  We establish trace inequalities for Riesz potentials on Herz-type spaces and
discuss the optimality of conditions imposed on specific parameters. We also
present some applications in the form of Sobolev-type inequalities, including
the Gagliardo-Nirenberg-Sobolev inequality and the fractional integration
theorem in the Herz space setting. In addition, we obtain a Sobolev embedding
theorem for Herz-type Sobolev spaces.
