Some remarks on \(L^{p}\) - \(L^{q}\) estimates for some singular fractional integral operators (Q882023)

In the paper \(L^p-L^q\) estimates for the operator \[ Af(x)= \int_{\mathbb{R}^d} f(x-y)\bigg( \prod_{i=1}^{d-1} | y_i| ^{\gamma_i-1} \bigg) d\mu(y), \;\; d\geq 2,\; \gamma_i>0, \] are studied, where \(\mu\) is the measure on \(R^d\) given by \[ \mu(E)= \int_{\mathbb{R}^{d-1}} \chi_E (x, \varphi(x)) \eta(x) dx \] with \(\varphi(x)= \sum_{i=1}^{d-1}\pm | x_i| ^{a_i}\) (\(1\neq a_i\in \mathbb{R}\)) and a smooth function \(\eta: \mathbb{R}^{d-1}\mapsto \mathbb{R}\) supported in \([-1,1]^{d-1}\).