Criteria of general weak type inequalities for integral transforms with positive kernels

Necessary and sufficient conditions are derived in order that an inequality of the form$$\begin{gathered} \varphi (\lambda )\theta (\beta \{ (x,t) \in X x (0,\infty ):\kappa (fdv)(x,t) > \lambda \} ) \leqslant \hfill \\ \leqslant c\int\limits_X {\psi \left( {\frac{{f(x)}}{{\eta (\lambda )}}} \right)\sigma (x)dv(x)} \hfill \\ \end{gathered}$$be fulfilled for some positivec independent of λ and a ν-measurable nonnegative functionf:X→R 1, where$$\kappa (fdv)(x,t) = \int\limits_X { f(y)k(x,y,t)dv(y), t \geqslant 0}$$k: XX[0,∞)→R 1 is a nonnegative measurable kernel, (X, d, μ) is a homogeneous type space, ϕη and ϕ are quasiconvex functions, ψ ∈ Δ2, andt -αθ(t) is a decreasing function for some α, 0<α<1. A similar problem was solved in Lorentz spaces with weights.