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Some capacitary strong type inequalities and related function spaces

Keng Hao Ooi, Nguyen Cong Phuc

TL;DR

The paper settles Adams's capacitary strong type inequality for all $\alpha\in(0,n)$ and $q\in[1,\infty)$, and develops a comprehensive framework linking capacitary quantities with function spaces via Köthe duality. It introduces and analyzes Kalton–Verbitsky and Sobolev-multiplier-type spaces, establishing isomorphisms between $\dot{KV}_q$, $\dot{\widetilde{O}}^{\alpha,s}_{q}$, and related $N$-spaces, and showing dualities with $\dot{M}^{\alpha,s}_{p,r}$ spaces. The results yield boundedness of Hardy–Littlewood and Calderón–Zygmund operators on the dual spaces and provide explicit norm representations via Wolff potentials and capacitary integrals, unifying capacitary, potential-theoretic, and Banach-function-space approaches. The inhomogeneous setting with Bessel potentials $H^{\alpha,s}$ and Cap$_{\alpha,s}$ extends the framework, ensuring analogous triplets and dualities. Overall, the work advances the understanding of capacitary inequalities and their implications for modern harmonic analysis and potential theory.

Abstract

We verify a conjecture of D. R. Adams on a capacitary strong type inequality that generalizes the classical capacitary strong type inequality of V. G. Maz'ya. As a result, we characterize related function spaces as Köthe duals to a class of Sobolev multiplier type spaces. Moreover, using tools from nonlinear potential theory, weighted norm inequalities, and Banach function space theory, we show that these spaces are also isomorphic to more concrete spaces that are easy to use and fit in well with the modern theory of function spaces of harmonic analysis.

Some capacitary strong type inequalities and related function spaces

TL;DR

The paper settles Adams's capacitary strong type inequality for all and , and develops a comprehensive framework linking capacitary quantities with function spaces via Köthe duality. It introduces and analyzes Kalton–Verbitsky and Sobolev-multiplier-type spaces, establishing isomorphisms between , , and related -spaces, and showing dualities with spaces. The results yield boundedness of Hardy–Littlewood and Calderón–Zygmund operators on the dual spaces and provide explicit norm representations via Wolff potentials and capacitary integrals, unifying capacitary, potential-theoretic, and Banach-function-space approaches. The inhomogeneous setting with Bessel potentials and Cap extends the framework, ensuring analogous triplets and dualities. Overall, the work advances the understanding of capacitary inequalities and their implications for modern harmonic analysis and potential theory.

Abstract

We verify a conjecture of D. R. Adams on a capacitary strong type inequality that generalizes the classical capacitary strong type inequality of V. G. Maz'ya. As a result, we characterize related function spaces as Köthe duals to a class of Sobolev multiplier type spaces. Moreover, using tools from nonlinear potential theory, weighted norm inequalities, and Banach function space theory, we show that these spaces are also isomorphic to more concrete spaces that are easy to use and fit in well with the modern theory of function spaces of harmonic analysis.

Paper Structure

This paper contains 12 sections, 28 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Banach function spaces
  3. Proof of Theorem \ref{['Main1']}
  4. Proof of Theorem \ref{['Main2']}
  5. Proof of Theorem \ref{['KV-q']}
  6. The spaces $\dot{M}^{\alpha,s}_{p, r}$ and $(\dot{M}^{\alpha,s}_{p, r})'$
  7. The spaces $\dot{N}^{\alpha,s}_{p', s/r}$ and $(\dot{N}^{\alpha,s}_{p', s/r})'$ and Proof of Theorem \ref{['Main3']}
  8. Proof of Theorem \ref{['Newnorm2']}
  9. The space $\dot{KV}_q$ and the triplet $\overline{C_{c}(\mathbb{R}^{n})}^{\dot{M}_{p,r}^{\alpha,s}}\text{--}(\dot{M}_{p,r}^{\alpha,s})'\text{--}\dot{M}_{p,r}^{\alpha,s}$
  10. The space $\dot{KV}_q$
  11. The triplet $\overline{C_{c}(\mathbb{R}^{n})}^{\dot{M}_{p,r}^{\alpha,s}}\text{--}(\dot{M}_{p,r}^{\alpha,s})'\text{--}\dot{M}_{p,r}^{\alpha,s}$
  12. The inhomogeneous case

Key Result

Theorem 1.1

Let $s>1$, $0<\alpha<\frac{n}{s}$, and $q\in [1,\infty)$. There exists a constant $A>0$ such that capstrong2 holds for all nonnegative functions $f\in L^\infty(\mathbb{R}^n)$ with compact support, or more generally for all nonnegative measurable functions $f$ such that $I_\alpha f \in L^q( {\rm cap}

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • ...and 42 more