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$θ$-Lebesgue spaces

Shouvik Datta Choudhury

Abstract

Traditional Lp spaces are fundamental in functional analysis, demarcated by the relationship $1/p + 1/q = 1$. This research pioneers the concept of $θ$-Lebesgue space, stemming from a simultaneous weakening of both the classical $L_p$ relation and its $θ$-variant, $1/(θ(p)) + 1/(θ(q)) = 1$. This conceptual shift addresses a gap in existing mathematical frameworks, aiming to create a space that encompasses a broader range of mathematical purpose. The primary objective is to rigorously demarcate the $θ$-Lebesgue space within this new context, explore its foundational properties, and articulate its theoretical significance in the realm of functional analysis.

$θ$-Lebesgue spaces

Abstract

Traditional Lp spaces are fundamental in functional analysis, demarcated by the relationship . This research pioneers the concept of -Lebesgue space, stemming from a simultaneous weakening of both the classical relation and its -variant, . This conceptual shift addresses a gap in existing mathematical frameworks, aiming to create a space that encompasses a broader range of mathematical purpose. The primary objective is to rigorously demarcate the -Lebesgue space within this new context, explore its foundational properties, and articulate its theoretical significance in the realm of functional analysis.
Paper Structure (5 sections, 24 theorems, 167 equations)

This paper contains 5 sections, 24 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Comparison
  3. $\theta$-Lebesgue Sequence Spaces
  4. Space of $\theta$-Bounded Sequences
  5. $\theta$-Hardy and $\theta$-Hilbert Inequalities

Key Result

Theorem 1

(Classical GT/factorization)[2]. Let $S, T$ be compact sets. For any bounded bilinear form $\omega: C(S) \times C(T) \rightarrow \mathbb{K}$ (here $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ ) there are probabilities $\lambda$ and $\mu$, respectively on $S$ and $T$, such that where $K$ is a numerical constant, the best value of which is denoted by $K_G$, more precisely by $K_G^{\mathbb{R}}$ or $K_G^{

Theorems & Definitions (53)

  • Theorem 1
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 2
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 43 more