Two-weight inequalities for Riesz potential and its commutators on weighted global Morrey-type spaces GM p,θ,φ ω (R n ) Cahit Avsar 1 , Canay Aykol 2,* , Javanshir J. Hasanov 3 , Ali M. Musayev 4 1,2 Department of Mathematics, Ankara University, Ankara, Turkey. 3,4 Azerbaijan State Oil and Industry University, Baku, Azerbaijan. * Corresponding author. E-mail: cahitavsar@gmail.com 1 , Canay.Aykol@science.ankara.edu.tr 2 , hasanovjavanshir@gmail.com 3 , emus1957@mail.ru 4 Abstract In this paper we prove the two-weight boundedness of Riesz potential and its commutators on weighted global Morrey-type spaces GM p,θ,ϕ ω (R n ). Also we give some applications of our results. 2010 Mathematics Subject Classification. 42B20. 42B25, 42B35. Keywords. Riesz potential, maximal operator, commutator, weighted Lebesgue space, weighted global Morrey-type space, BMO space. 1 Introduction If 0 <α<n, then the Riesz potential I α f of a locally integrable function f on R n is the function defined by I α f (x)=  R n f (y)dy |x − y| n−α , 0 < α < n. Riesz potential is one of the significant tools in harmonic analysis that has a background in PDEs. In fact, by using a fractional integral operator, the solution of Laplace equation Δu = f can be represented for a nice function f on R n (see [53]). The commutators of the Riesz potential is defined by the following equality [b, I α ]f (x)=  R n (b(x) − b(y))|x − y| α−n f (y)dy, 0 < α < n. Given a measurable function b the operator |b, I α | is defined by |b, I α |f (x)=  R n |b(x) − b(y)||x − y| α−n |f (y)|dy, 0 < α < n. It is well-known that the commutator is an important integral operator and plays a key role in harmonic analysis; especially in studying the regularity of solutions of elliptic, parabolic and ultra- parabolic PDEs of second order (see [14, 18, 28, 29, 47]). Advanced Studies: Euro-Tbilisi Mathematical Journal 16(1) (2023), pp. 33–50. DOI: 10.32513/asetmj/19322008236 Tbilisi Centre for Mathematical Sciences. Received by the editors: 06 April 2022. Accepted for publication: 21 November 2022.