All conditions for Stein-Weiss inequalities are necessary

The famous Stein-Weiss inequality on ${\mathbf{R}}^n\times {\mathbf{R}}^n$, also known as the doubly weighted Hardy-Littlewood-Sobolev inequality, asserts that $$\mid {\iint }_{{\mathbf{R}}^n\times {\mathbf{R}}^n}\frac{f(x)g(y)}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\mathrm{\mid }x-y{\mathrm{\mid }}^{\lambda }\mathrm{\mid }y{\mathrm{\mid }}^{\beta }}dxdy\mid \lesssim \mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\mathbf{R}}^n)}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^r({\mathbf{R}}^n)}$$ holds for any $f\in L^p({\mathbf{R}}^n)$ and $g\in L^r(\mathbf R^n)$underseveralconditionsontheparameters$n$,$p$,$r$,$\alpha$,$\beta$,and$\lambda$. Extending the above inequality to either different domains rather than ${\mathbf{R}}^n\times {\mathbf{R}}^n$ or classes of more general kernels rather than the classical singular kernel $\mathrm{\mid }x-y{\mathrm{\mid }}^{-\lambda }$ has been the subject of intensive studies over the last three decades. For example, Stein-Weiss inequalities on the upper half space, on the Heisenberg group, on homogeneous Lie group are known. Served as the first step, this work belongs to a set in which the following inequality on the product ${\mathbf{R}}^{n-k}\times {\mathbf{R}}^n$ is studied $$\mid {\iint }_{{\mathbf{R}}^n\times {\mathbf{R}}^{n-k}}\frac{f(x)g(y)}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\mathrm{\mid }x-y{\mathrm{\mid }}^{\lambda }\mathrm{\mid }y{\mathrm{\mid }}^{\beta }}dxdy\mid \lesssim \mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\mathbf{R}}^{n-k})}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^r({\mathbf{R}}^n)}\mathrm{.}$$ Toward the validity of the above new inequality, in this work, by constructing suitable counter-examples, we establish all conditions for the parameters $n$, $p$, $r$, $\alpha$, $\beta$, and $\lambda$ necessarily for the validity of the above proposed inequality. Surprisingly, these necessary conditions applied to the case $k=1$ suggest that the existing Stein-Weiss inequalities on the upper half space are yet in the optimal range of the parameter $\lambda$. This could reflect limitations of the methods often used. Comments on the Stein-Weiss inequality on homogeneous Lie groups as well as the reverse form for Stein-Weiss inequalities are also made.