Journal of Mathematical Sciences, Vol. 268, No. 3, December, 2022 COMPACTNESS OF FRACTIONAL TYPE INTEGRAL OPERATORS ON SPACES OF HOMOGENEOUS TYPE V. Kokilashvili A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University 2, Merab Aleksidze II Lane, Tbilisi 0193, Georgia A. Meskhi ∗ A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University 2, Merab Aleksidze II Lane, Tbilisi 0193, Georgia School of Mathematics, Kutaisi International University 5th Lane, K Building, 4600 Kutaisi, Georgia alexander.meskhi@tsu.ge, alexander.meskhi@kiu.edu.ge UDC 517.9 For a space (X, d, μ) of homogeneous type and a fractional type integral operator K α defined on (X, d, μ) we find a necessary and sufficient condition on the exponent q governing the compactness of K α from L p (X ) to L q (X ), where 1  p,q < ∞ and μ(X ) < ∞. Bibliography: 12 titles. Dedicated to the 85th birthday of Professor V. Maz’ya 1 Introduction Fractional integral operators have important applications to harmonic analysis, partial differen- tial equations, and the theory of Sobolev embeddings (cf., for example, [1, 2]). In this paper, we deal with the fractional type integral operator K α f (x) :=  X k(x, y)f (y) dμ(y), k(x, y)  0, (1.1) where the kernel k(x, y) satisfies the following condition. It is assumed that there are positive constants c 1 and c 2 such that for all x, y ∈ X , x = y, c 1 d(x, y) α−N  k(x, y)  c 2 d(x, y) α−N , 0 < α < N, and (X, d, μ) is a space of homogeneous type with finite measure μ of dimension N , i.e., the ∗ To whom the correspondence should be addressed. Translated from Problemy Matematicheskogo Analiza 118, 2022, pp. 109-115. 1072-3374/22/2683-0368 c  2022 Springer Nature Switzerland AG 368 DOI 10.1007/s10958-022-06202-2