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Trace Principle for Riesz Potentials on Herz-Type Spaces and Applications

M. Ashraf Bhat, G. Sankara Raju Kosuru

Abstract

We establish trace inequalities for Riesz potentials on Herz-type spaces and discuss the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo-Nirenberg-Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces.

Trace Principle for Riesz Potentials on Herz-Type Spaces and Applications

Abstract

We establish trace inequalities for Riesz potentials on Herz-type spaces and discuss the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo-Nirenberg-Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces.
Paper Structure (5 sections, 11 theorems, 43 equations)

This paper contains 5 sections, 11 theorems, 43 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Function spaces
  3. Trace Theorems
  4. Optimality Conditions
  5. Sobolev Inequalities

Key Result

Theorem 1.1

Let $1<p_1 \leq q_1 < \infty$ and $1<p_2 \leq q_2< \infty$ satisfy $\frac{p_2}{q_2} \leq \frac{p_1}{q_1}$ for all $1<p_1<p_2<\infty$. Then, the inequality holds true if and only if the measure $\mu$ satisfies $\mu(B) \lesssim [m(B)]^{q_2\left(\frac{1}{q_1}-\frac{\gamma}{n}\right)}$ for every ball $B \subset \mathbb{R}^n$, given that $~n\left(\frac{1}{q_1}-\frac{1}{q_2}\right) \leq \gamma< \frac{n

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 12 more