Characterizations of anisotropic high order Sobolev spaces

Abstract

We establish two types of characterizations for high order anisotropic Sobolev spaces. In particular, we prove high order anisotropic versions of Bourgain–Brezis–Mironescu’s formula and Nguyen’s formula.

Keywords

High order Sobolev spaces asymptotic characterization anisotropic Sobolev spaces

1. Introduction

The celebrated Bourgain–Brezis–Mironescu formula, appeared for the first time in [5,6], and provided a new characterization for functions in the Sobolev space $W^{1, p} (R^{N})$ , with $1 < p < \infty$ . More precisely, they proved:

Theorem A (Bourgain, Brezis and Mironescu, [5]).

Let $g \in L^{p} (R^{N}), 1 < p < \infty$ . Then $g \in W^{1, p} (R^{N})$ iff $\begin{matrix} \int_{R^{N}} \int_{R^{N}} \frac{| g (x) - g (y) |^{p}}{| x - y |^{p}} ρ_{n} (| x - y |) d x d y ⩽ C, \forall n ⩾ 1, \end{matrix}$ for some constant $C > 0$ . Moreover, $\begin{matrix} lim_{n \to \infty} \int_{R^{N}} \int_{R^{N}} \frac{| g (x) - g (y) |^{p}}{| x - y |^{p}} ρ_{n} (| x - y |) d x d y = K_{N, p} \int_{R^{N}} {| \nabla g (x) |}^{p} d x . \end{matrix}$ Here $\begin{array}{l} (1.1) & K_{N, p} = \int_{S^{N - 1}} | e \cdot σ |^{p} d σ \end{array}$ for any $e \in S^{N - 1}$ and $d σ$ is the surface measure on $S^{N - 1}$ . Here ${(ρ_{n})}_{n \in N}$ is a sequence of nonnegative radial mollifiers satisfying $\begin{array}{l} lim_{n \to \infty} \int_{τ}^{\infty} ρ_{n} (r) r^{N - 1} d r = 0 \forall τ > 0, lim_{n \to \infty} \int_{0}^{\infty} ρ_{n} (r) r^{N - 1} d r = 1 . \end{array}$

Starting from the previous result and since the theory of Sobolev spaces is a fundamental tool in many branches of modern mathematics, such as harmonic analysis, complex analysis, differential geometry and geometric analysis, partial differential equations, etc, there has been a substantial effort to characterize Sobolev spaces in different settings (see e.g., [1,2,8–12,14,16,20,22–25]).

Theorem A has been extended to the high order case by Bojarski, Ihnatsyeva and Kinnunen [3] using the high order Taylor remainder and by Borghol [4] using high order differences.

We note here, as a consequence of Theorem A, that we can characterize the Sobolev space $W^{1, p} (R^{N})$ as follows: Let $g \in L^{p} (R^{N}), 1 < p < \infty$ . Then $g \in W^{1, p} (R^{N})$ iff $\begin{matrix} (1.2) & sup_{0 < δ < 1} \underset{| x - y | < δ}{\int_{R^{N}} \int_{R^{N}}} \frac{| g (x) - g (y) |^{p}}{δ^{N + p}} d x d y < \infty . \end{matrix}$

Recently, Nguyen [17] (see also [18]), motivated by an estimate for the topological degree for the Gizburg–Landau equation ([7]), established some new characterizations of the Sobolev space $W^{1, p} (R^{N})$ which are closely related to Theorem A. More precisely, he used the dual form of (1.2) and proved the following results:

Theorem B (H.M. Nguyen, [17]).

Let $1 < p < \infty$ . Then the following hold:

(a) Let $g \in W^{1, p} (R^{N})$ . Then there exists a positive constant $C_{N, p}$ depending only on N and p such that $\begin{matrix} \underset{| g (x) - g (y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{| x - y |^{N + p}} d x d y ⩽ C_{N, p} \int_{R^{N}} {| \nabla g (x) |}^{p} d x, \forall δ > 0, \forall g \in W^{1, p} (R^{N}) . \end{matrix}$

(b) If $g \in L^{p} (R^{N})$ satisfies $\begin{matrix} sup_{0 < δ < 1} \underset{| g (x) - g (y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{| x - y |^{N + p}} d x d y < \infty, \end{matrix}$ then $g \in W^{1, p} (R^{N})$ .

(c) For any $g \in W^{1, p} (R^{N})$ , $\begin{matrix} lim_{δ \to 0} \underset{| g (x) - g (y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{| x - y |^{N + p}} d x d y = \frac{1}{p} K_{N, p} \int_{R^{N}} {| \nabla g (x) |}^{p} d x, \end{matrix}$ where $K_{N, p}$ is as in ( 1.1 ).

The previous result has been generalized in many ways and for different spaces (see e.g. [11,13,19,21]). We recall in particular the folowing result proved in [21]:

Theorem C (H.M. Nguyen, M. Squassina [21]).

Let $1 < p < \infty$ and $K \subset R^{N}$ be a convex, symmetric set containing the origin and with nonempty interior. Then, for every $g \in W_{K}^{1, p} (R^{N})$ , $\begin{matrix} lim_{δ \to 0} \underset{| g (x) - g (y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + p}} d x d y = \int_{R^{N}} ‖ \nabla g ‖_{Z_{p}^{*} K}^{p} d x, \end{matrix}$ where $‖ \cdot ‖_{K}$ is the norm in $R^{N}$ which admits as unit ball the set K, i.e. $‖ x ‖_{K} : = inf {λ > 0 | \frac{x}{λ} \in K}$ , $‖ \cdot ‖_{Z_{p}^{*} K}$ is the norm associated with the $L_{p}$ polar body of K, namely $\begin{matrix} (1.3) & ‖ v ‖_{Z_{p}^{*} K} = {(\frac{N + p}{p} \int_{K} | v \cdot x |^{p} d x)}^{1 / p}, v \in R^{N} \end{matrix}$ and $W_{K}^{1, p} (R^{N})$ is the associated Sobolev space.

The main purpose of this paper is to generalize Theorem A and Theorem C to high-order anisotropic Sobolev spaces. In order to describe our main results we recall the following notation ([4]): Let $f \in W^{k, p} (Ω)$ and $σ = (σ_{1}, \dots, σ_{N}) \in R^{N}$ , we denote $\begin{matrix} D^{k} f (x) (σ, \dots, σ) = \sum_{1 ⩽ i_{1}, \dots, i_{k} ⩽ N} σ_{i_{1}} \dots σ_{i_{k}} \frac{\partial^{k} f}{\partial x_{i_{1}} \dots \partial x_{i_{k}}} (x), a.e. x \in Ω \end{matrix}$ and, for every $m \in N$ $\begin{array}{l} (1.4) & R^{m} f (x, y) = \sum_{j = 0}^{m} {(- 1)}^{j} (\binom{m}{j}) f (\frac{m - j}{m} x + \frac{j}{m} y) . \end{array}$

Theorem 1.1.
Let $K \subset R^{N}$ be a convex, symmetric set containing the origin and with nonempty interior. Let $f \in W^{m, p} (R^{N})$ with $m \in N$ and $1 δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y = \frac{N + m p}{m^{m p + 1} p} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{array}$

Notice that taking $m = 1$ in the previous theorem we get Theorem C and taking $m = 2$ and $‖ \cdot ‖$ as the Euclidean norm we get [12, Theorem 1.1].

