Boundedness of quasilinear integral operators on the cone of monotone functions (Q511348)

Let \({\mathcal M}^+\) denote the set of nonnegative Lebesgue measurable function on \([0,\infty)\) and let \({\mathcal M}^\downarrow\) denote its subset of all nonincreasing functions on \([0,\infty)\). For given weights \(u,v,w,\varrho\in{\mathcal M}^+\) and \(0<p,r,q\leq \infty\), the weighted inequality \(\|Rf\|_{L_\varrho^p}\leq C\|f\|_{L_v^p}\) for \(f\in{\mathcal M}^\downarrow\) on Lebesgue spaces \(L_\varrho^p\) and \(L_v^p\) is characterized for each quasilinear integral operator \(R\) of one of the forms \[ Tf(x)=\left(\int_x^\infty k_1(y,x)w(y)\left(\int_0^y k_2(y,z)f(z)u(z)dz\right)^q dy\right)^{1/q}, \] \[ {\mathcal T}f(x)=\left(\int_0^x k_1(x,y)w(y)\left(\int_y^\infty k_2(z,y)f(z)u(z)dz\right)^q dy\right)^{1/q}, \] \[ Sf(x)=\left(\int_x^\infty k_1(y,x)w(y)\left(\int_y^\infty k_2(z,y)f(z)u(z)dz\right)^q dy\right)^{1/q}, \] \[ {\mathcal S}f(x)=\left(\int_0^x k_1(x,y)w(y)\left(\int_0^y k_2(y,z)f(z)u(z)dz\right)^q dy\right)^{1/q}, \] where the kernels \(k_i(x,y)\geq 0\) satisfy the condition \(k_i(x,y)\approx k_i(x,z)+k_i(z,y)\) for \(0\leq y\leq z\leq x\) and \(i=1,2\).