Example when Kantorovich condition would not hold

Question

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: $$ \rho_{\Phi_1,u_1}(T_Kf^*)\leq \rho_{\Phi_2,u_2}(f^*) $$

Answer 1

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 $p_1=(\frac1{p_2}-(1-\lambda))^{-1}$.