Our next result is the analogous of [4, Theorem 4] in our setting.
Theorem 1.2.
Let $K \subset R^{N}$ be a convex, symmetric set containing the origin with nonempty interior. Let ${(ρ_{ε})}_{ε}$ be a family of functions $ρ_{ε} : [0, \infty) \to [0, \infty)$ satisfying the following conditions $\begin{array}{l} (1.6) & \int_{0}^{\infty} r^{N - 1} ρ_{ε} (r) d r = 1 and lim_{ε \to 0} \int_{δ}^{\infty} r^{N - 1} ρ_{ε} (r) d r = 0 \forall δ > 0 \end{array}$ Let $f \in W^{m, p} (R^{N})$ with $m \in N$ and $1 < p < \infty$ , then $\begin{array}{l} lim_{ε \to 0} \int_{R^{N}} \int_{R^{N}} \frac{| R^{m} f (x, y) |^{p}}{‖ x - y ‖_{K}^{m p}} ρ_{ε} (‖ x - y ‖_{K}) d x d y \\ (1.7) & = \frac{N + m p}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{array}$

Our next result results can be considered a generalization to high-order anisotropic spaces of [12, Theorem 1.2]. For any $f \in W^{m, p} (R^{N})$ and $y \in R^{N}$ let $\begin{matrix} T_{y}^{m - 1} f (x) = \sum_{| α | ⩽ m - 1} D^{α} f (y) \frac{{(x - y)}^{α}}{α!} \end{matrix}$ and $\begin{matrix} R_{m - 1} f (x, y) = f (x) - T_{y}^{m - 1} f (x) . \end{matrix}$ Then we will prove the following result
Theorem 1.3.
Let $K \subset R^{N}$ be a convex, symmetric set containing the origin, with nonempty interior and $f \in W^{m, p} (R^{N})$ with $1 δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y = \frac{N + m p}{{(m!)}^{p} m p} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{matrix}$
Theorem 1.4.
Let $K \subset R^{N}$ be a convex, symmetric set containing the origin with nonempty interior and $f \in W^{m, p} (R^{N})$ with $1 < p < \infty$ and $m \in N$ . Then $\begin{matrix} lim_{ε \to 0} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x, y) |^{p}}{‖ x - y ‖_{K}^{m p}} ρ_{ε} (‖ x - y ‖_{K}) d x d y = \frac{N + m p}{{(m!)}^{p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{matrix}$ Here the family ${(ρ_{ε})}_{ε}$ is as in Theorem 1.2 .

The plan of the paper is the following: In Section 2, we will introduce some terminology. In Section 3 we will prove Theorems 1.2 and 1.1. Finally, in Section 4, we will establish Theorems 1.3 and 1.4.
2. Anisotropic spaces

Let $| x |$ be the Euclidean norm of $x \in R^{N}$ and let us fix a convex, symmetric subset $K \subset R^{N}$ containing the origin with nonempty interior. For a multi-index $α = (α_{1} \dots, α_{N})$ , $α_{i} ⩾ 0$ and a point $x = (x_{1}, \dots, x_{N}) \in R^{N}$ we denote by $\begin{matrix} x^{α} = \prod_{i = 1}^{N} x_{i}^{α_{i}} and | α | = \sum_{i = 1}^{N} α_{i} . \end{matrix}$ In the same way $\begin{matrix} D^{α} u = \frac{\partial^{| α |} u}{\partial_{x_{1}}^{α_{1}} \dots \partial_{x_{N}}^{α_{N}}} \end{matrix}$ is a weak partial derivative of order u. We also denote by $\nabla^{m} u$ a vector with components $D^{α} u$ , $| α | = m$ . As in Theorem C, we denote by $‖ \cdot ‖_{K}$ be the norm $‖ x ‖_{K} = inf {λ > 0 | x / λ \in K}$ and we observe that since all norms on $R^{N}$ are equivalent, there are $A, B > 0$ such that $\begin{matrix} (2.1) & A | \cdot | ⩽ ‖ \cdot ‖_{K} ⩽ B | \cdot | \end{matrix}$ Now let $y \in K$ and denote by E the vector with components $E_{α} = (1 / α!) y_{1}^{α_{1}} \cdot y_{N}^{α_{n}}$ , $| α | = m$ . Then, for every $m \in N$ and $1 ⩽ p < \infty$ the function $\begin{matrix} ‖ v ‖_{Z_{p, m}^{*} K} = {(\frac{N + m p}{m^{m p + 1} p} \int_{K} | v \cdot E |^{p} d y)}^{1 / p} \end{matrix}$ is a norm on a linear space of all vectors $v = {(v_{α})}_{| α | = m}$ [3]. Set $\begin{matrix} {[u]}_{W_{K}^{m, p} (R^{N})} = {(\int_{R^{N}} {‖ \nabla^{m} u ‖}_{Z_{p, m}^{*} K}^{p} d x)}^{\frac{1}{p}} . \end{matrix}$ Fix $1 ⩽ p < \infty$ and $m \in N$ , we denote by $W_{K}^{m, p} (R^{N})$ the space of $u \in L^{p} (R^{N})$ such that ${[u]}_{W_{K}^{m, p} (R^{N})} < \infty$ endowed with the norm $\begin{matrix} ‖ u ‖_{W_{K}^{m, p} (R^{N})} = {(‖ u ‖_{L^{p} (R^{N})} + {[u]}_{W_{K}^{m, p} (R^{N})})}^{\frac{1}{p}} . \end{matrix}$ Clearly for $m = 1$ the space defined above coincides with the one studied in [21], moreover taking $K = B (0, 1)$ it is easy to see that $W_{K}^{m, p} (R^{N}) = {\dot{W}}^{m, p} (R^{N})$ , here ${\dot{W}}^{m, p} (R^{N})$ denotes the homogenous Sobolev space as defined in [15, Definition 11.17].

We will use the following notation: Given two quantities f and g we write $f ≲ g$ if there exists $C > 0$ such that $f ⩽ C g$ .

3. Proofs of Theorems 1.1 and 1.2

Let $m ⩾ 1$ . Set $Δ_{h} f (x) = Δ_{h}^{1} f (x) : = f (x + h) - f (x)$ , we call m-th difference the quantity $Δ_{h}^{m} f (x) = Δ_{h} [Δ_{h}^{m - 1} f] (x)$ . By above definition, it is not difficult to show that for any positive integer m, we have $\begin{matrix} Δ_{h}^{m} f (x) = \sum_{j = 0}^{m} {(- 1)}^{m + j} (\begin{matrix} m \\ j \end{matrix}) f (x + j h) \end{matrix}$ and, by [4, Lemma 8], we also have $\begin{matrix} (3.1) & Δ_{h}^{m} f (x) = \int_{{[0, 1]}^{m}} D^{m} f (x + \sum_{j = 1}^{m} t_{j} h) (h, \dots, h) d t_{1} \dots d t_{m} \end{matrix}$ where ${[0, 1]}^{m}$ denotes the unit-cube in $R^{m}$ . Finally, it is easy to see that for every $x, h \in R^{N}$ $\begin{matrix} R^{m} f (x, x + m h) = {(- 1)}^{m} Δ_{h}^{m} f (x) \end{matrix}$ where $R^{m} f (x, y)$ is as in (1.4).

