Boundedness of some singular integrals operators in weighted generalized Grand Lebesgue spaces (Q2147956)

For \(x\in I\), the Calderón singular operator is defined by \[ C_a f(x):=\int_{I} \frac{a(x)-a(t)}{(x-t)^2}f(t)\,\textrm{d}t \] and the Hardy-Littlewood maximal operator is defined by \[ Mf(x):=\sup_{J\subset I} \frac{1}{|J|}\int_{J} |f(y)|\,\textrm{d}y, \] where \(I\) is a bounded interval in \(\mathbb{R}\), \(\bar{I}\) is its closure, \(a: \bar{I}\rightarrow \mathbb{R}\) is a Lip1 function on \(\bar{I}\) and \(J\subset I\) is a subinterval. The authors consider the boundedness of \(C_a\) and \(M\) in the generalized weighted Grand Lebesgue spaces \(L_w^{p),\delta}(I)\), \(1<p<\infty\), equipped with the norm \[ \|f\|_{L_w^{p),\delta}(I)}:=\sup_{0<\varepsilon<p-1} \delta(\varepsilon)^{\frac{1}{p-\varepsilon}} \left(\frac{1}{|I|}\int_{I} |f(y)|^{p-\varepsilon} w(y)\,\textrm{d}y \right)^{\frac{1}{p-\varepsilon}}. \]