I added the necessary integrability restriction on the exponents $\alpha_{j}$. Assume that $\sum_{j=1}^{N}\alpha_{j}<N-1$.

Okay. I got it your example. Next question is: You have showed that, given $f\in L^{2}$ is such that $f(x)=x^{1/4} g(x)$ and $x\mapsto \int \frac{\widehat{f}(y)}{|x-y|^{3/4}}\,dy$ is supported in $[1,1+\epsilon]$ then the form is bounded by $\int |f|^{2}/x^{1/2}$. What does that imply for (*)? I am also shaking off the nagging existence question :)

I should have clarified this in the question; I hope the estimate (*) holds for some $1\leq p\leq 2$. And I don't see how you estimated the kernel $H$ in your example. I don't think it is so simple.