3.1. Nguyen’s formula

The aim of this section is to prove Theorem 1.1. We start with the following:

Lemma 3.1.
Let $K \subset R^{N}$ be a convex, symmetric set containing the origin with nonempty interior and $f \in W^{m, p} (R^{N})$ with $1 0$ s.t. $\begin{matrix} \underset{| R^{m} f (x, y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y ⩽ C {‖ \nabla^{m} f ‖}_{L^{p} (R^{N})}^{p} . \end{matrix}$
Proof.
Using polar coordinates ( $y = x + t σ$ , $σ = \frac{y - x}{| y - x |}$ and $t = | x - y |$ ) we write $\begin{matrix} \underset{| R^{m} f (x, y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y = \int_{S^{N - 1}} \underset{| Δ_{\frac{t}{m} σ}^{m} f (x) | > δ}{\int_{R^{N}} \int_{R^{+}}} \frac{δ^{p}}{‖ σ ‖_{K}^{N + m p} t^{1 + m p}} d t d x d σ \end{matrix}$ Thus, since $A ⩽ ‖ σ ‖_{K} ⩽ B$ , it is enough to show that there exists a constant $C = C (m, N, p) > 0$ such that for every $σ \in S^{N - 1}$ $\begin{array}{l} (3.2) & \underset{| Δ_{\frac{t}{m} σ}^{m} f (x) | > δ}{\int_{R^{N}} \int_{R^{+}}} \frac{δ^{p}}{t^{1 + m p}} d t d x ⩽ C \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x . \end{array}$ We assume, without loss of generality, that $σ = e_{N} = (0, \dots, 0, 1)$ .1
¹
Fix $σ \in S^{N - 1}$ and let $B \in SO (N)$ such that $B σ = e_{N}$ . Then, by a change of variables and (3.2) $\begin{array}{l} \int_{S^{N - 1}} \underset{| Δ_{\frac{t}{m} σ}^{m} f (x) | > δ}{\int_{R^{N}} \int_{R^{+}}} \frac{δ^{p}}{t^{1 + m p}} d t d x d σ & = \int_{S^{N - 1}} \underset{| Δ_{\frac{t}{m} e_{N}}^{m} f (x) | > δ}{\int_{R^{N}} \int_{R^{+}}} \frac{δ^{p}}{t^{1 + m p}} d t d x d σ \\ ⩽ C | S^{N - 1} | \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x . \end{array}$

By (3.1), we have for any $x = (x^{'}, x_{N}) \in R^{N - 1} \times R$ $\begin{array}{l} | Δ_{\frac{t}{m} e_{N}}^{m} f (x) | & ≲ t^{m} | \int_{{[0, 1]}^{m}} \partial_{x_{N}}^{m} f (x^{'}, x_{N} + \frac{t}{m} \sum_{j = 1}^{m} s_{j}) d s_{1} \dots d s_{m} | \\ ≲ t^{m - 1} \int_{x_{N}}^{x_{N} + t} | \partial_{x_{N}}^{m} f (x^{'}, s) | d s \\ ≲ t^{m} M_{N} (\partial_{x_{N}}^{m} f) (x^{'}, x_{N}) \end{array}$ where $M_{N} (f)$ denotes the maximal function of f in direction $x_{N}$ , namely $\begin{matrix} M_{N} (f) (x) = M_{N} (f) (x^{'}, x_{N}) = sup_{t > 0} \frac{1}{t} \int_{x_{N}}^{x_{N} + t} | f (x^{'}, s) | d s . \end{matrix}$ So, there exists $C = C (N, m, p) > 0$ such that $\begin{array}{l} \underset{| Δ_{\frac{t}{m} σ}^{m} f (x) | > δ}{\int_{R^{N}} \int_{R^{+}}} \frac{δ^{p}}{t^{1 + m p}} d t d x & ⩽ \int_{R^{N}} \int_{0}^{\infty} 1_{C t^{m} M_{N} (\partial_{x_{N}}^{m} f) (x) > δ} \frac{δ^{p}}{t^{1 + m p}} d t d x \\ ≲ \int_{R^{N}} M_{N} (\partial_{x_{N}}^{m} f) {(x)}^{p} d x \\ ≲ \int_{R^{N}} {| \partial_{x_{N}}^{m} f (x) |}^{p} d x \\ ≲ {‖ \nabla^{m} f ‖}_{L^{p} (R^{N})}^{p} \end{array}$ where in the line before the last we used (see [26]) $\begin{matrix} \int_{R^{N - 1}} \int_{R} {| M_{N} (\partial_{x_{N}}^{m} f) (x^{'}, x_{N}) |}^{p} d x_{N} d x^{'} ≲ \int_{R^{N - 1}} \int_{R} {| \partial_{x_{N}}^{m} f (x^{'}, x_{N}) |}^{p} d x_{N} d x^{'} . \end{matrix}$ This gives the conclusion. □

We are now in position to prove Theorem 1.1. Proof of Theorem 1.1.
Notice that (1.5) can be re-written as: $\begin{matrix} lim_{δ \to 0} \underset{| R^{m} f (x, y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y = \int_{R^{N}} {‖ \nabla^{m} f ‖}_{Z_{p, m}^{*} K}^{p} d x . \end{matrix}$ By changing variables (writing $y = x + \sqrt[m]{δ} h σ$ , $σ = \frac{y - x}{| y - x |}$ , $h = \frac{1}{\sqrt[m]{δ}} | y - x |$ ), we obtain $\begin{matrix} \underset{| R^{m} f (x, y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y = \underset{| \frac{Δ_{\frac{\sqrt[m]{δ} h σ}{m}}^{m} f (x)}{h^{m} δ} | h^{m} > 1}{\int_{S^{N - 1}} \int_{R^{N}}} \int_{0}^{\infty} \frac{1}{‖ σ ‖_{K}^{N + m p} h^{1 + m p}} d h d x d σ \end{matrix}$ By Lemma 3.1 there exists $C = C (m, N, p) > 0$ such that for every $σ \in S^{N - 1}$ , $\begin{matrix} (3.3) & \underset{| \frac{Δ_{\frac{\sqrt[m]{δ} h σ}{m}}^{m} f (x)}{h^{m} δ} | h^{m} > 1}{\int_{R^{N}} \int_{0}^{\infty}} \frac{1}{h^{m p + 1}} d h d x ⩽ C \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x . \end{matrix}$ Moreover, for every $σ \in S^{N - 1}$ it holds $\begin{matrix} (3.4) & lim_{δ \to 0} \underset{| \frac{Δ_{\frac{\sqrt[m]{δ} h σ}{m}}^{m} f (x)}{h^{m} δ} | h^{m} > 1}{\int_{R^{N}} \int_{0}^{\infty}} \frac{1}{h^{m p + 1}} d h d x = \frac{1}{m^{m p + 1} p} \int_{R^{N}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d x . \end{matrix}$

To prove (3.4), we define $F_{δ} : S^{N - 1} \to R$ by $\begin{matrix} F_{δ} (σ) : = \frac{1}{‖ σ ‖_{K}^{N + m p}} \underset{| \frac{Δ_{\frac{\sqrt[m]{δ}}{m} h σ}^{m} f (x)}{h^{m} δ} | h^{m} > 1}{\int_{R^{N}} \int_{0}^{\infty}} \frac{1}{h^{m p + 1}} d h d x . \end{matrix}$ By Lemma 3.1 there exists $C = C (N, p, m) > 0$ such that for all $σ \in S^{N - 1}$ and for all $δ > 0$ : $\begin{matrix} (3.5) & F_{δ} (σ) ⩽ \frac{C}{A^{N + m p}} \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x . \end{matrix}$

