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<p>The present paper studies the Dirichlet spaces in balls and the upper-half Euclidean spaces. As main results, we prove a number of identical forms of the Dirichlet semi-norms in the mentioned contexts as generalizations of the classical results proved by Douglas and Ahlfors with the 2-D disc case. With the scalar-valued phase derivative concept the semi-norm identical forms, in all cases, are extended to include the energy average of the instantaneous frequency. The latter has impacts to frequency analysis for multivariate functions.</p>

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Semi-norms of Higher Dimensional Dirichlet Spaces and Impacts to Frequency Analysis of Multivariate Functions

  • 期刊论文

  • 发表日期：

    2026年7月6日

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  • Yan Yang,
  • Tao Qian

摘要

The present paper studies the Dirichlet spaces in balls and the upper-half Euclidean spaces. As main results, we prove a number of identical forms of the Dirichlet semi-norms in the mentioned contexts as generalizations of the classical results proved by Douglas and Ahlfors with the 2-D disc case. With the scalar-valued phase derivative concept the semi-norm identical forms, in all cases, are extended to include the energy average of the instantaneous frequency. The latter has impacts to frequency analysis for multivariate functions.