Fractional Differential Couples by Sharp Inequalities and Duality Equations

TL;DR

This paper investigates sharp inequalities and duality equations related to fractional derivatives and gradients, providing new insights into fractional advection-dispersion equations and their solutions.

Contribution

It establishes sharp Hardy-Rellich, Adams-Moser, and Morrey-Sobolev inequalities for fractional derivatives and analyzes distributional solutions of duality equations involving these operators.

Findings

Derived sharp inequalities for fractional derivatives and gradients.

Analyzed distributional solutions of duality equations with Radon measures and Morrey functions.

Provided theoretical foundations for fractional advection-dispersion equations.

Abstract

This paper presents a highly non-trivial two-fold study of the fractional differential couples - derivatives ($\nabla^{0<s<1}_+=(-\Delta)^\frac{s}{2}$) and gradients ($\nabla^{0<s<1}_-=\nabla (-\Delta)^\frac{s-1}{2}$) of basic importance in the theory of fractional advection-dispersion equations: one is to discover the sharp Hardy-Rellich ($sp<p<n$) $|$ Adams-Moser ($sp=n$) $|$ Morrey-Sobolev ($sp>n$) inequalities for $\nabla^{0<s<1}_\pm$; the other is to handle the distributional solutions $u$ of the duality equations $[\nabla^{0<s<1}_\pm]^\ast u=\mu$ (a nonnegative Radon measure) and $[\nabla^{0<s<1}_\pm]^\ast u=f$ (a Morrey function).

Equations

$$\nabla =({\partial }_{x_1},...,{\partial }_{x_n})$$

$$\Delta =\sum _{j=1}^n{\partial }_{x_j}^2$$

$${\nabla }^m=\{\begin{array}{ll}(-1{)}^{\frac{m}{2}}(-\Delta {)}^{\frac{m}{2}}& \text{for}m\text{even}\\ (-1{)}^{\frac{m-1}{2}}\nabla (-\Delta {)}^{\frac{m-1}{2}}& \text{for}m\text{odd}\end{array}\&0<m<n,$$

$$\int_{\Omega}\exp\Bigg{(}\frac{\beta|u(x)|}{\|\nabla^{m}u\|_{L^{\frac{n}{m}}}}\Bigg{)}^{\frac{n}{n-m}}\,\frac{dx}{|\Omega|}\leq c_{0,m,n}\ \ \forall\ \ u\in C^{m}_{c}(\Omega)$$

$$0\le \beta \le {\beta }_{0,m,n}=\{\begin{array}{ll}{\left(\frac{n}{{\omega }_{n-1}}\right)}^{\frac{n-m}{n}}\frac{{\pi }^{\frac{n}{2}}{2}^m\Gamma (\frac{m}{2})}{\Gamma (\frac{n-m}{2})}& \text{for}m\text{even}\\ {\left(\frac{n}{{\omega }_{n-1}}\right)}^{\frac{n-m}{n}}\frac{{\pi }^{\frac{n}{2}}{2}^m\Gamma (\frac{m+1}{2})}{\Gamma (\frac{n+1-m}{2})}& \text{for}m\text{odd}\end{array}\&0<m<n;$$

$$|u(x)|\leq\frac{\Big{(}\frac{n}{\omega_{n-1}}\Big{)}^{\frac{n-m}{n}}}{\beta_{0,m,n}}\int_{{{{\mathbb{R}}}^{n}}}{|y-x|^{m-n}|\nabla^{m}u(y)|}\,dy\ \ \ \ \forall\ \ u\in C_{c}^{\infty}.$$

$$\Vert {\nabla }^{m}u{\Vert }_{L^{1<p<\frac{n}{m}}}\text{or}\Vert {\nabla }^{m}u{\Vert }_{L^{\infty >p>\frac{n}{m}}}.$$

$$\left(\int_{{{{\mathbb{R}}}^{n}}}\bigg{(}\frac{|u(x)|}{|x|^{m}}\bigg{)}^{p}\,dx\right)^{\frac{1}{p}}\leq c_{mp<n}\|\nabla^{m}u\|_{L^{p}}\ \ \forall\ \ u\in C^{\infty}_{c},$$

$$c_{mp<n}=\begin{cases}\frac{2^{-m}\Gamma\big{(}\frac{n}{2p}-\frac{m}{2}\big{)}\Gamma\big{(}\frac{n(p-1)}{2p}\big{)}}{\Gamma\big{(}\frac{n(p-1)}{2p}+\frac{m}{2}\big{)}\Gamma\big{(}\frac{n}{2p}\big{)}}\ \ &\text{for}\ m\ \text{even}\\ \Big{(}\frac{2^{1-m}p}{n-p}\Big{)}\left(\frac{\Gamma(\frac{n}{2p}-\frac{m}{2})\Gamma(\frac{n(p-1)}{2p}+\frac{1}{2})}{\Gamma(\frac{n(p-1)}{2p}+\frac{m}{2})\Gamma(\frac{n}{2p}-\frac{1}{2})}\right)\ \ &\text{for}\ m\ \text{odd}\end{cases}\ \ \&\ \ 0<m<n.$$

$$\Vert u{\Vert }_{L^{\infty }}\le \frac{{\left(\frac{n(p-1)}{mp-n}\right)}^{\frac{p-1}{p}}}{{\beta }_{0,m,n}}|\Omega {|}^{\frac{mp-n}{pn}}\Vert {\nabla }^{m}u{\Vert }_{L^p}\forall u\in C_c^m(\Omega ).$$

$${\partial }_{t}u(x,t)$$

$$\displaystyle=-\vec{v}\cdot\nabla\rho(x,t)+\kappa\text{div}^{s}\big{(}\nabla u(x,t)\big{)}$$

$$\displaystyle=-\vec{v}\cdot\nabla u(x,t)+\kappa\text{div}\big{(}\nabla^{s}_{-}u(x,t)\big{)}$$

$$\{{\nabla }_{+}^{0<s<1},{\nabla }_{-}^{0<s<1}\}\text{corresponding naturally to}\{{\nabla }^{m=\text{even}},{\nabla }^{m=\text{odd}}\},$$

$${\nabla }_{\pm }^{0<s<1}\&[{\nabla }_{\pm }^{0<s<1}{]}^{\ast }$$

$$H^{0<s<1,1<p<\infty }\&H_{\pm }^{0<s<1,1<p<\infty }$$

$$[{\nabla }_{\pm }^{0<s<1}{]}^{\ast }u=\mu$$

$$[{\nabla }_{\pm }^{0<s<1}{]}^{\ast }u=f$$

$$\Vert u{\Vert }_{H^p}=\Vert u{\Vert }_{L^p}+\Vert \vec{R}u{\Vert }_{L^p}<\infty ,$$

