On Korn’s First Inequality in a Hardy-Sobolev Space

Abstract

Korn’s first inequality states that there exists a constant such that the ${\mathcal {L}}^{2}$-norm of the infinitesimal displacement gradient is bounded above by this constant times the ${\mathcal {L}}^{2}$-norm of the infinitesimal strain, i.e., the symmetric part of the gradient, for all infinitesimal displacements that are equal to zero on the boundary of a body ℬ. This inequality is known to hold when the ${\mathcal {L}}^{2}$-norm is replaced by the ${\mathcal {L}}^{p}$-norm for any $p\in (1,\infty )$. However, if $p=1$ or $p=\infty$ the resulting inequality is false. It was previously shown that if one replaces the ${\mathcal {L}}^{\infty}$-norm by the $\operatorname{BMO}$-seminorm (Bounded Mean Oscillation) then one maintains Korn’s inequality. (Recall that ${\mathcal {L}}^{\infty}({\mathcal {B}})\subset \operatorname{BMO}({\mathcal {B}}) \subset {\mathcal {L}}^{p}({\mathcal {B}})\subset {\mathcal {L}}^{1}({ \mathcal {B}})$, $1< p<\infty$.) In this manuscript it is shown that Korn’s inequality is also maintained if one replaces the ${\mathcal {L}}^{1}$-norm by the norm in the Hardy space ${\mathcal {H}}^{1}$, the predual of $\operatorname{BMO}$. One caveat: the results herein are only applicable to the pure-displacement problem with the displacement equal to zero on the entire boundary of ℬ.

1 Introduction

In this manuscript we consider Korn’s first inequality in the form

$$ {[\hspace{- 0.3 ex}] \nabla{\mathbf {u}} [\hspace{- 0.3 ex}]}_{{\mathbf {X}}} \le {\mathcal {K}}{[\hspace{- 0.3 ex}] {\mathbf {E}} [\hspace{- 0.3 ex}]}_{{ \mathbf {X}}} $$

(1.1)

for all infinitesimal displacements ${\mathbf {u}}\in C^{\infty}_{0}({\mathcal {B}};{\mathbb{R}^{n}})$, the space of vector-valued, infinitely differentiable functions with compact support on a body ℬ contained in ${\mathbb{R}^{n}}$. Here

$$ {\mathbf {E}}({\mathbf {x}}):=\tfrac{1}{2}\bigl(\nabla {\mathbf {u}}({\mathbf {x}})+ \left [\nabla {\mathbf {u}}({\mathbf {x}})\right ]^{{\mathrm {T}}}\bigr) $$

denotes the infinitesimal strain, ${[\hspace{- 0.3 ex}] \cdot [\hspace{- 0.3 ex}]}_{{\mathbf {X}}}$ denotes a norm (or seminorm) on a Banach space ${\mathbf {X}}$ of functions with values in the set of real $n$ by $n$ matrices ${\mathbb{M}}^{n\times n}$, $\nabla {\mathbf {u}}$ denotes the infinitesimal displacement gradient, the matrix of partial derivatives of the components of ${\mathbf {u}}$, ${\mathcal {K}}>0$ is a constant that does not depend on ${\mathbf {u}}$, ${\mathbf {H}}^{{\mathrm {T}}}$ denotes the transpose of ${\mathbf {H}}\in {\mathbb{M}}^{n\times n}$, and we assume that $C^{\infty}_{0}({\mathcal {B}};{\mathbb{M}}^{n\times n})\subset { \mathbf {X}}$. We note that if (1.1) is satisfied by all such ${\mathbf {u}}$ it follows that (1.1) holds on the closure of $C^{\infty}_{0}({\mathcal {B}};{\mathbb{M}}^{n\times n})$ in ${\mathbf {X}}$.

A simple example, which was of interest to Korn [24] (see, also, [25–27]) and which has an elementary proof, is ${\mathbf {X}}={\mathcal {L}}^{2}({\mathcal {B}};{\mathbb{M}}^{n\times n})$, the space of square-integrable, matrix-valued functions, i.e.,

$$ \int _{{\mathcal {B}}}\sum _{i,j=1}^{n} \left [ \frac{\partial u_{i}}{\partial x_{j}}\right ]^{2}{\mathrm {d}}{ \mathbf {x}}\le 2 \int _{{\mathcal {B}}}\sum _{i,j=1}^{n}\frac{1}{4} \left [ \frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}\right ]^{2}{\mathrm {d}}{ \mathbf {x}}$$

(see, e.g., Gurtin [21, pp. 38–39], Friedrichs [17, §2], or Horgan [22, p. 493]). More generally, (1.1) is valid if ${\mathbf {X}}={\mathcal {L}}^{p}({\mathcal {B}};{\mathbb{M}}^{n\times n})$, $1< p < \infty$:

$$ \int _{{\mathcal {B}}}|\nabla {\mathbf {u}}|^{p}\,{\mathrm {d}}{\mathbf {x}} \le {\mathcal {K}}(p) \int _{{\mathcal {B}}}|{\mathbf {E}}|^{p}\,{\mathrm {d}}{ \mathbf {x}}. $$

(1.2)

An interesting proof of (1.2) was obtained by T. W. Ting [39, pp. 295–297], who made use of the Riesz transforms

$$ ({\mathcal {R}}_{i} \varphi )({\mathbf {x}}) := c_{n}\lim _{\varepsilon \to 0^{+}}\int _{|{\mathbf {y}}-{\mathbf {x}}|>\varepsilon} \frac{(x_{i}-y_{i})\varphi ({\mathbf {y}})}{|{\mathbf {x}}-{\mathbf {y}}|^{n+1}} \,{\mathrm {d}}{\mathbf {y}}, \quad \text{$i=1,2,\ldots ,n$}. $$

(1.3)

The key to this proof was then the fact that the Riesz transforms are bounded linear maps from ${\mathcal {L}}^{p}({\mathbb{R}^{n}})$ to ${\mathcal {L}}^{p}({\mathbb{R}^{n}})$ for every $p\in (1,\infty )$.

