A two-parameter “Bergman space” inequality

Abstract:

For $f\in L^{1}([0,1]\times [0,1])$, define $\lambda _{R} \equiv \langle f,h_{(R)}\rangle$, where $h_{(R)}(x,y)=h_{(I)}(x)\cdot h_{(J)}(y)$ is a tensor product of one-parameter Haar functions. Let $1<p\leq q<\infty$ and $q\geq 2$. We prove a sufficient condition, which is close to necessary, on double sequences of weights $\{\mu _{R}\}_{R}$ and non-negative $v\in L^{1}([0,1]\times [0,1])$, which ensures that the inequality \begin{equation*} \left (\sum _{R}\vert {\lambda _{R}}\vert ^{q}\mu _{R}\right )^{1/q}\leq \left (\int _{[0,1]\times [0,1]}\vert {f}\vert ^{p} v dx\right )^{1/p}\end{equation*} holds for all $f\in L^{1}([0,1]\times [0,1])$. We extend our result to an inequality concerning two-parameter wavelet families.