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

Abstract:

For $1 < p \leqslant q < \infty$ and $w(x)$, $v(x)$ nonnegative functions on ${{\mathbf {R}}^n}$, we show that the weighted inequality \[ {\left ( {\int {|Tf{|^q}w} } \right )^{1/q}} \leqslant C{\left ( {\int {{f^p}v} } \right )^{1/p}}\] holds for all $f \geqslant 0$ if and only if both \[ \int {{{[T({\chi _Q}{v^{1 - p’}})]}^q}w \leqslant {C_1}{{\left ( {\int _Q {{v^{1 - p’}}} } \right )}^{q/p}} < \infty } \] and \[ {\int {{{[T({\chi _Q}w)]}^{p’}}{v^{1 - p’}} \leqslant {C_2}\left ( {\int _Q w } \right )} ^{p’/q’}} < \infty \] 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 $K(x) \leqslant CK(2x)$, $x \in {{\mathbf {R}}^n}$. 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.