We note that (1.2) is false if $p=1$ (see Ornstein [33] or Conti, Faraco, and Maggi [10]) or $p=\infty$ (see de Leeuw and Mirkil [12]). Although these values are the natural endpoints of $(1,\infty )$, we note that, for all $p\in (1,\infty )$ and every nonempty bounded open region ${\Omega }\subset {\mathbb{R}^{n}}$,

$$ {\mathcal {L}}^{\infty}({\Omega }) \subset \operatorname{BMO}({\Omega }) \subset { \mathcal {L}}^{p}({\Omega }), \qquad {\mathcal {L}}_{0}^{p}({\Omega }) \subset {\mathcal {H}}^{1}_{z}({\Omega }) \subset {\mathcal {L}}_{0}^{1}({ \Omega }), $$

where $\operatorname{BMO}({\Omega })$ (Bounded Mean Oscillation) is the space of functions that were first analyzed by John and Nirenberg [23], ${\mathcal {H}}^{1}_{z}({\Omega })$ is the Hardy space defined by

$$ {\mathcal {H}}_{z}^{1}({\Omega }) := \{\varphi \in {\mathcal {H}}^{1}({ \mathbb{R}}^{n}): \varphi ({\mathbf {x}})=0 \text{ for } {\mathbf {x}} \notin \overline{{\Omega }}\}, $$

(1.4)

${\mathcal {H}}^{1}({\mathbb{R}}^{n})$ is the usual Hardy space on ${\mathbb{R}^{n}}$ (see (1.5) below) and

$$ {\mathcal {L}}_{0}^{p}({\Omega }) := \{\phi \in {\mathcal {L}}^{p}({ \Omega }): \int _{{\Omega }} \varphi ({\mathbf {x}})\,{\mathrm {d}}{ \mathbf {x}}= 0\}. $$

Thus, ${\mathcal {H}}^{1}_{z}({\Omega })$ and its dual space $\operatorname{BMO}({\Omega })$ can be thought of as alternative endpoints for ${\mathcal {L}}^{p}$, $1< p<\infty$, as is the usual practice in Harmonic Analysis.

In a previous paper, [35], we showed that a general form of Korn’s inequality is valid in $\operatorname{BMO}({\Omega })$ (without boundary conditions or side constraints). In this manuscript we establish that Korn’s first inequality (1.1) is valid when ${\mathbf {X}}={\mathcal {H}}_{z}^{1}({\Omega })$. In Section 2.3, we first present an appropriate definition of ${\mathcal {H}}^{1}({\mathbb{R}}^{n})$. Rather than making use of what has become the standard definition (via maximal functions, see, e.g., Stein [38, pp. 90–101] or atoms, see Remark 2.1) we follow Fefferman and Stein [15, §2] (see, also, Stein and Weiss [36] and Stein [38, pp. 123–124]) and, equivalently, define

$$ {\mathcal {H}}^{1}({\mathbb{R}^{n}}) := \{\varphi \in {\mathcal {L}}^{1}({ \mathbb{R}}^{n}): \int _{{\mathbb{R}}^{n}} \varphi ({\mathbf {x}})\,{ \mathrm {d}}{\mathbf {x}}= 0,\ ({\mathcal {R}}_{i} \varphi ) \in {\mathcal {L}}^{1}({\mathbb{R}}^{n}),\ i=1,2, \ldots ,n\} $$

(1.5)

with norm

$$ \|\varphi \|_{{\mathcal {H}}}:= \int _{{\mathbb{R}}^{n}} |\varphi ({ \mathbf {x}})|\,{\mathrm {d}}{\mathbf {x}}+ \sum _{i=1}^{n} \int _{{ \mathbb{R}}^{n}} \bigl|({\mathcal {R}}_{i} \varphi )({\mathbf {x}})\bigr| \,{\mathrm {d}}{\mathbf {x}}, $$

(1.6)

where ${\mathcal {R}}_{i}$ denote the Riesz transforms given by (1.3). We then define a local Hardy space, ${\mathcal {H}}^{1}_{z}({\Omega })$ (see (1.4) above), associated with a nonempty open set ${\Omega }\subsetneqq {\mathbb{R}^{n}}$. As noted by Chang, Krantz, and Stein [5] (see, also, Chang, Dafni, and Stein [6]) this space is the “smallest” of the Hardy spaces associated with ${\Omega }$ and also the one best associated with the Dirichlet (displacement) problem. We then follow Auscher, Russ, and Tchamitchian [2] (see, also, Chen, Jiang, and Yang [7]) and define the Hardy-Sobolev space $W^{1,{\mathcal {H}}}({\Omega })$ (see (2.11)) as the space of locally integrable functions on ${\mathbb{R}^{n}}$ that are integrable on ${\Omega }$ and whose weak derivatives are each contained in ${\mathcal {H}}^{1}_{z}({\Omega })$. This space is, essentially, a subspace of the standard Sobolev space $W^{1,1}({\Omega })$ (and, if ${\Omega }$ bounded, contains the spaces $W_{0}^{1,p}({\Omega })$, $p>1$).

In Section 3 we state our main result, that (1.1) is valid when ${\mathbf {X}}$ is equal to the Hardy space ${\mathcal {H}}^{1}_{z}({\Omega })$ (see (1.4)–(1.6)). Our proof then initially follows that of Ting [39] in the setting of ${\mathcal {L}}^{p}$-spaces. The only significant difference is our use of the fact that the Riesz transforms are bounded linear maps from ${\mathcal {H}}^{1}({\mathbb{R}^{n}})$ to ${\mathcal {H}}^{1}({\mathbb{R}^{n}})$ rather than from ${\mathcal {L}}^{p}({\mathbb{R}^{n}})$ to ${\mathcal {L}}^{p}({\mathbb{R}^{n}})$.

In Section 4 we note that the proof of our main theorem also yields Korn’s first inequality in any Banach space upon which the Riesz transforms are bounded linear operators, in particular, in certain Besov spaces (see, e.g., [20, Corollary 6.7.2]). We thus show that (1.1) is valid in the standard Hölder spaces ${\mathbf {X}}=C^{0,\alpha}({\mathbb{R}^{n}})$, $\alpha \in (0,1)$, with seminorm

$$ {[\hspace{- 0.3 ex}] \varphi [\hspace{- 0.3 ex}]}_{C^{0,\alpha}({\mathbb{R}^{n}})} = \sup _{ \substack{{\mathbf {x}},{\mathbf {z}}\in {\mathbb{R}^{n}}\\{\mathbf {x}}\ne {\mathbf {z}}}} \frac{|\varphi ({\mathbf {x}})-\varphi ({\mathbf {z}})|}{|{\mathbf {x}}-{\mathbf {z}}|^{\alpha}}. $$

Finally, in Section 5 we speculate concerning the validity of some form of Korn’s second inequality in an appropriate Hardy-Sobolev space.