Without loss of generality, we suppose that $σ = e_{N}$ . Given $x^{'} \in R^{N - 1}$ and $δ \in (0, 1)$ we define $\begin{matrix} A (x^{'}, δ) : = {(x_{N}, h) \in R \times (0, \infty) | | \frac{Δ_{\frac{\sqrt[m]{δ}}{m} h e_{N}}^{m} f (x^{'}, x_{N})}{h^{m} δ} | h^{m} > 1}, \end{matrix}$ and $\begin{matrix} A (x^{'}) : = {(x_{N}, h) \in R \times (0, \infty) | | \partial_{x_{N}}^{m} f (x^{'}, x_{N}) | h^{m} > m^{m}} . \end{matrix}$ Using (3.1) it is easy to see that $\begin{matrix} lim_{δ \to 0} \frac{1}{h^{m p + 1}} 1_{A (x^{'}, δ)} (x_{N}, h) = \frac{1}{h^{m p + 1}} 1_{A (x^{'})} (x_{N}, h) a.e. (x^{'}, x_{N}, h) \in R^{N - 1} \times R \times [0, \infty), \end{matrix}$ and $\begin{matrix} \frac{1}{h^{m p + 1}} 1_{A (x^{'}, δ)} (x_{N}, h) ⩽ \frac{1}{h^{m p + 1}} 1_{{h^{m} M_{N} (\partial_{x_{N}}^{m} f) (x) > m^{m}}} (x, h) \in L^{1} (R^{N} \times [0, \infty)) . \end{matrix}$ By the Lebesgue dominated convergence theorem, we get (3.4). Using (3.5) and the Lebesgue dominated convergence theorem again, we can conclude that $\begin{array}{l} lim_{δ \to 0} \underset{| R^{m} f (x, y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y & = \int_{S^{N - 1}} \frac{1}{m^{m p + 1} p} \frac{1}{‖ σ ‖_{K}^{N + m p}} \int_{R^{N}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d x d σ \\ = \frac{1}{m^{m p + 1} p} \int_{R^{N}} \int_{S^{N - 1}} \frac{1}{‖ σ ‖_{K}^{N + m p}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d σ d x . \end{array}$ Now, notice that $\begin{array}{l} \int_{S^{N - 1}} \frac{1}{‖ σ ‖_{K}^{N + m p}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d σ \\ = (N + m p) \int_{S^{N - 1}} \int_{0}^{\frac{1}{‖ σ ‖_{K}}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} r^{N + m p - 1} d r d σ \\ = (N + m p) \int_{S^{N - 1}} \int_{0}^{\frac{1}{‖ σ ‖_{K}}} {| D^{m} f (x) (r σ, \dots, r σ) |}^{p} r^{N - 1} d r d σ \\ (3.6) & = (N + m p) \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y \end{array}$ where in the last equality we used the fact that $K = {y \in R^{N} | ‖ y ‖_{K} ⩽ 1}$ and the conclusion follows. □
Remark 3.1.
We explicitly note that Theorem 1.1 generalizes some already known results: for example taking $m = 1$ and $K = {x \in R^{n} | | x | ⩽ 1}$ we get [17, Lemma 3] and taking $m = 2$ and $K = {x \in R^{n} | | x | ⩽ 1}$ we get [12, Theorem 1.1]. Moreover, Theorem 1.1 generalizes [21, Theorem 1.1] in the case where the magnetic field A is zero and $m ⩾ 1$ .

3.2. BBM formula

In this section we prove Theorem 1.2.

Let ρ be a positive real function satisfying (1.6). The following result is proved in [4, Lemma 8]

Lemma 3.2.
Let $f \in W^{m, p} (R^{N})$ with $m ⩾ 2$ and $1 ⩽ p < \infty$ and let $ρ \in L^{1} (R)$ . Then $\begin{matrix} \int_{R^{N}} \int_{R^{N}} {| Δ_{h}^{m} f (x) |}^{p} | h |^{- m p} ρ (| h |) d x d h ⩽ \frac{‖ ρ ‖_{L^{1} (R)}}{| S^{N - 1} |} \int_{R^{N}} (\int_{S^{N - 1}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d σ) d x . \end{matrix}$

