf{|}^{q}w\right)}^{1/q}⩽C{\left(\int f^{p}v\right)}^{1/p}$ \] holds for all $f⩾0$ if and only if both \[ $\int {[T({\chi }_Q{v}^{1-p^{\prime }})]}^{q}w⩽C_1{\left({\int }_Q{v}^{1-p^{\prime }}\right)}^{q/p}>\mathrm{\infty }$ \] and \[ $\int {{[T({\chi }_{Q}w)]}^{p^{\prime }}{v}^{1-p^{\prime }}⩽C_2\left({\int }_{Q}w\right)}^{p^{\prime }/q^{\prime }}>\mathrm{\infty }$ \] hold for all dyadic cubes $Q$ . Here $T$ denotes a fractional integral or, more generally, a convolution operator whose kernel $K$ is a positive lower semicontinuous radial function decreasing in $|x|$ and satisfying $K(x)⩽CK(2x)$ , $x\in {\mathbf{R}}^n$ . Applications to degenerate elliptic differential operators are indicated. In addition, a corresponding characterization of those weights $v$ on ${\mathbf{R}}^n$ and $w$ on ${\mathbf{R}}_{+}^{n+1}$ for which the Poisson operator is bounded from $L^p(v)$ to $L^q(w)$ is given.