# Kernel for Hardy operator

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 $$M_1(x,y)\leq M_1(x,z)+M_1(z,y), \quad y, (with $$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.

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