$$\vec{R}u=(R_{1}u,\dots ,R_{n}u)\&R_{j}u(x)=\left(\frac{\Gamma (\frac{n+1}{2})}{{\pi }^{\frac{n+1}{2}}}\right)\text{p.v.}{\int }_{{\mathbb{R}}^n}\frac{x_j-y_j}{|x-y{|}^{n+1}}u(y)dy\text{a.e.}x\in {\mathbb{R}}^n.$$

$$\vec{f}=(f_1,\dots ,f_n)$$

$$\|{\vec{f}}\|_{L^{p}}={\big{\|}|\vec{f}|\big{\|}_{L^{p}}\approx}\sum_{j=1}^{n}\|f_{j}\|_{L^{p}}.$$

$$I_{s}u(x)=\left(\frac{\Gamma (\frac{n-s}{2})}{{\pi }^{\frac{n}{2}}{2}^s\Gamma (\frac{s}{2})}\right){\int }_{{\mathbb{R}}^n}|x-y{|}^{s-n}u(y)dy\text{a.e.}x\in {\mathbb{R}}^n.$$

$$0<s<1\le p<\frac{n}{s}$$

$$\left(\int_{{{{\mathbb{R}}}^{n}}}\big{(}|x|^{-s}|I_{s}u(x)|\big{)}^{p}\,dx\right)^{\frac{1}{p}}\lesssim\|u\|_{L^{p}}+\|\vec{R}u\|_{L^{p}}{}{\approx\|u\|_{H^{p}}}\qquad\ \forall\ \ u\in H^{p}.$$

$$(-\Delta {)}^{\frac{s}{2}}u(x)=\{\begin{array}{ll}{I}_{-s}u(x)=c_{n,s}{\int }_{{\mathbb{R}}^n}\frac{u(x+y)}{|y{|}^{n+s}}dy& \text{as}s\in (-1,0)\\ u(x)& \text{as}s=0\\ c_{n,s,+}\text{p.v.}{\int }_{{\mathbb{R}}^n}\frac{u(x+y)-u(x)}{|y{|}^{n+s}}dy& \text{as}s\in (0,1)\end{array}$$

$$\displaystyle\nabla^{s}u(x)=\Bigg{(}\frac{\partial^{s}u}{\partial x_{j}^{s}}\Bigg{)}_{j=1}^{n}=\vec{R}(-\Delta)^{\frac{s}{2}}u(x)=c_{n,s,-}\int_{\mathbb{R}^{n}}\frac{y\big{(}u(x)-u(x-y)\big{)}}{|y|^{n+1+s}}\,dy,$$

$$\begin{cases}c_{n,s}=\frac{\Gamma(\frac{n-s}{2})}{\pi^{\frac{n}{2}}2^{s}\Gamma(\frac{s}{2})}\\ c_{n,s,+}=\frac{s2^{s-1}\Gamma\big{(}\frac{n+s}{2}\big{)}}{\pi^{\frac{n}{2}}\Gamma\big{(}1-\frac{s}{2}\big{)}}\\ c_{n,s,-}=\frac{2^{s}\Gamma\big{(}\frac{n+s+1}{2}\big{)}}{\pi^{\frac{n}{2}}\Gamma\big{(}\frac{1-s}{2}\big{)}}.\end{cases}$$

$$\begin{cases}(-\Delta)^{\frac{s}{2}}u(x)=c_{+}\bigg{(}\frac{d^{s}}{dx^{s}_{+}}+\frac{d^{s}}{dx^{s}_{-}}\bigg{)}u(x)\\ \nabla^{s}u(x)=c_{-}\bigg{(}\frac{d^{s}}{dx^{s}_{+}}-\frac{d^{s}}{dx^{s}_{-}}\bigg{)}u(x)\\ \frac{d^{s}}{dx^{s}_{\pm}}u(x)=\frac{s}{\Gamma(1-s)}\int_{\pm\infty}^{0}\frac{t(u(x+t)-u(x))}{|t|^{2+s}}\,dt.\end{cases}$$

$${\nabla }_{+}^{s}u=(-\Delta {)}^{\frac{s}{2}}u\&{\nabla }_{-}^{s}u={\nabla }^{s}u=\vec{R}(-\Delta {)}^{\frac{s}{2}}u.$$

$$\text{id}=-\sum _{j=1}^n{R}_j^2=-\vec{R}\cdot \vec{R}.$$

Full text

LL was supported by the National Natural Science Foundation of China (# 11771446); JX was supported by NSERC of Canada (# 202979463102000).

1. 1 Introduction

2. 2 Fractional differential couples $\nabla^{0<s<1}_{\pm}$ and their dualities $[\nabla^{0<s<1}_{\pm}]^{\ast}$

3. 2.1 Fractional differential couples $\nabla^{0<s<1}_{\pm}$

4. 2.2 Dual fractional differential couples $[\nabla^{s}_{\pm}]^{\ast}$

5. 3 Sharp fractional differential-integral inequalities

6. 3.1 Optimal control for Riesz’s operator $\mathrm{I}_{0<\alpha<n}$

7. 3.2 Optimal domination for $\nabla^{0<s<1}_{\pm}$

8. 4 Fractional Hardy-Sobolev spaces and their dualities

9. 4.1 Fractional Hardy-Sobolev spaces $H^{s,p}$ and $H^{s,p}_{\pm}$

10. 4.2 Dual Hardy-Sobolev spaces $[H^{s,p}]^{\ast}$ and $[H^{s,p}_{\pm}]^{\ast}$

11. 5 Distributional solutions of duality equations

12. 5.1 Distributional solutions to $[\nabla^{s}_{\pm}]^{\ast}u=\mu$

13. 5.2 Morrey’s regularity for distributional solutions of $[\nabla^{s}_{\pm}]^{\ast}u=f$

1. Introduction

In his celebrated 1988 paper [3], Adams extends the Moser inequality in [23] from the first order to the higher order gradients in the Euclidean space ${\mathbb{R}}^{n\geq 2}$ - given the gradient

[TABLE]

and the Laplacian

[TABLE]

as well as

[TABLE]

there is a constant $c_{0,m,n}$ such that

[TABLE]

holds, where:

• $\rhd$

[TABLE]

• $\rhd$

$\Omega$ is a subdomain of $\mathbb{R}^{n}$ with finite $n$-measure $|\Omega|$ and its associate space $C_{c}^{m}(\Omega)$ stands for all $C^{m}$-functions supported in $\Omega$;

• $\rhd$

$\Gamma(\cdot)$ is the standard gamma function and induces ${\omega_{n-1}=}\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}$ - the area of the unit sphere $\mathbb{S}^{n-1}$ of ${{{\mathbb{R}}}^{n}}$;

• $\rhd$

(1.1) is established through the Adams-Riesz potential inequality (just under [3, (23)])

[TABLE]

Moreover, if $\beta>\beta_{0,m,n}$ then there is $u\in C^{m}_{c}(\Omega)$ such that the integral in (1.1) can be made as large as desired - in other words - $\beta_{0,m,n}$ is sharp.