2 Preliminaries

We shall write ${\mathbb{M}}^{n\times n}$, $n\ge 2$, for the (vector) space of $n$ by $n$ matrices with real entries. ${\mathbf {H}}^{{\mathrm {T}}}$ shall denote the transpose of ${\mathbf {H}}\in {\mathbb{M}}^{n\times n}$. Given an orthonormal basis ${\mathbf {e}}_{i}$, $i=1,2,\dots ,n$, for ${\mathbb{R}}^{n}$ we write $a_{i}={\mathbf {a}}\cdot {\mathbf {e}}_{i}$ for ${\mathbf {a}}\in {\mathbb{R}}^{n}$ and $F_{ij}={\mathbf {e}}_{i}\cdot {\mathbf {F}}{\mathbf {e}}_{j}$ for ${\mathbf {F}}\in {\mathbb{M}}^{n\times n}$. For any nonempty open set ${\Omega }\subset {\mathbb{R}}^{n}$, $n\ge 2$, we denote by ${\mathcal {L}}^{p}({\Omega })$, $p\in [1,\infty )$, the space of real-valued Lebesgue measurable functions $\psi$ whose ${\mathcal {L}}^{p}$-norm is finite:

$$ ||\psi ||^{p}_{p,{\Omega }} := \int _{{\Omega }}|\psi ({\mathbf {x}})|^{p} \,{\mathrm {d}}{\mathbf {x}}< \infty . $$

${\mathcal {L}}^{1}_{\operatorname{loc}}({\Omega })$ shall consist of those Lebesgue measurable functions that are integrable on every compact subset of ${\Omega }$. We shall write $C^{\infty}({\Omega })$ for the set of real-valued functions which, together with all of its partial derivatives (of all orders) are continuous on ${\Omega }$. $C^{\infty}_{0}({\Omega })$ shall denote those functions $\psi \in C^{\infty}({\Omega })$ for which there exists a compact set $K=K_{\psi }\subset {\Omega }$ such that $\psi ({\mathbf {x}})=0$ for ${\mathbf {x}}\not \in K$. When a function is vector-valued we shall instead write ${\mathbf {u}}\in {\mathcal {L}}^{p}({\Omega };{\mathbb{R}^{n}})$, ${\mathbf {u}}\in C^{\infty}({\Omega };{\mathbb{R}^{n}})$, or ${\mathbf {u}}\in C^{\infty}_{0}({\Omega };{\mathbb{R}^{n}})$. For $1\le p < \infty$, $W^{1,p}({\Omega };{\mathbb{R}}^{n})$ shall denote the usual Sobolev space of (Lebesgue) measurable (vector-valued) functions ${\mathbf {u}}\in {\mathcal {L}}^{p}({\Omega };{\mathbb{R}}^{n})$ whose distributional gradient $\nabla {\mathbf {u}}$ is also contained in ${\mathcal {L}}^{p}$ (see, e.g., [1, 14]).

2.1 Extensions

Let ${\Omega }\subset {\mathbb{R}^{n}}$, $n\ge 2$, denote a nonempty open set. Then for any function $\varphi \in C^{\infty}_{0}({\Omega })$ we define its extension $\varphi ^{e}\in C^{\infty}_{0}({\mathbb{R}}^{n})$ by

$$ \varphi ^{e}({\mathbf {x}}) := \textstyle\begin{cases} \varphi ({\mathbf {x}}), & \text{if }\ {\mathbf {x}}\in {\Omega }, \\ 0, & \text{if }\ {\mathbf {x}}\in {\mathbb{R}}^{n}\setminus {\Omega }. \end{cases} $$

(2.1)

Moreover, in the sequel we shall drop the superscript $e$ and write $\varphi$ for this extension. Similarly, given ${\mathbf {u}}\in C^{\infty}_{0}({\Omega };{\mathbb{R}^{n}})$, we shall write ${\mathbf {u}}\in C^{\infty}_{0}({\mathbb{R}^{n}};{\mathbb{R}^{n}})$ with the implicit assumption that ${\mathbf {u}}({\mathbf {x}})=\mathbf{0}$ for ${\mathbf {x}}\in {\mathbb{R}^{n}}\backslash {\Omega }$.

2.2 Riesz Transforms

Let $\varphi \in C^{\infty}_{0}({\mathbb{R}}^{n})$, $n\ge 2$. Define the Riesz transform of $\varphi$ (in the direction ${\mathbf {e}}_{i}$, $i=1,2,\ldots ,n$) by

$$ ({\mathcal {R}}_{i} \varphi )({\mathbf {x}}) := c_{n}\int _{{\mathbb{R}}^{n}} \frac{(x_{i}-y_{i})\varphi ({\mathbf {y}})}{|{\mathbf {x}}-{\mathbf {y}}|^{n+1}} \,{\mathrm {d}}{\mathbf {y}}:= c_{n}\lim _{\varepsilon \to 0^{+}}\int _{|{ \mathbf {y}}-{\mathbf {x}}|>\varepsilon} \frac{(x_{i}-y_{i})\varphi ({\mathbf {y}})}{|{\mathbf {x}}-{\mathbf {y}}|^{n+1}} \,{\mathrm {d}}{\mathbf {y}}, $$

(2.2)

where $c_{n}:=1/(\pi \omega _{n-1})$ and $\omega _{n}$ denotes the volume of the unit ball in ${\mathbb{R}}^{n}$. Then standard results (see, e.g., [37, pp. 29–33 and pp. 54–60] or Theorem 1 in [3] or [4]) show that, for every $\varphi \in {\mathcal {L}}^{p}({\mathbb{R}^{n}})$, $1\le p<\infty$, $({\mathcal {R}}_{i} \varphi )({\mathbf {x}})$ is well-defined at almost every ${\mathbf {x}}\in {\mathbb{R}}^{n}$; in addition, ${\mathcal {R}}_{i}$ is a bounded linear map from ${\mathcal {L}}^{p}({\mathbb{R}^{n}})$ to ${\mathcal {L}}^{p}({\mathbb{R}^{n}})$ for every $p\in (1,\infty )$.

2.2.1 Identities

We note that (see, e.g., [4, Theorem 1 and pp. 908–909]), given any $\varphi \in {\mathcal {L}}^{p}({\mathbb{R}^{n}})$, $1< p<\infty$,

$$ \varphi =-\sum _{i=1}^{n} \left ({\mathcal {R}}_{i}^{2}\varphi \right ). $$

(2.3)

Moreover, if we define the linear operator $\Lambda :W^{1,p}({\mathbb{R}^{n}})\to {\mathcal {L}}^{p}({\mathbb{R}^{n}})$, $1< p<\infty$, by

$$ \Lambda \varphi := \sum _{j=1}^{n} \left ({\mathcal {R}}_{j} \frac{\partial \varphi}{\partial x_{j}}\right ), $$

(2.4)

then

$$ \frac{\partial \varphi}{\partial x_{j}} = \Lambda ({\mathcal {R}}_{j} \varphi ), \qquad \triangle \varphi := \sum _{j=1}^{n} \frac{\partial ^{2}\varphi}{\partial x_{j}^{2}} = -\Lambda ^{2} \varphi $$

(2.5)

(provided $\nabla \varphi \in W^{1,p}({\mathbb{R}^{n}};{\mathbb{R}^{n}})$), and hence $\Lambda =\sqrt{-\triangle}$.

