Potentials on R ^N

Abstract

In this chapter we develop a new approach to truncated potentials. We introduce the extension of truncated potentials and prove necessary and sufficient conditions for boundedness from L ^p (R ⁿ ) into L ^q_v (R ⁿ ), when 1 < p < ∞, 0 < q < ∞ and α > n/p. A generalization of Sawyer’s result [258] is presented. Then a compactness criterion for this operator is proved, and upper and lower estimates of its distance from the class of compact operators are derived.