Upon examining $\|\nabla^{m}u\|_{L^{\frac{n}{m}}}$ in (1.1), we are automatically suggested to consider a variant of (1.1) for

[TABLE]

• $\rhd$

For the former, we use the $m$-form of [6, Corollary 1 & Theorem 4 (16)] to derive the sharp $m$-order Hardy-Rellich inequality

[TABLE]

where

[TABLE]

Of course, the case $m=1$ of (1.3) is the classical sharp Hardy inequality (cf. [14]).

• $\rhd$

For the latter, we use the $m$-form of Theorem 3.1(iii) (viewed as a sharp Morrey-Riesz inequality) and (1.2) to discover the sharp $m$-order Morrey-Sobolev inequality

[TABLE]

In particular, the case $m=1$ of (1.4) is the classical sharp Morrey-Sobolev inequality (cf. [38, Theorem 2.E.]).

Clearly, (1.1), (1.3) and (1.4) give a complete structure on utilizing the higher derivatives and gradients to sharply dominate the size of a derivative/gradient-free function. However, upon recognizing the fractional vector calculus considerably used in both Herbst’s study of the Klein-Gordon equation for a Coulomb potential [15] and Meerschaert-Mortensen-Wheatcraft’s investigation of the particle mass density $u(x,t)$ of a contaminant in some fluid at a point $x\in\mathbb{R}^{n}$ at time $t>0$ which solves the fractional advection-dispersion equation (with a constant average velocity $\vec{v}$ of contaminant particles and a positive constant $\kappa$)

[TABLE]

combining a fractional Fick’s law for flux with a classic mass balance - and reversely- a fractional mass balance with a classic Fickian flux [21], in the forthcoming sections we are driven to work out versions of (1.1), (1.3) and (1.4) for the fractional differential couples - derivatives and gradients:

[TABLE]

and their essential applications in the study of the distributional solutions to some fractional partial differential equations of dual character. More precisely,

• $\rhd$

§2 collects some fundamental facts on

[TABLE]

through the Stein-Weiss-Hardy inequalities and the Fefferman-Stein type decompositions (cf. [10, 7, 20]).

• $\rhd$

§3 utilizes Theorem 3.1 - an sharp embedding principle for the Riesz potentials to discover the fractional extensions of (1.1), (1.3) and (1.4) - Theorem 3.2.

• $\rhd$

§4 discusses the fractional Hardy-Sobolev spaces

[TABLE]

and their dualities generated by $\nabla^{0<s<1}_{\pm}$ - Theorems 4.1-4.2.

• $\rhd$

§5 studies the distributional solutions of the duality equations

[TABLE]

for a nonnegative Radon measure $\mu$ and their absolutely continuous forms

[TABLE]

under the hypothesis that $f$ is in the Morrey space $\mathrm{L}^{1\leq p<{\kappa}/{s},0<\kappa\leq n}$ (cf. [1])

• Theorems 5.1-5.2.

Notation. In what follows, $U\lesssim V$ (resp.  $U\gtrsim V$) means $U\leq cV$ (resp. $U\geq cV$) for a positive constant $c$ and $U\approx V$ amounts to $U\gtrsim V\gtrsim U$.

2. Fractional differential couples $\nabla^{0<s<1}_{\pm}$ and their dualities $[\nabla^{0<s<1}_{\pm}]^{\ast}$

2.1. Fractional differential couples $\nabla^{0<s<1}_{\pm}$

For $(n,p)\in\mathbb{N}\times[1,\infty)$ let $H^{p}$ be the real Hardy space of all functions $u$ in the Lebesgue space $L^{p}$ on the Euclidean space ${{{\mathbb{R}}}^{n}}$ with

[TABLE]

where $\vec{R}={(R_{1},\dots,R_{n})}$ is the vector-valued Riesz transform on ${{{\mathbb{R}}}^{n}}$, with

[TABLE]

Also, for a vector-valued function

[TABLE]

let

[TABLE]

Note that $H^{p}$ coincides with the classical Lebesgue space $L^{p}$ whenever $p\in(1,\infty)$ and the $(0,1)\ni s$-th order Riesz singular integral operator $I_{s}$ acting on a suitable function $u$ is defined by

[TABLE]

We refer the reader to Stein’s seminal texts [35, 36] for more about these basic notions. The Stein-Weiss-Hardy inequality (cf. [37] for $p>1$ and (5.5) in §5 for $p=1$) states that under

[TABLE]

we have

[TABLE]

Let $C_{c}^{\infty}$ be the collection of all infinitely differentiable functions compactly supported in ${{{\mathbb{R}}}^{n}}$. Note that $C_{c}^{\infty}\cap H^{p}$ is dense in $H^{p}$ for any $p\in[1,\infty)$. For any $u\in C_{c}^{\infty}$ let

[TABLE]

and

[TABLE]

where (cf. [8, Definition 1.1, Lemma 1.4] for $c_{n,s,+}$ and [20] for $c_{n,s,-}$)

[TABLE]

Especially, if $0<s<n=1$ then there are two $s$-dependent constants $c_{\pm}$ to make the following Liouville fractional derivative formulae (cf. [31]):

[TABLE]

Hence it is natural and reasonable to adopt the notations

[TABLE]

The operators $\nabla^{s}_{+}$ and $\nabla^{s}_{-}$ can be viewed as the fractional derivative and the fractional gradient due to

[TABLE]

Accordingly, for any $s\in(0,1)$, the Stein-Weiss-Hardy inequality (2.1) (cf. [28]) amounts to

[TABLE]

Here it is worth pointing out the following fundamentals:

• $\rhd$

If $0<s<1<p<\frac{n}{s}$, then the right-hand-side of (2.5) can be replaced by $\|\nabla^{s}_{\pm}u\|_{L^{p}}$. More precisely, on the one hand, the boundedness of $\vec{R}$ on $L^{p>1}$ and (2.5) give (cf. [30, Lemma 2.4])

[TABLE]

One the other hand, [30, Theorems 1.8-1.9] derives

[TABLE]

• $\rhd$

If $0<s<p=1\leq n$, then according to Spector’s [34, Theorem 1.4] the right-hand-side of (2.5) except $n=1$ (cf. (5.6)) can be replaced by $\|\nabla^{s}_{-}u\|_{L^{1}}$ - i.e. -

[TABLE]

which may be viewed as a rough extension of Shieh-Spector’s [31, Theorem 1.2] and the classic sharp Hardy’s inequality (cf. [11]) under $n\geq 2$:

[TABLE]

However, the right-hand-side of (2.5) cannot be replaced by $\|\nabla^{s}_{+}u\|_{L^{1}}$ (cf. [35, p.119], [28, Section 3.3] & [31, Section 1.1]).