The following result is the analogous of [4, Lemma 9] in our setting. Lemma 3.3.
Fix $m \in N$ and $1 < p < \infty$ . If $f \in C_{c}^{m + 1} (R^{N})$ then $\begin{array}{l} lim_{ε \to 0} \int_{R^{N}} \int_{R^{N}} \frac{| R^{m} f (x, y) |^{p}}{‖ x - y ‖_{K}^{m p}} ρ_{ε} (‖ x - y ‖_{K}) d x d y \\ (3.7) & = \frac{(N + m p)}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{array}$
Proof.
Let $S = ‖ f ‖_{W^{m + 1, \infty}}$ . Since $t \to | t |^{p}$ is uniformly continuous in $[0, (m + 1) S]$ then for any $δ > 0$ there exists $C = C (δ) > 0$ such that $\begin{matrix} (3.8) & | | s |^{p} - | t |^{p} | ⩽ C | s - t | + δ \forall s, t \in [0, (m + 1) S] . \end{matrix}$ Using (3.8), (2.1) and proceeding as in [4, Lemma 9] we get $\begin{array}{l} | {| Δ_{h}^{m} f (x) |}^{p} ‖ h ‖_{K}^{- m p} - {| D^{m} f (x) (\frac{h}{‖ h ‖_{K}}, \dots, \frac{h}{‖ h ‖_{K}}) |}^{p} | \\ ⩽ C ‖ h ‖_{K}^{- m} | Δ_{h}^{m} f (x) - D^{m} f (x) (h, \dots, h) | + δ \\ (3.9) & ⩽ C A^{- m - 1} S ‖ h ‖_{K} + δ, \end{array}$ for every $x \in R^{N}$ and for all $h \in R^{N} ∖ {0}$ . In particular, when $h = \frac{y - x}{m}$ , we get $\begin{array}{l} (3.10) & | m^{m p} {| R^{m} f (x, y) |}^{p} ‖ y - x ‖_{K}^{- m p} - {| D^{m} f (x) (\frac{y - x}{‖ y - x ‖_{K}}, \dots, \frac{y - x}{‖ y - x ‖_{K}}) |}^{p} | \\ ⩽ C A^{- m - 1} S ‖ y - x ‖_{K} + δ . \end{array}$ Let $A (x, y) = {(x, y) : at least one of \frac{m - j}{m} x + \frac{j}{m} y, j = 0, \dots, m, is in supp (f)}$ . Then we note that $\begin{matrix} \int_{R^{N}} \int_{R^{N}} \frac{| R^{m} f (x, y) |^{p}}{‖ y - x ‖_{K}^{m p}} ρ_{ε} (‖ y - x ‖_{K}) d x d y = \underset{A (x, y)}{\int_{R^{N}} \int_{R^{N}}} \frac{| R^{m} f (x, y) |^{p}}{‖ y - x ‖_{K}^{m p}} ρ_{ε} (‖ y - x ‖_{K}) d x d y . \end{matrix}$ Using (3.10), we get $\begin{array}{l} \iint_{A (x, y) \cap {| y - x | ⩽ 1}} \frac{m^{m p} | R^{m} f (x, y) |^{p}}{‖ y - x ‖_{K}^{m p}} ρ_{ε} (‖ y - x ‖_{K}) d x d y \\ ⩽ \int_{R^{N}} \int_{R^{N}} {| D^{m} f (x) (\frac{y - x}{‖ y - x ‖_{K}}, \dots, \frac{y - x}{‖ y - x ‖_{K}}) |}^{p} ρ_{ε} (‖ y - x ‖_{K}) d x d y \\ + C A^{- m - 1} S \iint_{A (x, y) \cap {| y - x | ⩽ 1}} ‖ y - x ‖_{K} ρ_{ε} (‖ y - x ‖_{K}) \\ + δ \iint_{A (x, y) \cap {| y - x | ⩽ 1}} ρ_{ε} (‖ y - x ‖_{K}) d x d y \end{array}$ Note that $\begin{array}{l} \iint_{A (x, y) \cap {| y - x | ⩽ 1}} ‖ y - x ‖_{K} ρ_{ε} (‖ y - x ‖_{K}) & = \int_{| h | ⩽ 1} ‖ h ‖_{K} ρ_{ε} (‖ h ‖_{K}) \int_{A (x, x + h)} d x d h \\ ⩽ \int_{| h | ⩽ 1} ‖ h ‖_{K} ρ_{ε} (‖ h ‖_{K}) (m + 1) | supp (f) | d h \\ = (m + 1) supp (f) \int_{S^{N - 1}} ‖ σ ‖_{K}^{- N} d σ \int_{0}^{1} ρ_{ε} (s) s^{N} d s . \end{array}$ By (1.6) we automatically get that [22, Remark 4.3] ${lim}_{ε \to 0} \int_{0}^{1} ρ_{ε} (s) s^{N} d s = 0$ , thus $\begin{matrix} lim_{ε \to 0} \iint_{A (x, y) \cap {| y - x | ⩽ 1}} ‖ y - x ‖_{K} ρ_{ε} (‖ y - x ‖_{K}) = 0 . \end{matrix}$ Similarly, $\begin{matrix} δ \iint_{A (x, y) \cap {| y - x | ⩽ 1}} ρ_{ε} (‖ y - x ‖_{K}) d x d y ≲ δ \end{matrix}$ On the other hand, $\begin{array}{l} \iint_{A (x, y) \cap {| y - x | > 1}} \frac{m^{m p} | R^{m} f (x, y) |^{p}}{‖ y - x ‖_{K}^{m p}} ρ_{ε} (‖ y - x ‖_{K}) d x d y \\ ≲ ‖ f ‖_{p} \int_{| h | > 1} ρ_{ε} (‖ h ‖) d h \to 0 as ε \to 0 . \end{array}$ Hence, by sending $ε \to 0$ and then $δ \to 0$ , we can now conclude that $\begin{array}{l} \underset{ε \to 0}{lim sup} \int_{R^{N}} \int_{R^{N}} \frac{| R^{m} f (x, y) |^{p}}{‖ y - x ‖_{K}^{m p}} ρ_{ε} (‖ y - x ‖_{K}) d x d y \\ ⩽ \frac{1}{m^{m p}} \underset{ε \to 0}{lim sup} \int_{R^{N}} \int_{R^{N}} {| D^{m} f (x) (\frac{y - x}{‖ y - x ‖_{K}}, \dots, \frac{y - x}{‖ y - x ‖_{K}}) |}^{p} ρ_{ε} (‖ y - x ‖_{K}) d x d y . \end{array}$ We next compute the limit of the quantity on the right-hand side. We have $\begin{array}{l} \int_{R^{N}} {| D^{m} f (x) (\frac{x - y}{‖ x - y ‖_{K}}, \dots, \frac{x - y}{‖ x - y ‖_{K}}) |}^{p} ρ_{ε} (‖ x - y ‖_{K}) d y \\ = \int_{S^{N - 1}} \int_{0}^{\infty} {| D^{m} f (x) (σ, \dots, σ) |}^{p} \frac{1}{‖ σ ‖_{K}^{m p}} ρ_{ε} (r ‖ σ ‖_{K}) r^{N - 1} d r d σ \\ = \int_{S^{N - 1}} \frac{1}{‖ σ ‖_{K}^{N + m p}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d σ \\ (3.11) & = (N + m p) \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y, \end{array}$ where the last equality follows from (3.6). Thus, $\begin{array}{l} \underset{ε \to 0}{lim sup} \int_{R^{N}} \int_{R^{N}} \frac{| R^{m} f (x, y) |^{p}}{‖ y - x ‖_{K}^{m p}} ρ_{ε} (‖ y - x ‖_{K}) d x d y \\ ⩽ \frac{N + m p}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{array}$ Let $R > 0$ be such that $supp (f) \subset B_{R}$ , then $\begin{array}{l} \iint_{| y - x | ⩽ 1} {| D^{m} f (x) (\frac{y - x}{‖ y - x ‖_{K}}, \dots, \frac{y - x}{‖ y - x ‖_{K}}) |}^{p} ρ_{ε} (‖ y - x ‖_{K}) d x d y \\ = \underset{| y - x | ⩽ 1}{\int_{B_{R}} \int_{R^{N}}} {| D^{m} f (x) (\frac{y - x}{‖ y - x ‖_{K}}, \dots, \frac{y - x}{‖ y - x ‖_{K}}) |}^{p} ρ_{ε} (‖ y - x ‖_{K}) d y d x \\ ⩽ \int_{R^{N}} \int_{R^{N}} \frac{m^{m p} | R^{m} f (x, y) |^{p}}{‖ y - x ‖_{K}^{m p}} ρ_{ε} (‖ y - x ‖_{K}) d x d y \\ + \underset{| y - x | ⩽ 1}{\int_{B_{R}} \int_{R^{N}}} C_{δ} S ‖ y - x ‖_{K} ρ_{ε} (‖ y - x ‖_{K}) d y d x + \underset{| y - x | ⩽ 1}{\int_{B_{R}} \int_{R^{N}}} ρ_{ε} (‖ y - x ‖_{K}) d x d y \end{array}$ As above $\begin{array}{l} \underset{| y - x | ⩽ 1}{\int_{B_{R}} \int_{R^{N}}} ‖ y - x ‖_{K} ρ_{ε} (‖ y - x ‖_{K}) d y d x \to 0 as ε \to 0, \\ \underset{| y - x | ⩽ 1}{\int_{B_{R}} \int_{R^{N}}} ρ_{ε} (‖ y - x ‖_{K}) d x d y ≲ δ . \end{array}$ By (3.11) $\begin{array}{l} \underset{| y - x | ⩽ 1}{\int_{B_{R}} \int_{R^{N}}} {| D^{m} f (x) (\frac{y - x}{‖ y - x ‖_{K}}, \dots, \frac{y - x}{‖ y - x ‖_{K}}) |}^{p} ρ_{ε} (‖ y - x ‖_{K}) d y d x \\ = (N + m p) \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x \end{array}$ Thus, $\begin{array}{l} \frac{N + m p}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x \\ ⩽ \underset{ε \to 0}{lim sup} \int_{R^{N}} \int_{R^{N}} \frac{| R^{m} f (x, y) |^{p}}{‖ y - x ‖_{K}^{m p}} ρ_{ε} (‖ y - x ‖_{K}) d x d y, \end{array}$ We conclude that $\begin{array}{l} lim_{ε \to 0} \int_{R^{N}} \int_{R^{N}} \frac{| R^{m} f (x, y) |^{p}}{‖ y - x ‖_{K}^{m p}} ρ_{ε} (‖ y - x ‖_{K}) d x d y \\ = \frac{N + m p}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{array}$ □
Proof of Theorem 1.2.
So now we consider $f \in W^{m, p} (R^{N})$ and let $f_{n} \in C_{c}^{\infty} (R^{N})$ such that $f_{n} \to f$ in the $W^{m, p} (R^{N})$ norm. Then one has $\begin{array}{l} | {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} \\ - {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f_{n} (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} | \\ ⩽ {(\int_{R^{N}} \int_{R^{N}} {| R^{m} (f - f_{n}) (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} \\ ≲ {(\int_{R^{N}} (\int_{S^{N - 1}} {| D^{m} (f_{n} - f) (x) (σ, \dots, σ) |}^{p} d σ) d x)}^{\frac{1}{p}} . \end{array}$ Where we used Lemma 3.2 in the last inequality. Thus $\begin{array}{l} {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} \\ = {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f_{n} (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} + o (1) \end{array}$ here $o (1) \to 0$ as $n \to \infty$ uniformly on ε. So fix $ϵ > 0$ , then there exists $n_{0}$ big enough so that for $n ⩾ n_{0}$ , we have $‖ f - f_{n} ‖_{W^{m, p}} < ϵ$ and $\begin{array}{l} | {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} \\ - {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f_{n} (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} | < ϵ . \end{array}$ Then we have $\begin{array}{l} lim_{ε \to 0} | {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} \\ - {(\frac{N + m p}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x)}^{\frac{1}{p}} | \\ ⩽ lim_{ε \to 0} | {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} \\ - {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f_{n} (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} | \\ + lim_{ε \to 0} | {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f_{n} (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} \\ - {(\frac{N + m p}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f_{n} (x) (y, \dots, y) |}^{p} d y d x)}^{\frac{1}{p}} | \\ + | {(\frac{N + m p}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f_{n} (x) (y, \dots, y) |}^{p} d y d x)}^{\frac{1}{p}} \\ - {(\frac{N + m p}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x)}^{\frac{1}{p}} | \\ ≲ lim_{ε \to 0} | {(\int_{R^{N}} \int_{R^{N}} {| R^{m} f_{n} (x, y) |}^{p} ‖ x - y ‖_{K}^{- m p} ρ_{ε} (‖ x - y ‖_{K}) d y d x)}^{\frac{1}{p}} \\ (3.12) & - {(\frac{N + m p}{m^{m p}} \int_{R^{N}} \int_{K} {| D^{m} f_{n} (x) (y, \dots, y) |}^{p} d y d x)}^{\frac{1}{p}} | + 2 ϵ \end{array}$ So the conclusion follows from Theorem 3.3. □