2.3 The Hardy Spaces ${\mathcal {H}}^{1}({\mathbb{R}^{n}})$ and ${\mathcal {H}}^{1}_{z}({\Omega })$

The Hardy space ${\mathcal {H}}^{1}({\mathbb{R}^{n}})$, $n\ge 2$, shall be defined as¹ (see, e.g., [15, p. 144] or [38, pp. 123–124])

$$ {\mathcal {H}}^{1}({\mathbb{R}^{n}}) := \{\varphi \in {\mathcal {L}}^{1}({ \mathbb{R}}^{n}): \int _{{\mathbb{R}}^{n}} \varphi ({\mathbf {x}})\,{ \mathrm {d}}{\mathbf {x}}= 0,\ ({\mathcal {R}}_{i} \varphi ) \in {\mathcal {L}}^{1}({\mathbb{R}}^{n}),\ i=1,2, \ldots ,n\} $$

(2.6)

with norm

$$ \|\varphi \|_{{\mathcal {H}}}:= \int _{{\mathbb{R}}^{n}} |\varphi ({ \mathbf {x}})|\,{\mathrm {d}}{\mathbf {x}}+ \sum _{i=1}^{n} \int _{{ \mathbb{R}}^{n}} \bigl|({\mathcal {R}}_{i} \varphi )({\mathbf {x}})\bigr| \,{\mathrm {d}}{\mathbf {x}}. $$

(2.7)

Remark 2.1

An equivalent formulation for ${\mathcal {H}}^{1}({\mathbb{R}^{n}})$ can be obtained from the so-called atomic decomposition of C. Fefferman²: There is a constant $C=C(n)>1$ such that each $\varphi \in {\mathcal {H}}^{1}({\mathbb{R}^{n}})$ can be written as

$$ \varphi = \sum _{k=1}^{\infty }\lambda _{k} a_{k}, \qquad C^{-1}\sum _{k=1}^{ \infty }|\lambda _{k}| \le \|\varphi \|_{{\mathcal {H}}} \le C\sum _{k=1}^{ \infty }|\lambda _{k}|, $$

where $\lambda _{k}=\lambda _{k}^{\varphi}>0$, the (Lebesgue) measurable functions $a_{k}=a_{k}^{\varphi}:{\mathbb{R}}^{n}\to {\mathbb{R}}$ are each supported on a hypercube $Q_{k}=Q_{k}^{\varphi}$ with sides parallel to the axes,

$$ |a_{k}({\mathbf {x}})|\le |Q_{k}|^{-1}, \qquad \int _{Q_{k}} a_{k}({ \mathbf {x}})\,{\mathrm {d}}{\mathbf {x}}=0, $$

and $|Q_{k}|$ here denotes the volume of $Q_{k}$.

We note that the Riesz transforms are bounded linear operators on ${\mathcal {H}}^{1}({\mathbb{R}^{n}})$ (see, e.g., [15, pp. 149–150] or [37, p. 232]). Thus, there exists a constant $\rho =\rho (n)>0$ such that, for all $\varphi \in {\mathcal {H}}^{1}({\mathbb{R}^{n}})$,

$$ \|{\mathcal {R}}_{i} \varphi \|_{{\mathcal {H}}} \le \rho (n) \|\varphi \|_{{\mathcal {H}}}\ \text{ for $i=1,2,\ldots ,n$}. $$

(2.8)

Given any nonempty open set ${\Omega }\subset {\mathbb{R}^{n}}$, $n\ge 2$, the Hardy space ${\mathcal {H}}_{z}^{1}({\Omega })$ (see, e.g., [2, p. 55] or [7, p. 336]) is defined by

$$ {\mathcal {H}}_{z}^{1}({\Omega }) := \{\varphi \in {\mathcal {H}}^{1}({ \mathbb{R}}^{n}): \varphi ({\mathbf {x}})=0 \text{ for } {\mathbf {x}} \notin \overline{{\Omega }}\} $$

(2.9)

with norm (2.7). If ${\mathbf {F}}:{\mathbb{R}^{n}}\to {\mathbb{M}}^{n\times n}$ satisfies $F_{ij}\in {\mathcal {H}}_{z}^{1}({\Omega })$, for $i,j=1,2,\ldots ,n$, we write ${\mathbf {F}}\in {\mathcal {H}}_{z}^{1}({\Omega };{\mathbb{M}}^{n\times n})$ with

$$ \|{\mathbf {F}}\|_{{\mathcal {H}}} := \sum _{i,j=1}^{n} \|F_{ij}\|_{{ \mathcal {H}}}. $$

(2.10)

2.4 The Hardy-Sobolev Spaces $W^{1,{\mathcal {H}}}({\Omega };{\mathbb{R}^{n}})$ and $W_{0}^{1,{\mathcal {H}}}({\Omega };{\mathbb{R}^{n}})$

Given any nonempty open set ${\Omega }\subset {\mathbb{R}^{n}}$, $n\ge 2$, we define the Hardy-Sobolev spaces

$$ \begin{aligned} W^{1,{\mathcal {H}}}({\Omega };{\mathbb{R}}^{n}) &:= \left \{{\mathbf {u}} \in {\mathcal {L}}^{1}_{\operatorname{loc}}({\mathbb{R}^{n}};{\mathbb{R}}^{n}): { \mathbf {u}}\in {\mathcal {L}}^{1}({\Omega };{\mathbb{R}}^{n}),\ \nabla {\mathbf {u}}\in {\mathcal {H}}_{z}^{1}({\Omega };{\mathbb{M}}^{n \times n})\right \}, \\ W^{1,{\mathcal {H}}}_{0}({\Omega };{\mathbb{R}}^{n}) &:= \text{ the closure of $C^{\infty}_{0}({\Omega };{\mathbb{R}^{n}})$ in $W^{1,{\mathcal {H}}}({\Omega };{\mathbb{R}}^{n})$} \end{aligned} $$

(2.11)

with norm

$$ \|{\mathbf {u}}\|_{1,{\mathcal {H}}}:= \int _{{\Omega }} |{\mathbf {u}}({ \mathbf {x}})|\,{\mathrm {d}}{\mathbf {x}}+ \|\nabla {\mathbf {u}}\|_{{ \mathcal {H}}}. $$

(2.12)

Remark 2.2

It is clear from (2.6), (2.9), (2.11), and (2.12) that $W^{1,{\mathcal {H}}}({\Omega };{\mathbb{R}}^{n})\subset W^{1,1}({ \Omega };{\mathbb{R}}^{n})$ and $W_{0}^{1,{\mathcal {H}}}({\Omega };{\mathbb{R}}^{n})\subset W^{1,1}_{0}({ \Omega };{\mathbb{R}}^{n})$ (see, e.g., [1, p. 60]).