2.2. Dual fractional differential couples $[\nabla^{s}_{\pm}]^{\ast}$

Suppose that $C^{\infty}$ is the space of all infinitely differentiable functions on ${{{\mathbb{R}}}^{n}}$. Denote by ${\mathcal{S}}$ the Schwartz class on ${{{\mathbb{R}}}^{n}}$ consisting of all functions $f$ in $C^{\infty}$ such that

[TABLE]

Also, write ${\mathcal{S}}^{\prime}$ for the Schwartz tempered distribution space - the dual of ${\mathcal{S}}$ endowed with the weak-$\ast$ topology. According to [32, 20], given $s\in(0,1)$, if we let

[TABLE]

and ${\mathcal{S}}_{s}^{\prime}$ be the dual space of ${\mathcal{S}}_{s}$ (i. e., the space of all continuous linear functionals on ${\mathcal{S}}_{s}$), then for any

[TABLE]

we can define below $\nabla^{s}_{\pm}u$ as a distribution in ${\mathcal{S}}^{\prime}$:

[TABLE]

where the action of $\nabla^{s}_{\pm}$ on any function $\phi\in{\mathcal{S}}$ is determined by the Fourier transform

[TABLE]

according to

[TABLE]

If $\phi\in C_{c}^{\infty}$, then (2.7) goes back to (2.2)-(2.3)-(2.4) (cf. [32, 8, 20]). Moreover, the above equalities in (2.6) are well defined because $\nabla^{s}_{+}$ and $\nabla^{s}_{j}$ send ${\mathcal{S}}$ to ${\mathcal{S}}_{s}$ (cf. [32, 8] for $\nabla^{s}_{+}$ and [20, Lemma 2.6] for $\nabla^{s}_{j}$).

Based on the foregoing discussion, we may describe the dual/adjoint operators of $\nabla^{s}_{\pm}$ and one of their most important consequences.

• $\rhd$

The adjoint operator \big{[}(-\Delta)^{\frac{s}{2}}\big{]}^{\ast} of $(-\Delta)^{\frac{s}{2}}$ is itself, namely,

[TABLE]

which can be understood in the sense of

[TABLE]

This is reasonable, because for nice function pair $(f,\phi)\in(C_{c}^{\infty})^{2}$ we have (cf. [33])

[TABLE]

and

[TABLE]

• $\rhd$

Upon setting

[TABLE]

then $-\text{div}^{s}$ exists as the adjoint operator \big{[}\nabla^{s}_{-}\big{]}^{\ast} of $\nabla^{s}_{-}$ - in short -

[TABLE]

Note that (cf. [30, Theorem 1.3])

[TABLE]

and (cf. [9, Lemma 2.5])

[TABLE]

• $\rhd$

Recall that $\mathrm{BMO}$ stands for the John-Nirenberg class of all locally integrable functions $f$ on ${{{\mathbb{R}}}^{n}}$ with bounded mean oscillation (cf. [16])

[TABLE]

where the supremum is taken over all Euclidean balls $B\subseteq{{{\mathbb{R}}}^{n}}$ with

[TABLE]

Of remarkable interest is that the Fefferman-Stein decomposition (cf. [10, 39])

[TABLE]

can be written as the following form (cf. [20, Theorem 4.4])

[TABLE]

where

[TABLE]

Note that if $W^{1,n}$ stands for the Sobolev space of all locally integrable functions $f$ with $\|\nabla f\|_{L^{n}}<\infty$ then there are (cf. [20, Theorem 4.4])

[TABLE]

and (cf. [7, theorem 1])

[TABLE]

So, I_{s}\big{(}[\mathring{H}^{s,1}_{-}]^{\ast}\big{)} exists as a solution to the Bourgain-Brazis question (cf. [7, p.396]) - What are the function spaces $X$, $W^{1,n}\subseteq X\subseteq{\mathop{\mathrm{missing}}{\,BMO\,}}$, such that every $F\in X$ has a decomposition $F=\sum_{j=1}^{n}R_{j}Y_{j}$ where $Y_{j}\in L^{\infty}$?.

3. Sharp fractional differential-integral inequalities

3.1. Optimal control for Riesz’s operator $\mathrm{I}_{0<\alpha<n}$

The following is of independent interest.

Theorem 3.1**.**

Let

[TABLE]

Then the following assertions are true.

• (i)

If $\alpha p<{n}$, then

[TABLE]

• (ii)

If $\alpha p=n$, $\Omega\subseteq{{{\mathbb{R}}}^{n}}$ is a domain with volume $|\Omega|<\infty$ and $L^{p}_{c}(\Omega)$ stands for the class of all $f\in L^{p}$ with support contained in $\Omega$, then there is a constant $c_{\alpha p=n}$ depending only on $\alpha$ and $n$ such that

[TABLE]

Here $\frac{n}{\omega_{n-1}}$ is sharp in the sense that if $\Omega$ is a Euclidean ball and $\beta>\frac{n}{\omega_{n-1}}$ then the last integral inequality cannot hold without forcing $c_{\alpha p=n}$ to depend only on $\alpha$ and $n$.

• (iii)

If $\alpha p>{n}$ and $\Omega\subseteq{{{\mathbb{R}}}^{n}}$ is a domain with volume $|\Omega|<\infty$, then

[TABLE]

Moreover, the constant $c_{\alpha p>n}$ is sharp in the sense that if $\Omega$ is a Euclidean ball then

[TABLE]

Proof.

(i) This is regarded as the sharp Stein-Weiss-Hardy inequality. The sharp constant $c_{\alpha p<n}$ is obtained in Herbst [15]; see also [6, 27, 13] for more information.

(ii) This is just the sharp Adams inequality in [3, Theorem 2] whose argument is still valid for $n=1$ and $\frac{\omega_{n-1}}{n}=2$.

(iii) This is totally brand-new. In the sequel let $p^{\prime}=\frac{p}{p-1}.$ For any $f\in L^{p}$ supported on $\Omega$ and for any $x\in{{{\mathbb{R}}}^{n}}$, we utilize the Hölder inequality to derive that

[TABLE]

Note that the Fubini theorem and $(\alpha-n)p^{\prime}+n>0$ imply

[TABLE]

Thus we arrive at the desired inequality

[TABLE]

To prove that

[TABLE]

is sharp, let us consider the case

[TABLE]

and the function

[TABLE]

where $\beta$ satisfies

[TABLE]

On the one hand, a direct calculation gives

[TABLE]

On the other hand, by the fact $\alpha+\beta>\alpha-\frac{n}{p}>0$, we get

[TABLE]

Combining the last two formulae gives

[TABLE]

Now the problem turns to calculate

[TABLE]

Consider the function

[TABLE]

Note that

[TABLE]

and

[TABLE]

So, this, combined with

[TABLE]

shows that $h$ attains its sharp value at the point

[TABLE]

Consequently,

[TABLE]

This in turn implies

[TABLE]

Accordingly, when $\Omega$ is a Euclidean ball of ${{{\mathbb{R}}}^{n}}$, it holds that

[TABLE]

∎

3.2. Optimal domination for $\nabla^{0<s<1}_{\pm}$

Interestingly and naturally, with

[TABLE]

replaced by the fractional version

[TABLE]

Theorem 3.1 induces the following new assertion.

