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On Extended Family of Fractional Sobolev Spaces Via Ultradistributions of Slow Growth

Anslem Amaonyeiro, Murphy E Egwe

Abstract

This paper considers a new version of fractional Sobolev spaces $\widetilde{\mathcal{W}}_{\mathcal{U}}^{β,p}(\mathbb{C}^{n})$ defined using the concept of tempered ultradistributions with respect to the spaces of ultradifferentiable functions $\mathcal{U}(\mathbb{C})$. The space $\widetilde{\mathcal{W}}_{\mathcal{U}}^{β,p}(\mathbb{C}^{n})$ is a natural generalization of the classical Sobolev space with integer order, where some additional conditions of growth control have been introduced. We analyze some possible definitions and their roles in the structure theory. We prove some density and compact embedding results, investigating the possibility of the extension domains. Some of the results we present here are extensions of the existing ones with some additional conditions. The construction of the new family of fractional Sobolev space is considered within the framework of Fourier transform of ultradistributions of slow growth.

On Extended Family of Fractional Sobolev Spaces Via Ultradistributions of Slow Growth

Abstract

This paper considers a new version of fractional Sobolev spaces defined using the concept of tempered ultradistributions with respect to the spaces of ultradifferentiable functions . The space is a natural generalization of the classical Sobolev space with integer order, where some additional conditions of growth control have been introduced. We analyze some possible definitions and their roles in the structure theory. We prove some density and compact embedding results, investigating the possibility of the extension domains. Some of the results we present here are extensions of the existing ones with some additional conditions. The construction of the new family of fractional Sobolev space is considered within the framework of Fourier transform of ultradistributions of slow growth.

Paper Structure

This paper contains 7 sections, 14 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Structural Notion of Fractional Sobolev Space
  4. Notion of Ultradistributions of Slow Growth
  5. Some Properties of Fractional Sobolev Spaces
  6. On Fractional Sobolev Spaces of Tempered Ultradistributions $\widetilde{\mathcal{W}}_{\mathcal{U}}^{\beta,p}(\mathbb{C})$
  7. Some Results on Density, Extension, and Embeddings in $\widetilde{\mathcal{W}}_{\mathcal{U}}^{\beta,p}(\mathbb{C})$

Key Result

Lemma 2.11

(2) Let $\mu\in \mathcal{W}^{\beta,p}(\mathbb{R}^{n})$. Then $\Vert\mu_{\varepsilon}-\mu\Vert_{\mathcal{W}^{\beta,p}(\Omega)}\longrightarrow 0$ as $\varepsilon$, where $\mu_{\varepsilon}(x)=(\mu \ast\xi_{\varepsilon})(y),\quad y\in\mathbb{R}^n$.

Theorems & Definitions (43)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 33 more