Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces

Abstract

Our aim is to establish Sobolev type inequalities for fractional maximal functions ${M_{\mathbb{H},\nu }}f$ and Riesz potentials ${I_{\mathbb{H},\alpha}}f$ in weighted Morrey spaces of variable exponent on the half space $\mathbb{H}$. We also obtain Sobolev type inequalities for a C¹ function on $\mathbb{H}$. As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents Φ(x, t) = t^p(x) + (b(x)t)^q(x), where p(·) and q(·) satisfy log-Hölder conditions, p(x) > q(x) for $x \in \mathbb{H}$, and b(·) is nonnegative and Hölder continuous of order θ ∈ (0,1].