Theorem 3.2**.**

Let $0<s<1<p<\infty$ and

[TABLE]

Then the following assertions are true.

• (i)

If $sp<p<n$, then

[TABLE]

• (ii)

If $sp=n$ and $\Omega\subseteq{{{\mathbb{R}}}^{n}}$ is a domain with volume $|\Omega|<\infty$, then exists a positive constant $c_{sp=n,\pm}$ depending only on $s$ and $n$ such that

[TABLE]

Here

[TABLE]

is sharp in the sense that if $\Omega$ is a Euclidean ball and $\kappa>\kappa_{sp=n,\pm}$ then the last integral inequality cannot hold without forcing $c_{sp=n,\pm}$ to depend only on $s$ and $p$.

• (iii)

If $sp>n$ and $\Omega\subseteq{{{\mathbb{R}}}^{n}}$ is a domain with volume $|\Omega|<\infty$ then

[TABLE]

Moreover, the constant $\kappa_{sp>n,\pm}$ is sharp in the sense that if $\Omega$ is a Euclidean ball then

[TABLE]

Proof.

The sharp inequalities in (i), (ii) and (iii) are suitably called the sharp Hardy-Rellich, Adams-Moser and Morrey-Sobolev inequalities for the fractional order twin gradients $\nabla^{0<s<1}_{\pm}$, respectively. Since (i) follows readily from [6, Corollary 1 & Theorem 4 (16)], the definition of $\nabla^{s}_{\pm}$ and $I_{s}=(-\Delta)^{-\frac{s}{2}}$, it remains to verify (ii)-(iii).

Case - $\nabla^{s}_{+}$. Under this situation we have

[TABLE]

and

[TABLE]

This, along with Theorem 3.1(ii)/(iii), directly gives the desired conclusion in (ii)/(iii) for $\nabla^{s}_{+}$ and the corresponding sharp case.

Case - $\nabla^{s}_{-}$. From the hypothesis

[TABLE]

it follows that

[TABLE]

and hence

[TABLE]

Also, according to [33, (5.6)&(4.4)] we have

[TABLE]

thereby finding

[TABLE]

which exists as a fractional variant of (1.2). In light of (3.2) and Theorem 3.1(ii)/(iii), we obtain the desired inequality in Theorem 3.2(ii)/(iii).

To see that $\kappa_{sp\geq n,-}$ is sharp, we consider two situations below.

• $\rhd$

$sp=n$. Without loss of generality we may assume that $\Omega$ is the origin-centered unit ball $\mathbb{B}^{n}$. If for some $\kappa>{\kappa}_{sp=n,-}$ it holds that

[TABLE]

then we are about to construct suitable functions $u$ to show that (3.3) forces $\kappa\leq{\kappa}_{sp=n,-}$, thereby revealing that ${\kappa}_{sp=n,-}$ is the sharp number to guarantee Theorem 3.2(ii).

Being somewhat motivated by [3, pp.391-392] and [12, p.7], for $r\in(0,1)$ we let $\mathbb{B}^{n}_{r}$ be the origin-centered ball with radius $r$ and

[TABLE]

Then

[TABLE]

Consequently, we use the first equation in (3.4), (3.1) and the polar-coordinate-system to achieve that if $x\in\mathbb{B}^{n}_{r}$ then

[TABLE]

where

[TABLE]

is independent of the variable $x$ after a rotation. Since $U(0)=1$, we write

[TABLE]

For the error term $T(|x|,r)$, observing that

[TABLE]

we therefore derive that, for $\epsilon>0$ there is a sufficiently small $r_{0}>0$ such that

[TABLE]

So, we have

[TABLE]

This, along with (3.3) and the second formula of (3.4), gives

[TABLE]

which in turns implies that if $0<r\leq r_{0}$ then

[TABLE]

Letting $\epsilon\downarrow 0$ and $r\downarrow 0$ yields

[TABLE]

as desired.

• $\rhd$

$sp>n$. Let

[TABLE]

Notice that $u_{\beta}$ can be approximated by functions in $C_{c}^{\infty}$ and

[TABLE]

So, by (3.1) and the calculations in the proof of Theorem 3.1(iii), we obtain

[TABLE]

and

[TABLE]

This in turn implies

[TABLE]

and so $\kappa_{sp>n,-}$ is sharp.

∎

4. Fractional Hardy-Sobolev spaces and their dualities

4.1. Fractional Hardy-Sobolev spaces $H^{s,p}$ and $H^{s,p}_{\pm}$

Suppose $0<s<1\leq p<\infty.$ Since both $\nabla^{s}_{+}u$ and $\nabla^{s}_{-}u$ are well defined when $u\in{\mathcal{S}}_{s}^{\prime}$, the study for the case $p=1$ of (2.5) in [20] motivates us to consider the fractional Hardy-Sobolev space

[TABLE]

Note that

[TABLE]

So, $[\cdot]_{H^{s,p}}$ is properly a norm on quotient space of $H^{s,p}$ modulo the space of all real constants, and consequently this quotient space is a Banach space.

Upon introducing

[TABLE]

we find immediately

[TABLE]

Indeed, as shown in the next theorem, when $s\in(0,1)$ and $p\in(1,\infty)$, these three spaces are equal to each other and they all have the Schwartz class ${\mathcal{S}}$ and

[TABLE]

as dense subspaces.

Theorem 4.1**.**

Let $0<s<1<p<\infty$. Then

[TABLE]

Moreover, both ${\mathcal{S}}_{\infty}$ and ${\mathcal{S}}$ are dense in $H^{s,p}$ and $H^{s,p}_{\pm}$.

Proof.

