J ournal of Mathematical I nequalities Volume 5, Number 4 (2011), 491–506 NECESSARY AND SUFFICIENT CONDITIONS FOR THE BOUNDEDNESS OF THE RIESZ POTENTIAL IN MODIFIED MORREY SPACES VAGIF S. GULIYEV,JAVANSHIR J. HASANOV AND YUSUF ZEREN (Communicated by A. Kufner) Abstract. We prove that the fractional maximal operator M α and the Riesz potential operator I α ,0 < α < n are bounded from the modified Morrey space  L 1,λ (R n ) to the weak modified Morrey space W  L q,λ (R n ) if and only if, α /n  1 - 1/q  α /(n - λ ) and from  L p,λ (R n ) to  L q,λ (R n ) if and only if, α /n  1/ p - 1/q  α /(n - λ ) . As applications, we establish the boundedness of some Sch ¨ odinger type operators on mod- ified Morrey spaces related to certain nonnegative potentials belonging to the reverse H¨ older class. As an another application, we prove the boundedness of various operators on modified Morrey spaces which are estimated by Riesz potentials. Introduction For x ∈ R n and t > 0, let B(x, t ) denote the open ball centered at x of radius t and ∁ B(x, t )= R n \ B(x, t ) . One of the most important variants of the Hardy-Littlewood maximal function is the so-called fractional maximal function defined by the formula M α f (x)= sup t >0 |B(x, t )| -1+α/n  B(x,t ) | f (y)|dy , 0  α < n, where |B(x, t )| is the Lebesgue measure of the ball B(x, t ) . The fractional maximal function M α f coincides for α = 0 with the Hardy-Little- wood maximal function Mf ≡ M 0 f and is intimately related to the Riesz potential operator I α f (x)=  R n f (y)dy |x - y| n-α , 0 < α < n (see, for example, [1] and [23]). The operators M α and I α play important role in real and harmonic analysis (see, for example [26, 29, 34, 35]). Mathematics subject classification (2010): Primary 42B20, 42B25, 42B35. Keywords and phrases: Riesz potential, fractional maximal function, modified Morrey space, Hardy- Littlewood-Sobolev inequality, Sch¨ odinger type operator. The research of V. Guliyev and J. Hasanov was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project EIF-2010-1(1)-40/06-1. The research of V. Guliyev was partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 110T695). c  , Zagreb Paper JMI-05-43 491