Continuity properties for logarithmic potentials of functions in Morrey spaces of variable exponent (Q1012454)

Let \(B(x,r)\) denote the open ball of centre \(x\) and radius \(r\) in \(\mathbb{R}^{n}\). Also, let \(p:\mathbb{R}^{n}\rightarrow [ 1,\infty )\), let \(0\leq \nu \leq n\) and \(\phi :(0,\infty )\rightarrow (0,\infty )\). Then \(L^{p(\cdot ),\nu ,\phi }(\mathbb{R}^{n})\) denotes the space of all measurable functions \(f\) on \(\mathbb{R}^{n}\) such that, for some \(\lambda >0\), we have \(r^{-\nu }\phi (r)\int_{B(x,r)}\left| f(y)/\lambda \right|^{p(y)}dy\leq 1\) for all \(x\) and \(r\). This paper establishes continuity properties for logarithmic potentials of functions in \(L^{p(\cdot ),\nu,\phi }(\mathbb{R}^{n})\) for certain choices of \(p(\cdot )\), \(\nu \) and \(\phi \).