Remark 2.3

Suppose that ${\Omega }\subset {\mathbb{R}^{n}}$ is a (strongly) Lipschitz domain (see, e.g., [14, p. 127]). Then results of [2, p. 56] show that each ${\mathbf {u}}\in W^{1,{\mathcal {H}}}_{0}({\Omega };{\mathbb{R}}^{n})$ satisfies ${\mathbf {u}}=\mathbf{0}$ on $\partial {\Omega }$ (in the sense of trace).

3 Korn’s Inequality

Theorem 1

Fix $n\ge 2$. Then there exists a constant ${\mathcal {K}}={\mathcal {K}}(n)>0$ such that, for any nonempty open set ${\Omega }\subset {\mathbb{R}}^{n}$ and every ${\mathbf {u}}\in W^{1,{\mathcal {H}}}_{0}({\Omega };{\mathbb{R}}^{n})$,

$$ \|\nabla {\mathbf {u}}\|_{{\mathcal {H}}}\le {\mathcal {K}}\|{\mathbf {E}}\|_{{ \mathcal {H}}}, $$

(3.1)

where

$$ {\mathbf {E}}({\mathbf {x}}):=\tfrac{1}{2}\bigl(\nabla {\mathbf {u}}({\mathbf {x}})+ \left [\nabla {\mathbf {u}}({\mathbf {x}})\right ]^{{\mathrm {T}}}\bigr). $$

Proof

(cf. [39, Proof of Theorem 1]) Let ${\mathbf {u}}\in C^{\infty}_{0}({\Omega };{\mathbb{R}^{n}})$. Extend ${\mathbf {u}}$ to all of ${\mathbb{R}^{n}}$ by defining ${\mathbf {u}}({\mathbf {x}})=\mathbf{0}$ for ${\mathbf {x}}\in {\mathbb{R}^{n}}\backslash {\Omega }$ (cf. §2.1). Thus, ${\mathbf {u}}\in C^{\infty}_{0}({\mathbb{R}^{n}};{\mathbb{R}^{n}})$. It follows that each component of ${\mathbf {u}}$ as well as each of its derivatives (of all orders) are contained in ${\mathcal {L}}^{p}({\mathbb{R}^{n}})$, $1\le p < \infty$, and so the Riesz transform identities in §2.2.1 are valid for such functions. Next, consider the well-known identity (which follows from the equality of the mixed partial derivatives), $i,j,k\in \{1,2,\ldots ,n\}$,

$$ \frac{\partial}{\partial x_{j}} \left [ \frac{\partial u_{i}}{\partial x_{k}}\right ] = \frac{\partial ^{2} u_{i}}{\partial x_{j}\partial x_{k}} = \frac{\partial E_{ik}}{\partial x_{j}} + \frac{\partial E_{ij}}{\partial x_{k}} - \frac{\partial E_{jk}}{\partial x_{i}}, \qquad E_{ij}:=\frac{1}{2} \left (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}\right ). $$

(3.2)

If we now make use of (2.5)₁ to write the partial derivatives in (3.2)₁ in an alternative form, then we conclude that

$$ \Lambda \left ({\mathcal {R}}_{j}\left [ \frac{\partial u_{i}}{\partial x_{k}}\right ]\right ) = \Lambda \bigl({\mathcal {R}}_{j} E_{ik} +{\mathcal {R}}_{k} E_{ij} -{\mathcal {R}}_{i} E_{jk}\bigr). $$

(3.3)

We next compose (3.3) with the linear operator $\Lambda$ to conclude, upon rearranging terms and making use of the identity (2.5)₂,

$$ \triangle \psi = 0, \qquad \psi := \left ({\mathcal {R}}_{j}\left [ \frac{\partial u_{i}}{\partial x_{k}}\right ] -{\mathcal {R}}_{j} E_{ik} -{\mathcal {R}}_{k} E_{ij} +{\mathcal {R}}_{i} E_{jk}\right ). $$

(3.4)

We now note that each term in (3.4)₂ is the Riesz transform of a function in $C^{\infty}_{0}({\mathbb{R}}^{n})$ and hence (see Appendix A) is a bounded function on ${\mathbb{R}^{n}}$. It then follows from Liouville’s theorem (see, e.g., [13, p. 30]) that $\psi \equiv c$, a constant on ${\mathbb{R}^{n}}$. However, each term in (3.4)₂ is the Riesz transform of a function that is contained in ${\mathcal {L}}^{p}({\mathbb{R}^{n}})$, $1< p < \infty$, and hence is itself contained in ${\mathcal {L}}^{p}({\mathbb{R}^{n}})$. Therefore, $\psi =c\in {\mathcal {L}}^{p}({\mathbb{R}^{n}})$ and so $\psi = 0$. Thus,

$$ {\mathcal {R}}_{j}\left (\frac{\partial u_{i}}{\partial x_{k}}\right ) = {\mathcal {R}}_{j} E_{ik} +{\mathcal {R}}_{k} E_{ij} -{\mathcal {R}}_{i} E_{jk} $$

(3.5)

and hence, upon taking the Riesz transform, ${\mathcal {R}}_{j}$, of (3.5) we find that

$$ {\mathcal {R}}_{j}^{2}\left (\frac{\partial u_{i}}{\partial x_{k}} \right ) = {\mathcal {R}}_{j}^{2} E_{ik} +{\mathcal {R}}_{j}{\mathcal {R}}_{k} E_{ij} -{\mathcal {R}}_{j}{\mathcal {R}}_{i} E_{jk}. $$

(3.6)

We next sum (3.6) over $j$ to conclude, with the aid of the identity (2.3), that

$$ \frac{\partial u_{i}}{\partial x_{k}} = E_{ik} -\sum _{j=1}^{n}\bigl[{ \mathcal {R}}_{j}{\mathcal {R}}_{k} E_{ij} - {\mathcal {R}}_{j}{\mathcal {R}}_{i} E_{jk}\bigr]. $$

(3.7)

We then take the ${\mathcal {H}}^{1}$-norm of both sides of (3.7) to conclude with the aid of the triangle inequality and the fact that the Riesz transforms are bounded on ${\mathcal {H}}^{1}({\mathbb{R}^{n}})$, (2.8),

$$ \begin{aligned} \left \|\frac{\partial u_{i}}{\partial x_{k}}\right \|_{{\mathcal {H}}} & \le \left \|E_{ik}\right \|_{{\mathcal {H}}} +\sum _{j=1}^{n}\bigl( \left \|{\mathcal {R}}_{j}{\mathcal {R}}_{k} E_{ij}\right \|_{{\mathcal {H}}} + \left \|{\mathcal {R}}_{j}{\mathcal {R}}_{i} E_{jk}\right \|_{{ \mathcal {H}}}\bigr) \\ &\le \left \|E_{ik}\right \|_{{\mathcal {H}}} +\rho \sum _{j=1}^{n} \left \|{\mathcal {R}}_{k} E_{ij}\right \|_{{\mathcal {H}}} +\rho \sum _{j=1}^{n} \left \|{\mathcal {R}}_{i} E_{jk}\right \|_{{\mathcal {H}}} \\ &\le \left \|E_{ik}\right \|_{{\mathcal {H}}} +\rho ^{2}\sum _{j=1}^{n} \left \|E_{ij}\right \|_{{\mathcal {H}}} +\rho ^{2}\sum _{j=1}^{n} \left \|E_{jk}\right \|_{{\mathcal {H}}}. \end{aligned} $$

