Journal of Global Optimization

, Volume 57, Issue 4, pp 1385–1399 | Cite as

Maximal, potential and singular operators in vanishing generalized Morrey spaces



We introduce vanishing generalized Morrey spaces \({V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}\) with a general function \({\varphi(x, r)}\) defining the Morrey-type norm. Here \({\Pi \subseteq \Omega}\) is an arbitrary subset in Ω including the extremal cases \({\Pi = \{x_0\}, x_0 \in \Omega}\) and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces \({V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}\) we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type \({V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}\) -theorem for the potential operator I α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \({\varphi(x, r)}\). No monotonicity type condition is imposed on \({\varphi(x, r)}\). In case \({\varphi}\) has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function \({\varphi}\). The proofs are based on pointwise estimates of the modulars defining the vanishing spaces


Morrey spaces Vanishing generalized Morrey spaces Maximal operator Singular operator Potential operator 

Mathematics Subject Classification



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© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Luleå University of TechnologyLuleåSweden
  2. 2.Narvik University CollegeNarvikNorway

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