<abstract xmlns="http://www.w3.org/1999/xhtml"> In this paper we study the fractional maximal operator Mα, 0 ≤ α &lt; Q and the Riesz potential operator ℑα, 0 &lt; α &lt; Q on the Heisenberg group in the modified Morrey spaces L͂p,λ(ℍn), where Q = 2n + 2 is the homogeneous dimension on ℍn. We prove that the operators Mα and ℑα are bounded from the modified Morrey space L͂1,λ(ℍn) to the weak modified Morrey space WL͂q,λ(ℍn) if and only if, α/Q ≤ 1 - 1/q ≤ α/(Q - λ) and from L͂p,λ(ℍn) to L͂q,λ(ℍn) if and only if, α/Q ≤ 1/p - 1/q ≤ α/(Q - λ).In the limiting case <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10309-012-0013-8_1.png"></inline-graphic></inline-formula> we prove that the operator Mα is bounded from L͂p,λ(ℍn) to L∞(ℍn) and the modified fractional integral operator Ĩα is bounded from L͂p,λ(ℍn) to BMO(ℍn).As applications of the properties of the fundamental solution of sub-Laplacian Ը on ℍn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of ℑα from the Besov-modified Morrey spaces BL͂spθ,λ(ℍn) to BL͂spθ,λ(ℍn). </abstract>

In this paper we study the fractional maximal operator Mα, 0 ≤ α < Q and the Riesz potential operator ℑα, 0 < α < Q on the Heisenberg group in the modified Morrey spaces L͂p,λ(ℍn), where Q = 2n + 2 is the homogeneous dimension on ℍn. We prove that the operators Mα and ℑα are bounded from the modified Morrey space L͂1,λ(ℍn) to the weak modified Morrey space WL͂q,λ(ℍn) if and only if, α/Q ≤ 1 - 1/q ≤ α/(Q - λ) and from L͂p,λ(ℍn) to L͂q,λ(ℍn) if and only if, α/Q ≤ 1/p - 1/q ≤ α/(Q - λ).In the limiting case we prove that the operator Mα is bounded from L͂p,λ(ℍn) to L∞(ℍn) and the modified fractional integral operator Ĩα is bounded from L͂p,λ(ℍn) to BMO(ℍn).As applications of the properties of the fundamental solution of sub-Laplacian Ը on ℍn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of ℑα from the Besov-modified Morrey spaces BL͂spθ,λ(ℍn) to BL͂spθ,λ(ℍn).

Riesz potential on the Heisenberg group and modified Morrey spaces

Institute of Mathematics and Mechanics, Baku, Azerbaijan

Nakhchivan Teacher-Training Institute, Nakhchivan, Azerbaijan

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

{"article-title":"Riesz potential on the Heisenberg group and modified Morrey spaces"}

In this paper we study the fractional maximal operator Mα, 0 ≤ α < Q and the Riesz potential operator ℑα, 0 < α < Q on the Heisenberg group in the modified...

Riesz potential on the Heisenberg group and modified Morrey spaces

Published Online: May 17, 2013

Page range: 189 - 212

DOI: https://doi.org/10.2478/v10309-012-0013-8

Keywords
Heisenberg group, Riesz potential, fractional maximal function, fractional integral, modified Morrey space, BMO space

This content is open access.

Riesz potential on the Heisenberg group and modified Morrey spaces

Published Online: May 17, 2013

Page range: 189 - 212

DOI: https://doi.org/10.2478/v10309-012-0013-8

KeywordsHeisenberg group, Riesz potential, fractional maximal function, fractional integral, modified Morrey space, BMO space

This content is open access.

Keywords
Heisenberg group, Riesz potential, fractional maximal function, fractional integral, modified Morrey space, BMO space