Let $K \in M\_+(R\_+^2), f \in M\_+(R\_+)$. Consider operator


$$
(T\_k)(x)=\int\_{R\_+}K(x,y)f(y)dy, \quad y\in R\_+.
$$


Denote by $f^\*(t)=\inf\{\lambda>0: \alpha x \in R\_+: \mu\_f(y)>\lambda\}$ the non-increasing rearrangement of $f$. Here $\mu\_f(y)=\{\alpha x\in R\_+: |f(x)|>y\}$.


Let $\Phi(x)=\int\_0^x \phi(y)\,dy$, $x \in \mathbb{R}\_+$, be an N-function, and let $u$ be locally integrable on $\mathbb{R}\_+$. Consider the gauge norm
$$
\rho\_{\Phi,u}(f)=\inf\{\lambda>0: \int\_{\mathbb{R}\_+}\Phi\left(\frac{|f(x)|}{\lambda}\right)u(x)\,dx\leq 1\},
$$where $f \in M\_+(R\_+)$.


I am trying to find an example of such $u\_1, u\_2$ when Kantorovich conditions (stated that the $l\_q$ norm of the kernel is finite) would not be true, but the following inequality would hold:
$$
\rho\_{\Phi\_1,u\_1}(T\_Kf^\*)\leq \rho\_{\Phi\_2,u\_2}(f^\*)
$$



Example when Kantorovich condition would not hold
Maybe the simplest classical example is a weakly singular kernel


$$K(x,y) = |x-y|^{-\lambda}$$


with some fixed $\lambda\in(0,1)$.


In this example $\int\_{\mathbb R^2}K(x,y)^qdx=\infty$ for every $q>0$ by Fubini-Tonelli (and also all mixed norms from my earlier comment are infinite).


However, for constant $u$ and $v$ and $\Phi\_k(t)=|t|^{p\_k}$, a famous classical theorem of Hardy-Littlewood implies that $T\_K$ is bounded if $1<p\_2<\frac1{1-\lambda}$ and $p\_1=(\frac1{p\_2}-(1-\lambda))^{-1}$.


