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Some sharp bounds for Hardy type operators on mixed radial-angular type function spaces

Ronghui Liu, Yanqi Yang, Shuangping Tao

Abstract

In this paper, we are devoted to studying some sharp bounds for Hardy type operators on mixed radial-angular type function spaces. In addition, we will establish the sharp weak-type estimates for the fractional Hardy operator and its conjugate operator, respectively.

Some sharp bounds for Hardy type operators on mixed radial-angular type function spaces

Abstract

In this paper, we are devoted to studying some sharp bounds for Hardy type operators on mixed radial-angular type function spaces. In addition, we will establish the sharp weak-type estimates for the fractional Hardy operator and its conjugate operator, respectively.
Paper Structure (5 sections, 11 theorems, 198 equations)

This paper contains 5 sections, 11 theorems, 198 equations.

Table of Contents

  1. Some classical conclusions of Hardy type operators
  2. Some mixed radial-angular type function spaces
  3. Some sharp bounds of Hardy type operators on mixed radial-angular type function spaces
  4. Sharp weak-type estimates for the fractional Hardy operator
  5. Sharp Bounds of weighted Hardy-Littlewood averages

Key Result

Theorem 2.2

Let $1<p_i, q_i, \tilde{q}_{i}<\infty$, $n_i\in\mathbb N$, $x_i\in\mathbb R^{n_i}$, $r_i\in (0,\infty)$, $i=1,\ldots,m$. If $f\in L^{\vec{p}}_{rad}L^{\vec{\tilde{q}}}_{ang}(\mathbb R^{\vec{n}}, r^{\vec{\alpha}})$, where $r^{\vec{\alpha}}=r_1^{{\alpha}_1}\times\cdots \times r_2^{{\alpha}_m}$ and ${\a

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 25 more