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

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    $\begingroup$ Could you please define what you mean by "Kantorovich conditions"? There are numerous sufficient boundedness conditions for positive integral operators between weighted Orlicz spaces in literature (but no necessary and sufficient ones). Also the term "locally inferable" does not seem to be so standard to me. $\endgroup$ – Martin Väth Sep 27 '20 at 19:05
  • $\begingroup$ Thank you for noticing my typo. It should integrable. As for kantorovich condition I have in mind condition that states that the $l_q$ norm of the kernel is finite. I have added this to the question. $\endgroup$ – user124297 Sep 29 '20 at 1:38
  • $\begingroup$ You mean $\int K(x,y)^qd(x,y)<\infty$? For which $q$ and how is that dependent from $\Phi_k$ and $u_k$? Just to make clear what I mean: In case $\Phi_k(u)=|u|^{p_k}$ a simple sufficient criterion would be the finiteness of the mixed norm $\int\left(\int K(x,y)^{p_2'}dy\right)^{p_1}dx$ which is already rather different than the first criterion. $\endgroup$ – Martin Väth Sep 29 '20 at 17:05
  • $\begingroup$ Yes, that is exactly what I meant. Thank you. My though was to represent $u_1, u_2$ as power weights functions. But still cannot figure it out. $\endgroup$ – user124297 Sep 30 '20 at 2:52
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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 $p_1=(\frac1{p_2}-(1-\lambda))^{-1}$.

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