Boundedness and Compactness Criteria for a Generalized Truncated Potential

Boundedness and compactness criteria are established for a generalized truncated Riesz potential $K_{\alpha} f(x,t) =\int _{|y|\leq 2|x|} (|x-y| +t)^{\alpha -n} f(y)\,dy$, $t\in [0,\infty)$, $x\in \mathbb R^n$, that acts from $L^p (\mathbb R^n)$ to $L_{\nu }^q (\mathbb R_+^{n+1})$, where ${1<p<\infty }$, ${0<q<\infty}$, ${\alpha >n/p}$, and $\nu$ is a positive Borel measure on $\mathbb R_+^{n+1}$. Also, two-sided estimates for a measure of noncompactness of the operator $K_{\alpha }$ are obtained.