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 \({\mathscr {H}}^{\alpha ,\eta }_{p,\lambda }({\mathbb {R}}^{n+1}_{+})\).

中文翻译：

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通过卷积运算符扩展Besov-Q类型空间及其应用

在本文中，我们使用常规小波来研究一类Besov-Q空间的扩展问题\（{\ dot {B}} ^ {\ alpha，\ lambda} _ {p，p}（{\ mathbb {R }} ^ {n}）\）。我们引入了通过\（\ psi \）生成的扩展运算符\（\ Pi _ {\ psi} \），并证明\（{\ dot {B}} ^ {\ alpha，\ lambda} _ {p，p }（{\ mathbb {R}} ^ {n}）\）可以扩展到函数空间\（{\ mathscr {C}} ^ {\ alpha} _ {p，\ lambda}（{\ mathbb {R} } ^ {n + 1} _ {+}）\）通过\（\ Pi _ {\ psi} \）。相反，受复制公式的启发，我们在分布的意义上构造了跟踪运算符\（\ pi _ {\ phi} \）。我们得到\（{\ mathscr {C}} ^ {\ alpha} _ {p，\ lambda}（{\ mathbb {R}} ^ {n +} _ {+}）\）也可以拉回以\（{\点{B}} ^ {\α，\拉姆达} _ {P，P}（{\ mathbb {R}} ^ {N}）\）的操作下\（ \ pi _ {\ phi} \）。作为应用程序，我们建立了某些谐波函数空间\（{\ mathscr {H}} ^ {\ alpha，\ eta} _ {p，\ lambda}（{\ mathbb {R}} ^ {n +1} _ {+}）\）。