Finally, we sum over $i$ and $k$ to get (see (2.10))

$$ \left \|\nabla {\mathbf {u}}\right \|_{{\mathcal {H}}} = \sum _{i,k=1}^{n} \left \|\frac{\partial u_{i}}{\partial x_{k}}\right \|_{{\mathcal {H}}} \le (1+2n\rho ^{2})\left \|{\mathbf {E}}\right \|_{{\mathcal {H}}}, \quad {\mathbf {E}}=\tfrac{1}{2}\left [\nabla {\mathbf {u}}+(\nabla { \mathbf {u}})^{{\mathrm {T}}}\right ] $$

for all ${\mathbf {u}}\in C^{\infty}_{0}({\Omega };{\mathbb{R}^{n}})$. Since such ${\mathbf {u}}$ are dense in $W^{1,{\mathcal {H}}}_{0}({\Omega };{\mathbb{R}}^{n})$, this is the desired result. □

4 Korn’s Inequality in Other Spaces

In this section we note that our proof of Korn’s inequality in a Hardy-Sobolev space yields a similar result in other spaces upon which the Riesz transforms are bounded linear operators. The proof is the same as the one we have given, modulo a change in norm/seminorm.

Theorem 2

Let ${\mathbf {X}}={\mathbf {X}}({\mathbb{R}^{n}})$, $n\ge 2$, be a Banach space with seminorm (or norm) ${[\hspace{- 0.3 ex}] \cdot [\hspace{- 0.3 ex}]}_{{\mathbf {X}}({\mathbb{R}^{n}})}$. Suppose the Riesz transforms are bounded linear operators on ${\mathbf {X}}$ with respect to this seminorm, i.e., there exists a constant $\rho =\rho (n)>0$ such that for all $\varphi \in {\mathbf {X}}({\mathbb{R}^{n}})$

$$ {[\hspace{- 0.3 ex}] {\mathcal {R}}_{i} \varphi [\hspace{- 0.3 ex}]}_{{ \mathbf {X}}({\mathbb{R}^{n}})} \le \rho (n) {[\hspace{- 0.3 ex}] \varphi [ \hspace{ - 0.3 ex}]}_{{\mathbf {X}}({\mathbb{R}^{n}})}\ \textit{ for $i=1,2,\ldots ,n$}. $$

(4.1)

Then there exists ${\mathcal {K}}={\mathcal {K}}(n)>0$ such that, for any nonempty open set ${\Omega }\subset {\mathbb{R}}^{n}$ and every ${\mathbf {u}}\in C^{\infty}_{0}({\Omega };{\mathbb{R}}^{n})$,

$$ {[\hspace{- 0.3 ex}] \nabla{\mathbf {u}} [\hspace{- 0.3 ex}]}_{{\mathbf {X}}({ \mathbb{R}^{n}})} \le {\mathcal {K}}(n) {[\hspace{- 0.3 ex}] {\mathbf {E}} [ \hspace{ - 0.3 ex}]}_{{\mathbf {X}}({\mathbb{R}^{n}})}. $$

A standard result (see, e.g., [20, Corollary 6.7.2]) yields the required bound, (4.1), in certain Besov spaces and, in particular the special case of the Hölder spaces ${\mathbf {X}}=C^{0,\alpha}({\mathbb{R}^{n}})$, $\alpha \in (0,1)$ (see, e.g., Remark 6.5.2 and Definition 6.3.4 in [20]).

Corollary 1

Fix $n\ge 2$ and $\alpha \in (0,1)$. Then there exists a constant ${\mathcal {K}}={\mathcal {K}}(n,\alpha )>0$ such that, for any nonempty open set ${\Omega }\subset {\mathbb{R}}^{n}$ and every ${\mathbf {u}}\in C_{0}^{1,\alpha}({\Omega };{\mathbb{R}^{n}})$,

$$ {[\hspace{- 0.3 ex}] \nabla{\mathbf {u}} [\hspace{- 0.3 ex}]}_{C^{0,\alpha}({ \mathbb{R}^{n}})} \le {\mathcal {K}}(n,\alpha ) {[\hspace{- 0.3 ex}] { \mathbf {E}} [\hspace{- 0.3 ex}]}_{C^{0,\alpha}({\mathbb{R}^{n}})}. $$

Here

$$ \begin{gathered} {[\hspace{- 0.3 ex}] \varphi [\hspace{- 0.3 ex}]}_{C^{0,\alpha}({\mathbb{R}^{n}})} = \sup _{ \substack{{\mathbf {x}},{\mathbf {z}}\in {\mathbb{R}^{n}}\\{\mathbf {x}}\ne {\mathbf {z}}}} \frac{|\varphi ({\mathbf {x}})-\varphi ({\mathbf {z}})|}{|{\mathbf {x}}-{\mathbf {z}}|^{\alpha}}, \\ {[\hspace{- 0.3 ex}] {\mathbf {F}} [\hspace{- 0.3 ex}]}_{C^{0,\alpha}({ \mathbb{R}^{n}})} = \max _{1\le i,j\le n} {[\hspace{- 0.3 ex}] F_{ij} [ \hspace{ - 0.3 ex}]}_{C^{0,\alpha}({\mathbb{R}^{n}})} \ \textit{ for ${\mathbf {F}}\in C^{0,\alpha}({\mathbb{R}^{n}};{\mathbb{M}}^{n\times n})$}, \\ C_{0}^{1,\alpha}({\Omega };{\mathbb{R}^{n}}):= \textit{ the closure of $C^{\infty}_{0}({\Omega };{\mathbb{R}^{n}})$ in $C^{1,\alpha}({\Omega };{\mathbb{R}^{n}})$}, \end{gathered} $$

and a function defined on ${\Omega }$ is assumed to be extended to ${\mathbb{R}^{n}}\backslash {\Omega }$ as the zero function (see (2.1)).

