ОБ ОГРАНИЧЕННОСТИ И КОМПАКТНОСТИ ОДНОГО КЛАССА ОПЕРАТОРОВ ТИПА СВЕРТКИ В ПРОСТРАНСТВАХ МОРРИ

Abstract

In a Morrey space, we study the integral operator 𝐻𝑏 with the kernel 𝑏(𝑥, 𝑦)ℎ(𝑥 − 𝑦), where ℎ is the summable function. The function 𝑏 will be referred to as a characteristic. For such operator, a sufficient condition of boundedness is obtained. This condition is obviously fulfilled in the case of the essentially bounded function 𝑏(𝑥, 𝑦). It is shown that if the characteristic 𝑏(𝑥, 𝑦) is essentially bounded and has a given behavior at infinity, then the operator 𝐻𝑏 is compact in a Morrey space. As a consequence, we have obtained the criterion for the Fredholm property of the operator, which is the sum of the identity operator and the operator 𝐻𝑏 . Namely, it is shown that such operator is Fredholm if and only if its symbol does not vanish, and the index of the operator is equal to zero.