4. Characterizations of the higher order Sobolev spaces via the Taylor remainder

We recall that $\begin{matrix} T_{y}^{m - 1} f (x) = \sum_{| α | ⩽ m - 1} D^{α} f (y) \frac{{(x - y)}^{α}}{α!} \end{matrix}$ and $\begin{matrix} R_{m - 1} f (x, y) = f (x) - T_{y}^{m - 1} f (x) . \end{matrix}$ Proceeding as in [12] and by an easy induction we get $\begin{matrix} (4.1) & R_{m - 1} f (x, x + h e_{N}) = h^{m} \int_{{[0, 1]}^{m}} \partial_{x_{N}}^{m} f (x^{'}, x_{N} + \prod_{i = 1}^{m} t_{i} h) \prod_{i = 1}^{m} t_{i}^{m - i} d t_{1} \dots d t_{m} \end{matrix}$ thus $\begin{matrix} (4.2) & | R_{m - 1} f (x, x + h e_{N}) | ⩽ \frac{h^{m}}{m!} M_{N} (\partial_{x_{N}}^{m} f) (x) . \end{matrix}$

4.1. Proof of Theorem 1.3

Using (4.2) the proof of the following Lemma is very similar to the one of Lemma 3.1 so we omit it.

Lemma 4.1.
Let $K \subset R^{N}$ be a convex, symmetric set containing the origin with nonempty interior. Let $f \in W^{m, p} (R^{N})$ , $1 0$ such that $\begin{matrix} \underset{| R_{m - 1} f (x, y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y ⩽ C \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x, \forall δ > 0 . \end{matrix}$

