Abstract:
In this article, we introduce the HK-Sobolev space $HK^{\alpha, \natural}_\zeta(\mathbf{G})$ over a Gelfand pair within the framework of a second countable hypergroup, employing the Fourier transform on the hypergroup. We discuss Kuelbs-Steadman space $KS^p$ in Hypergroup and prove that $KS^p(\mathbf{G})$ is a Banach algebra under a suitable convolution. Additionally, we also address the dominated convergence theorem in the $KS^p$ space over the hypergroup. Several Sobolev embedding-type results are discussed in the HK-Sobolev space $HK^{\alpha, \natural}_\zeta(\mathbf{G})$. Finally, we explore Rellich-Kondrashov theorem within this specific context.
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