Remark 4.1

The Riesz transforms are known ([15, Corollary 2]) to be bounded linear operators on $\operatorname{BMO}({\mathbb{R}}^{n})$. However, the closure of $C^{\infty}_{0}({\mathbb{R}^{n}})$ in $\operatorname{BMO}({\mathbb{R}}^{n})$ in the $\operatorname{BMO}$-seminorm is the space $\operatorname{VMO}({\mathbb{R}}^{n})$ of Sarason [34] ($\operatorname{VMO}({\mathbb{R}}^{n})\subsetneqq \operatorname{BMO}({\mathbb{R}}^{n})$). Thus, Theorem 2 is valid for all ${\mathbf {u}}\in W^{1,1}({\Omega };{\mathbb{R}^{n}})$ with $\nabla {\mathbf {u}}\in \operatorname{VMO}_{z}({\Omega };{\mathbb{M}}^{n\times n})$ and ${\mathbf {X}}=\operatorname{BMO}({\mathbb{R}}^{n};{\mathbb{M}}^{n\times n})$. However, for bounded regions ${\Omega }$, Theorem 3.3 in [35] yields a stronger result: for every ${\mathbf {u}}\in W_{\operatorname{loc}}^{1,1}({\Omega };{\mathbb{R}}^{n})$ with $\nabla {\mathbf {u}}\in \operatorname{BMO}({\Omega };{\mathbb{M}}^{n\times n})$,

$$ {[\hspace{- 0.3 ex}] \nabla{\mathbf {u}} [\hspace{- 0.3 ex}]}_{\operatorname{BMO}({\Omega })} \le {\mathcal {K}}{[\hspace{- 0.3 ex}] {\mathbf {E}} [\hspace{- 0.3 ex}]}_{ \operatorname{BMO}({\Omega })}. $$

5 Discussion; Korn’s Second Inequality

Our main result is, essentially, a proof of Korn’s inequality in a Hardy space when zero displacements are prescribed on the entire boundary of the body. When the Hardy space, ${\mathcal {H}}={\mathcal {H}}^{1}_{z}({\Omega })$, is replaced by one of the standard ${\mathcal {L}}^{p}$-spaces, $1< p<\infty$, many other versions of Korn’s inequality are known (see, e.g., [16, §12], [21, p. 38], and [32]). In particular, suppose that $p=2$, ${\Omega }$ is (strongly) Lipschitz³, and that the boundary of ${\Omega }$ satisfies $\partial {\Omega }= \overline{{\mathcal {S}}} \cup \overline{{\mathcal {D}}}$, where ${\mathcal {S}}\cap {\mathcal {D}}=\varnothing$ and ${\mathcal {S}}$ and ${\mathcal {D}}$ are relatively open. Then it is well known that there exists a constant ${\mathcal {K}}={\mathcal {K}}(n,{\Omega })$ (or ${\mathcal {K}}={\mathcal {K}}(n,{\Omega },{\mathcal {D}})$) such that, for every ${\mathbf {u}}\in W^{1,2}({\Omega };{\mathbb{R}^{n}})$:

$$ \int _{{\mathcal {B}}}|\nabla {\mathbf {u}}|^{2}\,{\mathrm {d}}{\mathbf {x}} \le {\mathcal {K}}\int _{{\mathcal {B}}}|{\mathbf {E}}|^{2}\,{\mathrm {d}}{ \mathbf {x}}, $$

(5.1)

• (A) provided ${\mathcal {D}}$ is nonempty and ${\mathbf {u}}=\mathbf{0}$ on ${\mathcal {D}}$; or

• (B) provided ${\mathbf {u}}$ satisfies

  $$ \int _{{\Omega }}\bigl[\nabla {\mathbf {u}}-(\nabla {\mathbf {u}})^{{ \mathrm {T}}}\bigr]\,{\mathrm {d}}{\mathbf {x}}=\mathbf{0}. $$

It is also known that (5.1) is not valid⁴ if one replaces the ${\mathcal {L}}^{2}$-norms by the corresponding norms in ${\mathcal {L}}^{1}$. However, we suspect that one might instead replace these norms by a norm for a Hardy space, i.e., estimate (3.1) with $\|\cdot \|_{{\mathcal {H}}}$ replaced by⁵

$$ \|\varphi \|_{{\mathcal {H}}_{r}} := \inf \bigl\{ \|\Phi \|_{{ \mathcal {H}}^{1}({\mathbb{R}^{n}})}: \Phi \in {\mathcal {H}}^{1}({ \mathbb{R}^{n}}),\ \text{ $\Phi =\varphi $ on ${\Omega }$}\bigr\} $$

and for ${\mathbf {u}}$ that satisfies (A) or (B) and $\nabla {\mathbf {u}}$ in an appropriate Hardy space (see, e.g., [2]). A Korn inequality in the spaces ${\mathcal {H}}_{z}^{p}({\Omega })$ and ${\mathcal {H}}_{r}^{p}({\Omega })$, $p\in (0,1)$, is also of interest (see, e.g., [7]).

Finally, we note that there are other similar inequalities, such as a generalized Korn inequality for (so-called) incompatible tensor fields (see, e.g., [19, 29–31]) and geometric rigidity (see, e.g., [9, 11, 18]) that are valid for $1< p<\infty$, but not for $p=1$ or $p=\infty$. We are curious if there are versions of such inequalities that are valid in ${\mathcal {H}}^{1}$ and its dual space $\operatorname{BMO}$.

Appendices

Appendix A: The Riesz Transform of a Smooth Function with Compact Support

We herein show that the Riesz transform of a smooth function with compact support is a bounded function on ${\mathbb{R}^{n}}$. Fix $\varphi \in C^{\infty}_{0}({\mathbb{R}}^{n})$. Then there exists an $R>1$ such that $\varphi ({\mathbf {y}})=0$ for all ${\mathbf {y}}$ with $|{\mathbf {y}}|\ge R$.

1.1 A.1 The Local Behavior

We first bound $({\mathcal {R}}_{i} \varphi )({\mathbf {x}})$ when $|{\mathbf {x}}|\le 2R$. Then $|{\mathbf {y}}-{\mathbf {x}}| \le |{\mathbf {y}}| +|{\mathbf {x}}| \le 2R+ |{ \mathbf {y}}|$. Now, consider

$$ ({\mathcal {R}}^{\varepsilon}_{i} \varphi )({\mathbf {x}}) :=\int _{ \varepsilon < |{\mathbf {y}}-{\mathbf {x}}|} \frac{(x_{i}-y_{i})\varphi ({\mathbf {y}})}{|{\mathbf {x}}-{\mathbf {y}}|^{n+1}} \,{\mathrm {d}}{\mathbf {y}}= \int _{\varepsilon < |{\mathbf {y}}-{\mathbf {x}}|< 3R} \frac{(x_{i}-y_{i})\varphi ({\mathbf {y}})}{|{\mathbf {x}}-{\mathbf {y}}|^{n+1}} \,{\mathrm {d}}{\mathbf {y}}, $$

since $\varphi ({\mathbf {y}})=0$ when $|{\mathbf {y}}|\ge R$. We write $\varphi ({\mathbf {y}})=\varphi ({\mathbf {x}})+[\varphi ({\mathbf {y}})- \varphi ({\mathbf {x}})]$ and make use of the mean-value theorem to get $|\varphi ({\mathbf {x}})-\varphi ({\mathbf {y}})|\le \sup |\nabla \varphi | |{\mathbf {x}}-{\mathbf {y}}|$ and hence

$$ \left |({\mathcal {R}}^{\varepsilon}_{i} \varphi )({\mathbf {x}})\right | \le |\varphi ({\mathbf {x}})| \left |\int _{\varepsilon < |{\mathbf {z}}|< 3R} \frac{z_{i}}{|{\mathbf {z}}|^{n+1}}\,{\mathrm {d}}{\mathbf {z}}\right | + \sup |\nabla \varphi | \int _{\varepsilon < |{\mathbf {z}}|< 3R} \frac{1}{|{\mathbf {z}}|^{n-1}}\,{\mathrm {d}}{\mathbf {z}}, $$