Notice that any $u\in{\mathcal{S}}$ satisfies $(-\Delta)^{\frac{s}{2}}u\in{\mathcal{S}}_{s}$ (cf. [32]). Of course, any function in ${\mathcal{S}}_{s}$ belongs to $L^{1<p<\infty}$. We therefore obtain

[TABLE]

Given $p\in(1,\infty)$, upon recalling boundedness of the Riesz transforms $R_{j}$ on $L^{p}$ (cf. [35]) and the identity

[TABLE]

we achieve

[TABLE]

thereby reaching

[TABLE]

This in turn implies

[TABLE]

Consequently, we obtain

[TABLE]

It suffices to show the density of ${\mathcal{S}}_{\infty}$ in $H^{s,p}_{+}$. If $u\in H^{s,p}_{+}$, then

[TABLE]

Due to the density of ${\mathcal{S}}_{\infty}$ in $L^{p}$ (cf. the proof of [20, Lemma 2.9(iii)]), we can find a sequence $\{f_{j}\}_{j\in{\mathbb{N}}}$ in ${\mathcal{S}}_{\infty}$ such that

[TABLE]

For any $j\in{\mathbb{N}}$, we write

[TABLE]

Upon noticing

[TABLE]

we obtain

[TABLE]

Thus, any $u\in H^{s,p}_{+}$ can be approximated by the ${\mathcal{S}}_{\infty}$-functions $\{u_{j}\}_{j\in{\mathbb{N}}}$. ∎

4.2. Dual Hardy-Sobolev spaces $[H^{s,p}]^{\ast}$ and $[H^{s,p}_{\pm}]^{\ast}$

In this subsection, we are about to show that these dual spaces can be characterized by

[TABLE]

solving the fractional differential equation

[TABLE]

Theorem 4.2**.**

Let $0<s<1<p<\infty$ and $p^{\prime}=\frac{p}{p-1}.$ Then for any distribution $T\in{\mathcal{S}}^{\prime}$ the following three assertions are equivalent:

• (i)

$T\in[H^{s,p}]^{\ast}=[H^{s,p}_{+}]^{\ast}=[H^{s,p}_{-}]^{\ast}$;

• (ii)

$\exists$* $T_{0}\in L^{p^{\prime}}$ such that $T=[\nabla_{+}^{s}]^{\ast}T_{0}$ in ${\mathcal{S}}^{\prime}$;*

• (iii)

$\exists$* $(T_{1},\dots,T_{n})\in(L^{p^{\prime}})^{n}$ such that $T=[\nabla_{-}^{s}]^{\ast}(T_{1},\dots,T_{n})$ in ${\mathcal{S}}^{\prime}$.*

Proof.

Note that Theorem 4.1 implies

[TABLE]

So, we begin with showing that (ii) implies (i) by considering $H^{s,p}_{+}$. If (ii) is valid, i.e., if

[TABLE]

then

[TABLE]

and hence

[TABLE]

Accordingly, using the density of ${\mathcal{S}}$ in $H^{s,p}_{+}$, we see that $T$ induces a bounded linear functional on $H^{s,p}_{+}$. This proves that

[TABLE]

and (i) holds due to (4.1).

Conversely, in order to show that (i) implies (ii), upon assuming

[TABLE]

we are required to find

[TABLE]

Inspiring by [7, Proposition 1, pp. 399-400], we consider the operator

[TABLE]

Evidently, the above-defined linear operator $A_{+}$ is bounded and hence closed. Thus, if

[TABLE]

then

[TABLE]

and hence

[TABLE]

This in turn implies that the operator $A_{+}$ is injective. Moreover, due to

[TABLE]

the operator $A_{+}$ has actually a continuous inverse from $L^{p}$ to $H^{s,p}_{+}$. Accordingly, by the closed range theorem (see [40, p. 208, Corollary 1]), we know that the adjoint operator

[TABLE]

is surjective. In particular, if

[TABLE]

then there exists

[TABLE]

Consequently, for any $\phi\in{\mathcal{S}}$, we have

[TABLE]

namely,

[TABLE]

This completes the argument for that (i) implies (ii).

Next, we show that (iii) implies (i) by considering $H^{s,p}_{-}$. If

[TABLE]

then for any $\phi\in{\mathcal{S}}$ we have

[TABLE]

whence

[TABLE]

Since ${\mathcal{S}}$ is dense in $H^{s,p}_{-}$, it follows that $T$ induces a bounded linear functional on $H^{s,p}_{-}$. This shows (iii)$\Longrightarrow$(i).

Conversely, in order to show (i)$\Longrightarrow$(iii), assuming

[TABLE]

we are about to verify that

[TABLE]

To this end, we consider the bounded linear operator

[TABLE]

Now we validate that the just-defined operator $A_{-}$ is injective. If

[TABLE]

then, for any $\psi\in{\mathcal{S}}_{\infty}$, we apply the Fourier transform to derive

[TABLE]

thereby giving

[TABLE]

This, along with the density of ${\mathcal{S}}_{\infty}$ in $L^{p^{\prime}}$ (cf. the proof of [20, Lemma 2.9(iii)]), further gives

[TABLE]

Accordingly, $A_{-}$ is an injective map from $H^{s,p}_{-}$ onto $A_{-}(H^{s,p}_{-})$ (the closed range of $A_{-}$) $\subseteq(L^{p})^{n}$. This, along with

[TABLE]

ensures that $A_{-}$ has a continuous inverse from $A_{-}(H^{s,p}_{-})$ to $H^{s,p}_{-}$. Upon applying the closed range theorem (see [40, p. 208, Corollary 1]) we get that the adjoint operator

[TABLE]

is surjective, thereby finding

[TABLE]

Upon utilizing the Hahn-Banach theorem to extend $\vec{T}_{o}$ to

[TABLE]

we have

[TABLE]

whence

[TABLE]

This completes the argument for (i)$\Longrightarrow$(iii). ∎

Let div be the classical divergence operator whose action on a vector-valued function $\vec{Y}$ is given by

[TABLE]

As a limiting case $s\uparrow 1$ of Theorem 4.2, we have the following conclusion.

Proposition 4.3**.**

Let $p\in(1,\infty)$. Then $L^{p}=\vec{R}\cdot(L^{p})^{n}$ - namely -

[TABLE]

Consequently, for any $Y\in L^{p}$, there exist $(Y_{0},Y_{1},\dots,Y_{n})\in(L^{p})^{1+n}$ such that

[TABLE]

Proof.

