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Study of Composition Operators in Certain Functional Spaces

Mahdi Tahar Brahimi

TL;DR

This work systematically studies composition operators across several functional-analytic spaces, focusing on BV_p^1(I) and (homogeneous) Besov spaces, with two core aims: to characterize when superposition operators map spaces into themselves and to extend Peetre-type results to broader scales. The approach blends Littlewood–Paley theory, finite-difference norms, and distributional realizations to obtain norm-inequalities, derivative formulas for nested compositions, and functional-calculus results. The main contributions include a generalization of a basic inequality for the derivative of compositions (Theorem 2.15) accompanied by a constructive algorithm for multiple compositions (Lemma 2.14), and a Peetre-type extension to BV_p^α spaces that clarifies when $f\circ g$ preserves Besov/BV regularity. Together with a parallel exploration in homogeneous Besov spaces and their realizations (including Peetre’s framework), the results provide a comprehensive view of how nonlinear composition interacts with refined function spaces, enabling precise control of norm growth and regularity. The findings have implications for nonlinear analysis and PDEs where composition operators arise naturally, offering tools for functional calculus and regularity theory in non-smooth settings.

Abstract

In this thesis we study three problems. The first is the superposition of the operators and their proprities, such as boundedness,continuity,regularity and the inequalities of the norms of the composition of functions in some functional spaces. The second is to generalize some results of the composition of more than two functions, and the third is to give a generalization of Peetre's theorem.

Study of Composition Operators in Certain Functional Spaces

TL;DR

This work systematically studies composition operators across several functional-analytic spaces, focusing on BV_p^1(I) and (homogeneous) Besov spaces, with two core aims: to characterize when superposition operators map spaces into themselves and to extend Peetre-type results to broader scales. The approach blends Littlewood–Paley theory, finite-difference norms, and distributional realizations to obtain norm-inequalities, derivative formulas for nested compositions, and functional-calculus results. The main contributions include a generalization of a basic inequality for the derivative of compositions (Theorem 2.15) accompanied by a constructive algorithm for multiple compositions (Lemma 2.14), and a Peetre-type extension to BV_p^α spaces that clarifies when preserves Besov/BV regularity. Together with a parallel exploration in homogeneous Besov spaces and their realizations (including Peetre’s framework), the results provide a comprehensive view of how nonlinear composition interacts with refined function spaces, enabling precise control of norm growth and regularity. The findings have implications for nonlinear analysis and PDEs where composition operators arise naturally, offering tools for functional calculus and regularity theory in non-smooth settings.

Abstract

In this thesis we study three problems. The first is the superposition of the operators and their proprities, such as boundedness,continuity,regularity and the inequalities of the norms of the composition of functions in some functional spaces. The second is to generalize some results of the composition of more than two functions, and the third is to give a generalization of Peetre's theorem.
Paper Structure (40 sections, 46 theorems, 150 equations)

This paper contains 40 sections, 46 theorems, 150 equations.

Table of Contents

  1. Some preliminary results
  2. Littlewood-Paley series
  3. Partition of unity
  4. Decomposition of a function $f \in \mathcal{S}^{\prime }(\mathbb{R}^n)$
  5. Decomposition of $\Delta_k(f \cdot g)$
  6. Finite difference operators $\Delta_h^m$
  7. Norms in Besov spaces
  8. By Littlewood-Paley theory
  9. By finite differences
  10. Functions of bounded p-variation
  11. General notions
  12. Functions of bounded p-variation as distributions
  13. Composition of Operators in $BV_p^1(I)$
  14. Reminder
  15. Properties of the spaces $BV_p^1(I)$
  16. ...and 25 more sections

Key Result

Lemma 1.1.3

There exist $\varphi, \psi \in C_0^\infty(\mathbb{R}^n)$, with $\text{supp}(\psi) \subset C_{-1}$, $\text{supp}(\varphi) \subset C_0$, such that

Theorems & Definitions (75)

  • Definition 1.1.1
  • Example 1.1.2
  • Lemma 1.1.3
  • Definition 1.2.1
  • Lemma 1.2.2
  • Definition 1.2.3
  • Remark 1.2.4
  • Definition 1.3.1
  • Proposition 1.3.2: ULLR
  • Definition 1.3.3
  • ...and 65 more