<abstract xmlns="http://www.w3.org/1999/xhtml"> We prove that the anisotropic fractional maximal operator Mα,σ and the anisotropic Riesz potential operator Iα,σ, 0 &lt; α &lt; ∣σ∣ are bounded from the anisotropic modified Morrey space L̃1,b,σ(Rn) to the weak anisotropic modified Morrey space WL̃q,b,σ(Rn) if and only if, α/|σ|≤1-1/q≤α/(|σ|(1-b)) and from L̃p,b,σ(Rn) to L̃q,b,σ(Rn) if and only if, α/|σ| ≤ 1/p-1/q≤α ((1-b) |σ|). In the limiting case <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/auom_111_Fig_1.png"></inline-graphic></inline-formula> we prove that the operator Mα,σ is bounded from L̃p,b,σ(Rn) to L∞ (Rn) and the modified anisotropic Riesz potential operator Ĩα,σ is bounded from L̃p,b,σ(Rn) to BMOσ(Rn). </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 space L̃1,b,σ(Rn) to the weak anisotropic modified Morrey space WL̃q,b,σ(Rn) if and only if, α/|σ|≤1-1/q≤α/(|σ|(1-b)) and from L̃p,b,σ(Rn) to L̃q,b,σ(Rn) if and only if, α/|σ| ≤ 1/p-1/q≤α ((1-b) |σ|). In the limiting case we prove that the operator Mα,σ is bounded from L̃p,b,σ(Rn) to L∞ (Rn) and the modified anisotropic Riesz potential operator Ĩα,σ is bounded from L̃p,b,σ(Rn) to BMOσ(Rn).

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

Department of Mathematics Azerbaijan State Pedagogical University, Baku, Azerbaijan

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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...

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

Published Online: Sep 19, 2013

Page range: 111 - 130

DOI: https://doi.org/10.2478/auom-2013-0026

Keywords
anisotropic Riesz potential, anisotropic fractional maximal function, anisotropic modified Morrey space, anisotropic BMO space

This content is open access.

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

Published Online: Sep 19, 2013

Page range: 111 - 130

DOI: https://doi.org/10.2478/auom-2013-0026

Keywordsanisotropic Riesz potential, anisotropic fractional maximal function, anisotropic modified Morrey space, anisotropic BMO space

This content is open access.

Keywords
anisotropic Riesz potential, anisotropic fractional maximal function, anisotropic modified Morrey space, anisotropic BMO space