Given $1<p<\infty$. Thanks to the boundedness of $\vec{R}$ on $L^{p}$ and the identity

[TABLE]

we have that any $f\in L^{p}$ enjoys the desired property

[TABLE]

As a consequence, for any $Y\in L^{p}$ we can find a vector-valued function

[TABLE]

such that

[TABLE]

Also, if

[TABLE]

then

[TABLE]

Since ${\mathcal{S}}$ is dense in $[L^{p}]^{\ast}=L^{\frac{p}{p-1}}$, we deduce that the last two equalities hold in $L^{p}$. ∎

Remark 4.4**.**

Whenever $s=p=1$ we define

[TABLE]

Just like ${\mathcal{S}}_{\infty}$ is dense in $H^{1}$, we have also the density of ${\mathcal{S}}_{\infty}$ in $H^{1,1}$ (cf. [20, Proposition 2.12]). But for functions in ${\mathcal{S}}_{\infty}$ the Fourier transform easily derives

[TABLE]

Thus, $H^{1,1}$ can be equivalently defined to be the space of all locally integrable functions on ${{{\mathbb{R}}}^{n}}$ satisfying $[f]_{H^{1,1}}<\infty$. In analogy to Theorem 4.2 and Proposition 4.3, we have:

• (i)

H^{1,1}=\vec{R}\cdot\big{(}H^{1,1}\big{)}^{n}* - namely -*

[TABLE]

This is due to the fact that any $Z\in H^{1,1}$ can be written as

[TABLE]

• (ii)

Given a distribution $T\in\mathcal{S}^{\prime}$,

[TABLE]

This follows from the endpoint $s=1$ of **[20, Theorem 4.3(i)]** (cf. **[25, Lemma 4.1]** for the dual of the endpoint Sobolev space $\dot{W}^{1,1}$) and the basic formula

[TABLE]

• (iii)

Thanks to (i) and the fact that any

[TABLE]

satisfies

[TABLE]

we get that

[TABLE]

5. Distributional solutions of duality equations

5.1. Distributional solutions to $[\nabla^{s}_{\pm}]^{\ast}u=\mu$

For any $\alpha\in(0,n)$ and nonnegative Radon measure $\mu$ on ${{{\mathbb{R}}}^{n}}$, define

[TABLE]

and

[TABLE]

Observe that

[TABLE]

As a straightforward application of Theorem 4.2, we can characterize distributional solutions to the following fractional duality equations

[TABLE]

Upon extending [24, Theorems 3.1-3.2-3.3] - if $\mu$ is a nonnegative Radon measure on $\mathbb{R}^{n\geq 2}$ then

[TABLE]

we obtain

Theorem 5.1**.**

Let $0<s<1<p<\infty$ and $\mu$ be a nonnegative Radon measure on $\mathbb{R}^{n}$. Then either

[TABLE]

or

[TABLE]

holds if and only if

[TABLE]

Proof.

Let us start with the case $p\in(1,\frac{n}{n-s}]$. Clearly, if $\mu=0$, then

[TABLE]

ensures

[TABLE]

and

[TABLE]

Thus it is enough to show the only-if-part.

Consider first the operator $[\nabla^{s}_{+}]^{\ast}$ and assume that (5.2) holds for some $u_{0}\in L^{p}$. For any $\phi\in{\mathcal{S}}_{\infty}$, we utilize the Fourier transform to derive

[TABLE]

and hence

[TABLE]

which, along with the fact that ${\mathcal{S}}_{\infty}$ is dense in

[TABLE]

gives

[TABLE]

From this and the observation (5.1) it follows that

[TABLE]

However, this is impossible unless $\mu=0$.

Consider next the operator $[\nabla^{s}_{-}]^{\ast}$. Assume that (5.3) holds - namely -

[TABLE]

is a distributional solution of

[TABLE]

For any $\psi\in{\mathcal{S}}_{\infty}$, by the fact $I_{s}\psi\in{\mathcal{S}}_{\infty}$, the definition of

[TABLE]

and the self-adjointness of $(-\Delta)^{\frac{s}{2}}$, we obtain

[TABLE]

which, together with the aforementioned density of ${\mathcal{S}}_{\infty}$ in

[TABLE]

and the boundedness of $R_{j}$ on $L^{p}$, yields

[TABLE]

Similarly to the argument for the operator $[\nabla^{s}_{+}]^{\ast}$, the fact $I_{s}\mu\in L^{p}$ and (5.1) again derive $\mu=0$.

Next, we handle the case $p\in(\frac{n}{n-s},\infty)$. Clearly, the only-if-part follows from the same argument as the case $p\in(1,\frac{n}{n-s}]$. So, it remains to verify the if-part under

[TABLE]

According to Theorem 4.2, we only need to validate that such a measure $\mu$ induces a bounded linear functional on $H^{s,p^{\prime}}_{+}$, where $p^{\prime}=\frac{p}{p-1}$. To this end, for any $\phi\in{\mathcal{S}}$, by the fact

[TABLE]

and the Fubini theorem, we write

[TABLE]

so the Hölder inequality gives

[TABLE]

Combining this with the density of ${\mathcal{S}}$ in $H^{s,p^{\prime}}_{+}$ (cf. Theorem 4.1) leads to that $\mu$ can be extended to a bounded linear functional on $H^{s,p^{\prime}}_{+}$.

∎

5.2. Morrey’s regularity for distributional solutions of $[\nabla^{s}_{\pm}]^{\ast}u=f$

In accordance with the basic identity

[TABLE]

and [29, Theorem 1.1] - if $\Omega$ is an open subset of ${{{\mathbb{R}}}^{n}}$,

[TABLE]

and $u\in H^{s,p}$ is a distributional solution to the following fractional $p$-Laplace equation with a natural variation structure

[TABLE]

i.e.,

[TABLE]

then $u\in C^{s+\alpha}_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ for some positive constant $\alpha$ depending on $p$ only, we are led to settle Morrey’s regularity for the distributional solutions of the fractional duality equations

[TABLE]

For any $(p,\kappa)\in[1,\infty)\times(0,n]$, the Morrey space $\mathrm{L}^{p,{\kappa}}$ was introduced by Morrey [22] and used to study the solution of some quasi-linear elliptic partial differential equations, where $\mathrm{L}^{p,{\kappa}}$ comprises all Lebesgue measurable functions $f$ on $\mathbb{R}^{n}$ with

[TABLE]

In particular, when $(p,{\kappa})\in[1,\infty)\times\{n\}$, the space $\mathrm{L}^{p,n}$ is just the classical Lebesgue space $L^{p}$.

For $(p,\kappa)\in(1,\infty)\times(0,n)$, let $\mathrm{H}^{p,{\kappa}}$ be the space of all Lebesgue measurable functions $f$ on ${{{\mathbb{R}}}^{n}}$ such that

[TABLE]

where the infimum is taken over all nonnegative functions $\omega$ on ${{{\mathbb{R}}}^{n}}$ satisfying

[TABLE]

Here and hereafter, for any given $\alpha\in(0,n)$, the symbol $\Lambda^{\alpha}_{(\infty)}(E)$ denotes the $\alpha$-th order Hausdorff capacity of a subset $E\subseteq{{{\mathbb{R}}}^{n}}$, given by

[TABLE]

According to [5], we have the duality

[TABLE]

From [26, (5.1)] and [2, Corollary & Proposition 5], we have that if

[TABLE]

then

[TABLE]

Consequently, if

[TABLE]

then

[TABLE]

and hence (5.4) is used to produce the Stein-Weiss-Hardy inequality at the endpoint $p=1$:

[TABLE]

This, along with (cf. [20, (1.3)-(1.4)])

[TABLE]

derives

[TABLE]

which may be viewed as an improvement of the case $p=1$ of [11, Theorem 1.1].