We are now ready to prove Theorem 1.3. Proof of Theorem 1.3.
Using a change of variables (writing $y = x + \sqrt[m]{δ} h σ$ , $σ = \frac{y - x}{| y - x |}$ , $h = \frac{1}{\sqrt[m]{δ}} | y - x |$ ), we obtain $\begin{matrix} \underset{| R_{m - 1} f (x, y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y = \underset{| \frac{R_{m - 1} f (x, x + \sqrt[m]{δ} h σ)}{h^{m} δ} | h^{m} > 1}{\int_{S^{N - 1}} \int_{R^{N}} \int_{0}^{\infty}} \frac{1}{‖ σ ‖_{K}^{N + m p}} \frac{1}{h^{m p + 1}} d h d x d σ . \end{matrix}$ We define the auxiliary function $F_{δ} : S^{N - 1} \to R$ by $\begin{matrix} F_{δ} (σ) : = \frac{1}{‖ σ ‖_{K}^{N + m p}} \underset{| \frac{R_{m - 1} f (x, x + \sqrt[m]{δ} h σ)}{h^{m} δ} | h^{m} > 1}{\int_{R^{N}} \int_{0}^{\infty}} \frac{1}{h^{m p + 1}} d h d x . \end{matrix}$ We first prove that for all $σ \in S^{N - 1}$ , $\forall δ > 0$ $\begin{matrix} (4.3) & F_{δ} (σ) ⩽ \frac{1}{A^{N + m p}} C (m, N, p) \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x \end{matrix}$ that is $\begin{matrix} \underset{| \frac{R_{m - 1} f (x, x + \sqrt[m]{δ} h σ)}{h^{m} δ} | h^{m} > 1}{\int_{R^{N}} \int_{0}^{\infty}} \frac{1}{h^{m p + 1}} d h d x ⩽ C (m, N, p) \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x . \end{matrix}$ Without loss of generality, we assume that $σ = e_{N} = (0, \dots, 0, 1)$ . Hence, we need to verify that $\begin{matrix} (4.4) & \underset{| \frac{R_{m - 1} f (x, x + \sqrt[m]{δ} h e_{N})}{h^{m} δ} | h^{m} > 1}{\int_{R^{N}} \int_{0}^{\infty}} \frac{1}{h^{m p + 1}} d h d x ⩽ C (m, N, p) \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x . \end{matrix}$ By (4.2) one has $\begin{matrix} | \frac{R_{m - 1} f (x, x + \sqrt[m]{δ} h e_{N})}{h^{m} δ} | ⩽ \frac{1}{m!} M_{N} (\partial_{x_{N}}^{m} f) (x) \end{matrix}$ and therefore $\begin{array}{l} \underset{| \frac{R_{m - 1} f (x, x + \sqrt[m]{δ} h e_{N})}{h^{m} δ} | h^{m} > 1}{\int_{R^{N}} \int_{0}^{\infty}} \frac{1}{h^{m p + 1}} d h d x & ⩽ \underset{h^{m} M_{N} (\partial_{x_{N}}^{m} f) (x) > m!}{\int_{R^{N}} \int_{0}^{\infty}} \frac{1}{h^{m p + 1}} d h d x \\ ⩽ \frac{{(m!)}^{p}}{m p} \int_{R^{N}} {| M_{N} (\partial_{x_{N}}^{m} f) (x) |}^{p} d x \\ ⩽ C (m, N, p) \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x . \end{array}$ Next we show that $\begin{matrix} (4.5) & F_{δ} (σ) \to \frac{1}{{(m!)}^{p} m p} \frac{1}{‖ σ ‖_{K}^{N + m p}} \int_{R^{N}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d x as δ \to 0 for every σ \in S^{N - 1} . \end{matrix}$ It is enough to show $\begin{matrix} \int_{R^{N}} \int_{0}^{\infty} G_{δ} (x, h) d h d x \to \frac{1}{{(m!)}^{p} m p} \int_{R^{N}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d x as δ \to 0, \end{matrix}$ where $\begin{matrix} G_{δ} (x, h) : = \frac{1}{h^{m p + 1}} 1_{{| \frac{R_{m - 1} f (x, x + \sqrt[m]{δ} h σ)}{h^{m} δ} | h^{m} > 1}} (x, h) . \end{matrix}$ With loss of generality, we suppose that $σ = e_{N} = (0, \dots, 0, 1)$ . Noting that for all $σ \in S^{N - 1}$ : $G_{δ} (x, h) \to \frac{1}{h^{2 p + 1}} 1_{{| D^{m} f (x) (σ, \dots, σ) | h^{m} > m!}}$ as $δ \to 0$ for a.e. $(x, h) \in R^{N} \times [0, \infty)$ , and $\begin{matrix} G_{δ} (x, h) ⩽ \frac{1}{h^{m p + 1}} 1_{{h^{m} M_{N} (\partial_{x_{N}}^{m} f) (x) > m!}} (x, h) \in L^{1} (R^{N} \times [0, \infty)) . \end{matrix}$ Hence, by the Lebesgue dominated convergence theorem, we get (4.5). Once again, by the Lebesgue dominated convergence theorem, we conclude that $\begin{array}{l} lim_{δ \to 0} \underset{| R_{m - 1} f (x, y) | > δ}{\int_{R^{N}} \int_{R^{N}}} \frac{δ^{p}}{‖ x - y ‖_{K}^{N + m p}} d x d y \\ = \int_{S^{N - 1}} \frac{1}{{(m!)}^{p} m p} \frac{1}{‖ σ ‖_{K}^{N + m p}} \int_{R^{N}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d x d σ \\ = \frac{1}{{(m!)}^{p} m p} \int_{R^{N}} \int_{S^{N - 1}} \frac{1}{‖ σ ‖_{K}^{N + m p}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} d σ d x \\ = \frac{N + m p}{{(m!)}^{p} m p} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x \end{array}$ □

4.2. Proof of Theorem 1.4

The mollifiers $ρ_{ε} \in L_{loc}^{1} (0, \infty)$ are nonnegative functions satisfying, $\begin{matrix} \int_{0}^{\infty} ρ_{ε} (r) r^{N - 1} d r = 1 and lim_{ε \to 0} \int_{δ}^{\infty} ρ_{ε} (r) r^{N - 1} d r = 0 for all δ > 0 . \end{matrix}$

Lemma 4.2.

Let $K \subset R^{N}$ be a convex, symmetric set containing the origin with nonempty interior. Let $f \in W^{m, p} (R^{N})$ , $1 0$ such that for all $ε > 0$ the following inequality holds $\begin{matrix} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x, y) |^{p}}{‖ x - y ‖_{K}^{m p}} ρ_{ε} (‖ x - y ‖_{K}) d x d y ⩽ C \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x . \end{matrix}$

Proof.

By density we can assume that $f \in C_{c}^{\infty} (R^{N})$ . We have $\begin{array}{l} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x, y) |^{p}}{‖ x - y ‖_{K}^{m p}} ρ_{ε} (‖ x - y ‖_{K}) d x d y \\ = \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ = \int_{R^{N}} \frac{ρ_{ε} (‖ h ‖_{K})}{‖ h ‖_{K}^{m p}} \int_{R^{N}} {| R_{m - 1} f (x + h, x) |}^{p} d x d h . \end{array}$ Since by [3, (2.17)] there exists $C = C (m, N, p) > 0$ such that $\begin{array}{l} \int_{R^{N}} {| R_{m - 1} f (x + h, x) |}^{p} d x ⩽ C | h |^{m p} \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x, \end{array}$ we have $\begin{matrix} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x, y) |^{p}}{‖ x - y ‖_{K}^{m p}} ρ_{ε} (‖ x - y ‖_{K}) d x d y ⩽ \frac{1}{A^{m p}} C \int_{R^{N}} ρ_{ε} (‖ h ‖_{K}) d h \int_{R^{N}} {| \nabla^{m} f (x) |}^{p} d x . \end{matrix}$ On the other hand $\begin{array}{l} \int_{R^{N}} ρ_{ε} (‖ h ‖_{K}) d h & = \int_{S^{N - 1}} \int_{0}^{\infty} ρ_{ε} (r ‖ σ ‖_{K}) r^{N - 1} d r d σ \\ = \int_{S^{N - 1}} \int_{0}^{\infty} ρ_{ε} (s) {(\frac{s}{‖ σ ‖_{K}})}^{N - 1} \frac{1}{‖ σ ‖_{K}} d s d σ \\ ⩽ \frac{1}{A^{m p}}, \end{array}$ and the conclusion follows. □

Proof of Theorem 1.4.

