In this paper it is proved that anisotropic fractional maximal operator M α,σ , 0 ≤ α < |σ| is bounded on anisotropic generalized Morrey spaces M p,ϕ,σ , where |σ| = n i=1 σ i is the homogeneous dimension of R n . We find the conditions on the pair (ϕ 1 , ϕ 2 ) which ensure the Spanne-Guliyev type boundedness of the operator M α,σ from anisotropic generalized Morrey space M p,ϕ 1 ,σ to M q,ϕ 2 ,σ , 1 < p ≤ q < ∞, 1/p − 1/q = α/|σ|, and from the space M 1,ϕ 1 ,σ to the weak space W M q,ϕ 2 ,σ , 1 < q < ∞, 1 − 1/q = α/|σ|. We also find conditions on the ϕ which ensure the AdamsGuliyev type boundedness ofAs applications, we establish the boundedness of some Schödinger type operators on anisotropic generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.
We prove that the anisotropic fractional maximal operator Mα,σ and the anisotropic Riesz potential operator Iα,σ , 0 < α < |σ| are bounded from the anisotropic modified Morrey spacewe prove that the operator Mα,σ is bounded from L p,b,σ (R n ) to L∞(R n ) and the modified anisotropic Riesz potential operator Iα,σ is bounded from L p,b,σ (R n ) to BM Oσ(R n ) .
We study some embeddings into the anisotropic Morrey space L p,b,σ (R n ) and the modified anisotropic Morrey spaces L p,b,σ (R n ). As applications we get the boundedness of the anisotropic fractional maximal operator Mα,In this limiting case we prove that the modified anisotropic Riesz potential operator Iα is bounded from L p,b,σ (R n ) to BM Oσ(R n ).Mathematics Subject Classification 2010: 42B20, 42B25, 42B35.
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