Necessary and sufficient conditions for the boundedness of the anisotropic Riesz potential in anisotropic modified Morrey spaces

Abstract

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 ) .

Some Embeddings Into the Anisotropic Morrey and Modified Anisotropic Morrey Spaces. Some Applications

“…In [9] we proved that the anisotropic Riesz potential operator I α,σ , 0 < α < |σ| is bounded from L 1,b,σ (R n ) to W L q,b,σ (R n ) if and only if, α/|σ| ≤ 1 − 1/q ≤ α/(|σ| (1 − b)) and from L p,b,σ (R n ) to L q,b,σ (R n ) if and only if, α/|σ| ≤ 1/p − 1/q ≤ α/ ((1 − b)|σ|). Note that in the case σ ≡ 1 and b = λ n this result was proved in [13].…”