Upon taking a function $\varphi$ satisfying

[TABLE]

we extend the real Hardy space $H^{p}$ from $p\in[1,\infty)$ to $p\in(0,\infty)$ via defining (cf. [36])

[TABLE]

Then (cf. [10, 36])

[TABLE]

Here and henceforth, $\mathrm{Lip}_{0<\alpha<1}$ is the $(0,1)\ni\alpha$-Lipschitz space of all functions $f$ on ${{{\mathbb{R}}}^{n}}$ satisfying

[TABLE]

Theorem 5.2**.**

Let

[TABLE]

If $f\in\mathrm{L}^{p,{\kappa}}$, then

[TABLE]

such that

[TABLE]

holds in the sense of

[TABLE]

Proof.

Suppose $f\in\mathrm{L}^{p,{\kappa}}$. Note that the desired regularity for

[TABLE]

follows from [18, Theorem 1.2] with $F_{0}=I_{s}f$. So, it remains to check the desired regularity for

[TABLE]

To this end, we define the measure $\mu_{f}$ by

[TABLE]

Then, for any $(x,r)\in{{{\mathbb{R}}}^{n}}\times(0,\infty)$, we utilize the Hölder inequality to derive

[TABLE]

thereby achieving

[TABLE]

The forthcoming demonstration consists of essentially two components.

Part 1 - the case $sp\geq{\kappa}$.

Under this condition we have

[TABLE]

We are inspired by the proof of [7, Proposition 1, pp. 399-400] (cf. [24, Theorem 3.2]) to set

[TABLE]

and

[TABLE]

endowed with the norm

[TABLE]

Note that $\|u\|_{X}=0$ if and only if $u$ is a constant function on ${{{\mathbb{R}}}^{n}}$. So, $X$ is treated as a quotient space modulo the space of constant functions. Since ${\mathcal{S}}_{\infty}\subset{\mathcal{S}}\cap X\subset X$ and ${\mathcal{S}}_{\infty}$ is dense in the Hardy space $\mathrm{H}^{\frac{n}{n+s-\frac{{\kappa}}{p}}}$ (cf. [2]), one easily deduces the density of ${\mathcal{S}}\cap X$ in $X$.

Consider the operator

[TABLE]

This operator is well defined in that the action of the operator $\nabla^{s}_{-}$ can be defined on the distribution space ${\mathcal{S}}_{s}^{\prime}$. Moreover, it is easy to see that $A$ is a bounded linear operator.

We can also show that the operator $A$ is injective. To this end, assuming that $u\in X$ satisfies

[TABLE]

we are required to show

[TABLE]

Note that

[TABLE]

Thus, for any $\psi\in{\mathcal{S}}_{\infty}$, we use the Fourier transform to derive

[TABLE]

thereby finding

[TABLE]

This shows

[TABLE]

In other words, $u$ is a polynomial on ${{{\mathbb{R}}}^{n}}$. However, if a polynomial $u$ is a bounded linear functional on ${\mathcal{S}}_{s}$, then $u$ must be a constant function, as desired.

The above analysis shows that the operator $A$ is injective and has a continuous inverse from $A(X)\subseteq Y$ to $X$. Upon applying the closed range theorem (see [40, p. 208, Corollary 1]), we deduce that the adjoint operator

[TABLE]

is surjective.

Next, we validate that any $f\in\mathrm{L}^{p,{\kappa}}$ belongs to $X^{\ast}$. Indeed, for any $\phi\in{\mathcal{S}}\cap X$, we apply [30, Theorem 1.12] to write

[TABLE]

Also, using $\phi\in{\mathcal{S}}$, we derive from [20, Lemma 2.6] that $\nabla^{s}_{j}\phi\in{\mathcal{S}}_{s}$, which easily implies that $R_{j}\nabla^{s}_{j}\phi$ is continuous on ${{{\mathbb{R}}}^{n}}$. From the fact

[TABLE]

it follows that

[TABLE]

while the second inequality of (5.8) holds because after a change of variable

[TABLE]

the function

[TABLE]

is strictly increasing on the interval $(0,s]$ and $\psi(s)=0$. By (5.8), [17, Theorem 1.1] and its remark, we can derive the continuity of the mapping

[TABLE]

with operator norm at most a constant multiple of $\||\mu_{f}\||_{n-\frac{{\kappa}}{p}}$. Combining these and boundedness of $R_{j}$ on $\mathrm{H}^{\frac{n}{n+s-\frac{{\kappa}}{p}}}$ yields

[TABLE]

Due to the density of ${\mathcal{S}}\cap X$ in $X$, we arrive at the conclusion that $f$ induces a bounded linear functional on $X$.

To continue, like proving Theorem 4.2(iii) we use the surjective property of $A^{\ast}$ and the Hahn-Banach extension theorem to obtain

[TABLE]

such that

[TABLE]

whence

[TABLE]

Part 2 - the case $sp<{\kappa}$.

This part is similar to the case $sp\geq{\kappa}$. To be precise, we take

[TABLE]

Define

[TABLE]

endowed with the norm

[TABLE]

Again, observing that $\|u\|_{X}=0$ if and only if $u$ is a constant, we also understood this $X$ as a quotient space. Though we do not know if ${\mathcal{S}}\cap X$ is dense in $X$, we use the space $\mathring{X}$ which is the closure of ${\mathcal{S}}\cap X$ in $X$.

Still we consider the operator

[TABLE]

and can show that $A$ is injective and has a continuous inverse from $A(\mathring{X})$ (the close range of $A$) to $\mathring{X}$. Consequently, the closed range theorem (cf. [40, p. 208, Corollary 1]) can be applied to derive that the adjoint operator

[TABLE]

is surjective.

Next, we validate that any $f\in\mathrm{L}^{p,{\kappa}}$ belongs to $(\mathring{X})^{\ast}$. Applying [19, Proposition 5.1] gives the continuity of the mapping

[TABLE]

Note that the boundedness of $R_{j}$ on ${\mathrm{H}^{q^{\prime},q(\frac{{\kappa}}{p}-s)}}$ is given in [4, Chapter 8]. So, upon using (5.7) and the Fubini theorem, we derive that any $\phi\in{\mathcal{S}}\cap X$ satisfies

[TABLE]

This implies that $f$ can be extended to a bounded linear functional on $\mathring{X}$, that is, $f\in(\mathring{X})^{\ast}$.

Because of $f\in(\mathring{X})^{\ast}$ and the surjective property of $A^{\ast}$, we can borrow the idea of verifying Theorem 4.2(iii) and use the Hahn-Banach extension theorem to find a vector-valued function

[TABLE]

such that

[TABLE]

thereby reaching

[TABLE]

∎