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Let $H_1$ is generalized Hardy operator, i.e. $$ (H_1 f)(x)=\int_0^xM_1(x,y)f(y)dy, $$ where $f$ is measurable function on $R_+$ and kernel $M_1$ satisfies

$M_1(x,y)=\int_0^xL(y,z)dz$ is increasing in $x$ an decreasing in $y$ and $M_1(x,y)\leq M_1(x,z)+M_1(z,y), \quad y<z<x$, (with $L \in M_+(R_+^2)$, i.e. measurable function on $R^2$).

I am looking for an example of kernel $M(x+y)$, such that M(t) is decreasing with $t$, which would satisfy conditions above.

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    $\begingroup$ How could $M(x+y)$ be increasing in $x$ and decreasing in $y$? $\endgroup$ – Michael Renardy Oct 1 at 21:22

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