(A.1)

where we have made use of the change of variables ${\mathbf {z}}={\mathbf {x}}-{\mathbf {y}}$. However, the first integral on the right-hand side of (A.1) is zero by symmetry, while a straightforward computation shows that the second integral on the right-hand side of (A.1) is equal to $n\omega _{n}[3R-\varepsilon ]$. Thus, in view of (2.2) and (A.1), the Riesz transforms satisfy

$$ \left |({\mathcal {R}}_{i} \varphi )({\mathbf {x}})\right | \le 3Rnc_{n} \omega _{n} \sup _{{\mathbf {y}}\in {\mathbb{R}^{n}}}|\nabla \varphi ({ \mathbf {y}})|, \quad \text{when}\quad |{\mathbf {x}}|\le 2R. $$

1.2 A.2 The Riesz Transform at Infinity

In this subsection we analyze the behavior at infinity of ${\mathcal {R}}_{i} \varphi$. Recall that $\varphi ({\mathbf {y}})=0$ when $|{\mathbf {y}}|\ge R$ and suppose that $|{\mathbf {x}}|>2R$. Consider

$$ ({\mathcal {R}}^{L}_{i} \varphi )({\mathbf {x}}) :=\int _{|{\mathbf {y}}-{ \mathbf {x}}|>1} \frac{(x_{i}-y_{i})\varphi ({\mathbf {y}})}{|{\mathbf {x}}-{\mathbf {y}}|^{n+1}} \,{\mathrm {d}}{\mathbf {y}}= \int _{|{\mathbf {y}}|< R} \frac{(x_{i}-y_{i})\varphi ({\mathbf {y}})}{|{\mathbf {x}}-{\mathbf {y}}|^{n+1}} \,{\mathrm {d}}{\mathbf {y}}. $$

Thus,

$$ \big|({\mathcal {R}}^{L}_{i} \varphi )({\mathbf {x}})\big| \le \sup | \varphi |\int _{|{\mathbf {y}}|< R} \frac{1}{|{\mathbf {x}}-{\mathbf {y}}|^{n}}\,{\mathrm {d}}{\mathbf {y}}. $$

However, for $|{\mathbf {x}}|>2R$ and $|{\mathbf {y}}|< R$, it follows that $|{\mathbf {x}}-{\mathbf {y}}|>|{\mathbf {x}}|-R>R$. Therefore, for such ${\mathbf {x}}$,

$$ \big|({\mathcal {R}}^{L}_{i} \varphi )({\mathbf {x}})\big| \le \sup | \varphi |\int _{|{\mathbf {y}}|< R} \frac{1}{\left (|{\mathbf {x}}|-R\right )^{n}}\,{\mathrm {d}}{\mathbf {y}}= \omega _{n} \frac{R^{n}}{\left (|{\mathbf {x}}|-R\right )^{n}}\sup | \varphi |, $$

(A.2)

where $\omega _{n}$ denotes the volume of the unit ball in ${\mathbb{R}^{n}}$.

Next, suppose that $|{\mathbf {x}}-{\mathbf {y}}|<1<R$. Then, since $|{\mathbf {x}}|>2R$, it follows that

$$ 2R < |{\mathbf {x}}| = |{\mathbf {x}}-{\mathbf {y}}+{\mathbf {y}}| \le |{ \mathbf {x}}-{\mathbf {y}}| +|{\mathbf {y}}| < R + |{\mathbf {y}}| $$

and hence $|{\mathbf {y}}|>R$. Thus, $\varphi ({\mathbf {y}})=0$ for such ${\mathbf {y}}$. Consequently,

$$ \int _{\varepsilon < |{\mathbf {y}}-{\mathbf {x}}|< 1} \frac{(x_{i}-y_{i})\varphi ({\mathbf {y}})}{|{\mathbf {x}}-{\mathbf {y}}|^{n+1}} \,{\mathrm {d}}{\mathbf {y}}=0, $$

which when combined with (A.2) and the definition of the Riesz transform (2.2) yields

$$ \left |({\mathcal {R}}_{i} \varphi )({\mathbf {x}})\right | \le \frac{c_{n}\omega _{n} R^{n}}{\left (|{\mathbf {x}}|-R\right )^{n}} \sup |\varphi | \le c_{n}\omega _{n} \sup _{{\mathbf {y}}\in {\mathbb{R}^{n}}}| \varphi ({\mathbf {y}})|, \quad \text{ when} \quad |{\mathbf {x}}|>2R. $$

Appendix B: Other Useful Identities

The paper by Calderón and Zygmund [4] has a number of other identities that the reader may find to be of interest. Let $\varphi \in {\mathcal {L}}^{p}({\mathbb{R}^{n}})$, $1< p<\infty$. Then ${\mathcal {R}}_{i}({\mathcal {R}}_{k}\varphi )={\mathcal {R}}_{k}({ \mathcal {R}}_{i}\varphi )$. Moreover, if $\varphi \in W^{1,p}({\mathbb{R}^{n}})$, then

$$ \frac{\partial}{\partial x_{k}}\left ({\mathcal {R}}_{i}\varphi \right ) ={\mathcal {R}}_{i}\left ( \frac{\partial \varphi}{\partial x_{k}} \right ). $$

Define the linear operator $\Lambda :W^{1,p}({\mathbb{R}^{n}})\to {\mathcal {L}}^{p}({\mathbb{R}^{n}})$, $1< p<\infty$, by (2.4). Then

$$ \Lambda \varphi = \sum _{j=1}^{n} \left [ \frac{\partial}{\partial x_{j}} \left ({\mathcal {R}}_{j}\varphi \right )\right ], \qquad \frac{\partial \varphi}{\partial x_{j}} = { \mathcal {R}}_{j}(\Lambda \varphi ) = \Lambda ({\mathcal {R}}_{j}\varphi ). $$