Let $K:\mathbb{R}^3\backslash\{0\}\times\mathbb{R}^3\backslash\{0\}\rightarrow\mathbb{C}$, such that $K(x,y)=K(y,x)$ and $K(x,y)=|x|^{-1}|y|^{-1}H(x,y)$, with $H$ locally bounded. 
Let $T$ be the (singular) integral operator with kernel $K$, i.e.
$$T(f)(x)=\int\_{\mathbb{R}^3}K(x,y)f(y)dy$$
Suppose that $T$ is unitary on $L^2(\mathbb{R}^3)$ and bounded from $w^{-1}L^1(\mathbb{R}^3)$ to $wL^{\infty}(\mathbb{R}^3)$, where $w(x)=1+|x|^{-1}$.


Can i deduce, by means of some interpolation tecnique, that $T$ is bounded from ${z\_p}^{-1}L^p(\mathbb{R}^3)$ to ${z\_p}L^q(\mathbb{R}^3)$? (with $1<p<2$, $1/p+1/q=1$and $z\_p$ an appropriate weight function depending on $p$)


EDIT: As Christian pointed out, one cand find $z$ such that the operator with integral kernel $K(x,y)/(z(x)z(y))$ is bounded from $L^2$ to $L^2$ and from $L^1$ to $L^{\infty}$.
Hence, a typical interpolation argument allows us to take $z\_p=z$ for every $p$.


However, i'm interested about the dependence of an "optimal" $z\_p$ on $p$. Hopefully, something like $z\_p=1+|x|^{-\alpha(p)}$, with $\alpha(p)>0$ and decreasing to $0$ as $p\rightarrow 2$.


Thank you for any suggestion



Interpolation between weighted $L^p$ spaces
I think what you need is in the following paper:


E. M. Stein and G. Weiss, Interpolation of operators with change of measures,
Transactions of the American Mathematical Society, Vol. 87 (1958), pp. 159-172 