First, we assume that $f \in C_{c}^{\infty} (R^{N})$ . We have $\begin{matrix} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x, y) |^{p}}{‖ x - y ‖_{K}^{m p}} ρ_{ε} (‖ x - y ‖_{K}) d x d y = \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x, x + h) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \end{matrix}$ By Taylor’s formula, we have that for every $δ > 0$ , there exists $C_{δ} > 0$ such that $\begin{matrix} {| R_{m - 1} f (x + h, x) |}^{p} ⩽ (1 + δ) \frac{1}{{(m!)}^{p}} {| D^{m} f (x) (h, \dots, h) |}^{p} + C_{δ} | h |^{(m + 1) p} . \end{matrix}$ Hence $\begin{array}{l} \underset{| h | ⩽ 1}{\int_{R^{N}} \int_{R^{N}}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ ⩽ \frac{1 + δ}{{(m!)}^{p}} \int_{R^{N}} \int_{R^{N}} {| D^{m} f (x) (h, \dots, h) |}^{p} \frac{ρ_{ε} (‖ h ‖_{K})}{‖ h ‖_{K}^{m p}} d x d h \\ + C_{δ} \underset{| h | ⩽ 1}{\int_{R^{N}}} | h |^{(m + 1) p} \frac{ρ_{ε} (‖ h ‖_{K})}{‖ h ‖_{K}^{m p}} \int_{{x or x + h \in supp (f)}} d x d h \\ ⩽ \frac{1 + δ}{{(m!)}^{p}} \int_{R^{N}} \int_{R^{N}} {| D^{m} f (x) (\frac{h}{| h |}, \dots, \frac{h}{| h |}) |}^{p} | h |^{m p} \frac{ρ_{ε} (‖ h ‖_{K})}{‖ h ‖_{K}^{m p}} d x d h \\ + 2 C_{δ} | supp (f) | \underset{| h | ⩽ 1}{\int_{R^{N}}} | h |^{(m + 1) p} \frac{ρ_{ε} (‖ h ‖_{K})}{‖ h ‖_{K}^{m p}} d h . \end{array}$ By (2.1) and using (1.6) it is easy to see that $\begin{array}{l} lim_{ε \to 0} \underset{| h | ⩽ 1}{\int_{R^{N}}} | h |^{(m + 1) p} \frac{ρ_{ε} (‖ h ‖_{K})}{‖ h ‖_{K}^{m p}} d h = 0 . \end{array}$ On the other hand, $\begin{array}{l} \frac{1}{{(m!)}^{p}} \int_{R^{N}} \int_{R^{N}} {| D^{m} f (x) (\frac{h}{| h |}, \dots, \frac{h}{| h |}) |}^{p} | h |^{m p} \frac{ρ_{ε} (‖ h ‖_{K})}{‖ h ‖_{K}^{m p}} d x d h \\ = \frac{N + m p}{{(m!)}^{p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{array}$ Also, since $f \in C_{c}^{\infty} (R^{N})$ , we have $\begin{array}{l} \underset{| h | ⩾ 1}{\int_{R^{N}} \int_{R^{N}}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h & ≲ \underset{| h | ⩾ 1}{\int_{R^{N}}} ρ_{ε} (‖ h ‖_{K}) d h \\ ≲ \int_{1}^{\infty} ρ_{ε} (s) s^{N - 1} d s \to 0 as ε \to 0 . \end{array}$ Letting $ε \to 0$ and $δ \to 0$ we conclude, $\begin{array}{l} \underset{ε \to 0}{lim sup} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x, y) |^{p}}{‖ x - y ‖_{K}^{m p}} ρ_{ε} (‖ x - y ‖_{K}) d x d y \\ ⩽ \frac{N + m p}{{(m!)}^{p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{array}$ Assume $supp (f) \subset B_{R}$ , then by Taylor’s formula and using the fact that for any $τ > 0$ there exists $C (τ) > 1$ such that for all $a, b \in R$ : $\begin{array}{l} (4.6) & | a |^{p} ⩽ (1 + τ) | b |^{p} + C (τ) | a - b |^{p} . \end{array}$ we get $\begin{matrix} \frac{1}{{(m!)}^{p}} {| D^{m} f (x) (h, \dots, h) |}^{p} ⩽ (1 + δ) {| R_{m - 1} f (x + h, x) |}^{p} + C_{δ, B} | h |^{(m + 1) p} . \end{matrix}$ Hence, $\begin{array}{l} \frac{1}{{(m!)}^{p}} \int_{B} \int_{| h | ⩽ 1} \frac{| D^{m} f (x) (h, \dots, h) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ ⩽ (1 + δ) \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ + C_{δ, B} \int_{B} \int_{| h | ⩽ 1} | h |^{(m + 1) p} \frac{ρ_{ε} (‖ h ‖_{K})}{‖ h ‖_{K}^{m p}} d x d h . \end{array}$ Proceeding as in (3.11) we get $\begin{array}{l} \frac{1}{{(m!)}^{p}} \int_{B} \int_{| h | ⩽ 1} {| D^{m} f (x) (\frac{h}{| h |}, \dots, \frac{h}{| h |}) |}^{p} | h |^{m p} \frac{ρ_{ε} (‖ h ‖_{K})}{‖ h ‖_{K}^{m p}} d x d h \\ = \frac{1}{{(m!)}^{p}} \int_{B} \int_{S^{N - 1}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} \frac{1}{‖ σ ‖_{K}^{N + m p}} d σ d x \end{array}$ By letting $ε \to 0$ , and then $δ \to 0$ , we get $\begin{array}{l} \frac{1}{{(m!)}^{p}} \int_{B} \int_{S^{N - 1}} {| D^{m} f (x) (σ, \dots, σ) |}^{p} \frac{1}{‖ σ ‖_{K}^{N + m p}} d σ d x \\ ⩽ \underset{ε \to 0}{lim inf} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h, \end{array}$ this proves the thesis if $f \in C_{c}^{\infty} (R^{N})$ .

In the general case $f \in W^{m, p} (R^{N})$ , we fix $τ > 0$ and let $C (τ) > 1$ be as in (4.6). By density, we can choose $g \in C_{c}^{\infty} (R^{N})$ such that $\begin{matrix} \int_{R^{N}} {| \nabla^{m} (f - g) (x) |}^{p} d x ⩽ \frac{τ}{C (τ)} \end{matrix}$ and $\begin{matrix} | \int_{R^{N}} \int_{K} {| D^{m} g (x) (y, \dots, y) |}^{p} d y d x - \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x | ⩽ τ . \end{matrix}$

Then $\begin{array}{l} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ ⩽ (1 + τ) \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} g (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ + C (τ) \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} (f - g) (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ ⩽ (1 + τ) \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} g (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ + C (τ) C (m, N, p) \int_{R^{N}} {| \nabla^{m} (f - g) (x) |}^{p} d x . \end{array}$ Letting $ε \to 0$ , we obtain $\begin{array}{l} \underset{ε \to 0}{lim sup} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ ⩽ (1 + τ) \frac{N + m p}{{(m!)}^{p}} \int_{R^{N}} \int_{K} {| D^{m} g (x) (y, \dots, y) |}^{p} d y d x + C (m, N, p) τ \\ ⩽ (1 + τ) \frac{N + m p}{{(m!)}^{p}} [\int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x + τ] + C (m, N, p) τ . \end{array}$ Since τ can be chosen arbitrarily, we deduce that $\begin{array}{l} \underset{ε \to 0}{lim sup} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ ⩽ \frac{N + m p}{{(m!)}^{p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{array}$ Also, if we switch the role of f and g in the above argument, then we get $\begin{array}{l} \frac{N + m p}{{(m!)}^{p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x \\ ⩽ \underset{ε \to 0}{lim inf} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h . \end{array}$ Hence, $\begin{array}{l} lim_{ε \to 0} \int_{R^{N}} \int_{R^{N}} \frac{| R_{m - 1} f (x + h, x) |^{p}}{‖ h ‖_{K}^{m p}} ρ_{ε} (‖ h ‖_{K}) d x d h \\ = \frac{N + m p}{{(m!)}^{p}} \int_{R^{N}} \int_{K} {| D^{m} f (x) (y, \dots, y) |}^{p} d y d x . \end{array}$ □

Footnotes

Acknowledgements

The authors would like to thank Quoc-Hung Nguyen and Professor Hoai-Minh Nguyen for their interest in our work and for stimulating discussions during the preparation of the manuscript. A.P. is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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