Extension of Besov-Q Type Spaces via Convolution Operators with Applications

Abstract

In this paper, we use regular wavelets to investigate the extension problem of a class of Besov-Q spaces $${\dot{B}}^{\alpha ,\lambda }_{p,p}({\mathbb {R}}^{n})$$ . We introduce an extension operator $$\Pi _{\psi }$$ generated via $$\psi $$ , and prove that $${\dot{B}}^{\alpha ,\lambda }_{p,p}({\mathbb {R}}^{n})$$ can be extended to function spaces $${\mathscr {C}}^{\alpha }_{p,\lambda }({\mathbb {R}}^{n+1}_{+})$$ via $$\Pi _{\psi }$$ . Conversely, inspired by the reproducing formula, we construct a trace operator $$\pi _{\phi }$$ in the sense of distributions. We obtain that $${\mathscr {C}}^{\alpha }_{p,\lambda }({\mathbb {R}}^{n+}_{+})$$ can also be pulled back to $${\dot{B}}^{\alpha ,\lambda }_{p,p}({\mathbb {R}}^{n})$$ under the operation of $$\pi _{\phi }$$ . As an application, we establish a characterization of some kind of harmonic function spaces $${\mathscr {H}}^{\alpha ,\eta }_{p,\lambda }({\mathbb {R}}^{n+1}_{+})$$ .