Abstract
We continue the study of the fractional variation following the distributional approach developed in the previous works Bruè et al. (2021), Comi and Stefani (2019), Comi and Stefani (2019). We provide a general analysis of the distributional space \(BV^{\alpha ,p}({\mathbb {R}}^n)\) of \(L^p\) functions, with \(p\in [1,+\infty ]\), possessing finite fractional variation of order \(\alpha \in (0,1)\). Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a \(BV^{\alpha ,p}\) function.
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1 Introduction
1.1 The fractional variation
For a parameter \(\alpha \in (0,1)\) and an exponent \(p\in [1,+\infty ]\), the space of \(L^p\) functions with bounded fractional variation is
where
is the (total) fractional variation of the function \(f\in L^p({\mathbb {R}}^n)\). Here and in the following, for sufficiently smooth functions and vector-fields, we let
and
be the fractional gradient and the fractional divergence operators respectively, where \(\mu _{n,\alpha }\) is a suitable renormalizing constant depending on n and \(\alpha \) only. The above fractional operators are dual, in the sense that
The fractional variation was considered by the first and the third authors in the work [7] in the geometric framework \(p=1\), also in relation with the naturally associated notion of fractional Caccioppoli perimeter. The fractional variation of an \(L^p\) function for an arbitrary exponent \(p\in [1,+\infty ]\) was then studied by the same authors in the subsequent paper [8], in connection with some embedding-type results arising from some optimal inequalities proved by the second author [31, 32].
Since the first appearance of the fractional gradient [16], the literature around \(\nabla ^\alpha \) and \(\mathrm{div}^\alpha \) has been rapidly growing in various directions, such as the study of PDEs [25, 26, 28, 29] and of functionals [4, 5, 17] involving these fractional operators, the discovery of new optimal embedding estimates [27, 31, 32] and the development of a distributional and asymptotic analysis in this fractional framework [6,7,8, 30]. We also refer the reader to the survey [33] and to the monograph [24].
At the present stage of the theory, the fine properties of functions having finite fractional variation are not completely understood and, to our knowledge, only some results [7] in the geometric regime \(p=1\) are available in the literature.
Besides providing a general treatment of the space \(BV^{\alpha ,p}({\mathbb {R}}^n)\), in the present paper we aim to develop the existing theory in this direction. On the one side, we study the relation between the fractional variation and the Hausdorff measure. On the other side, we establish the existence of the precise representative of a \(BV^{\alpha ,p}\) function.
1.2 The Hausdorff dimension of the fractional variation
The natural idea behind the definition of the space \(BV^{\alpha ,p}({\mathbb {R}}^n)\) is that a function \(f\in L^p({\mathbb {R}}^n)\) belongs to \(BV^{\alpha ,p}({\mathbb {R}}^n)\) if and only if there exists a finite vector-valued Radon measure \(D^\alpha f\in {\mathscr {M}}({\mathbb {R}}^n;{\mathbb {R}}^n)\) such that
for all \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\), generalizing the integration-by-parts formula (1.3).
In the classical integer case \(\alpha =1\), the variation of a function \(f\in BV({\mathbb {R}}^n)\) is known to satisfy
where \({\mathscr {H}}^{{s}}\) is the s-dimensional Hausdorff measure. If \(f=\chi _E\) for some measurable set \(E\subset {\mathbb {R}}^n\), then it actually holds that
where \({\mathscr {F}}E\) is the De Giorgi reduced boundary of E, see the monographs [3, 20].
Roughly speaking, formulas (1.4) and (1.5) mean that the variation measure of a BV function in \({\mathbb {R}}^n\) lives on sets with Hausdorff dimension \(n-1\) at least. By the analogy between the integer and the fractional settings, one may expect that a similar phenomenon should occur also for the fractional variation of order \(\alpha \in (0,1)\) on a set of Hausdorff dimension \(n-\alpha \) at least. In [7, Corollary 5.4], the first and the third authors confirmed this parallelism by showing that, for a measurable set \(E\subset {\mathbb {R}}^n\) such that \(\chi _E \in BV^{\alpha }({\mathbb {R}}^n)\) (or, more generally, for any measurable set having locally finite fractional Caccioppoli perimeter, see [7, Definition 4.1]), it holds that
where \(c_{n,\alpha }>0\) depends on n and \(\alpha \) only and \({\mathscr {F}}^\alpha E\) is the fractional analogue of the De Giorgi reduced boundary (1.5), the so-called fractional reduced boundary of E, see [7, Definition 4.7]. However, as shown in [7, Lemma 3.28] by the same authors, if \(f\in BV^\alpha ({\mathbb {R}}^n)\) then the function \(u=I_{1-\alpha }f\) (where \(I_s\) is the Riesz potential of order \(s\in (0,n)\), see below for the precise definition) does satisfy \(|Du|({\mathbb {R}}^n)<+\infty \), with
In particular, by combining (1.4) with the above (1.7), we immediately get that
for all \(f\in BV^\alpha ({\mathbb {R}}^n)\), thus ruling out the existence of a coarea formula in this fractional setting, see [7, Corollary 5.6].
Equations (1.6) and (1.8) illustrate the richness arising from the innocent-looking definition (1.2) and lead to the idea that the behavior of the fractional variation of a function \(f\in L^p({\mathbb {R}}^n)\) may depend on its integrability exponent \(p\in [1,+\infty ]\). Our first main result provides a rigorous formulation of this intuitive idea and can be stated as follows.
Theorem 1
(Absolute continuity properties of the fractional variation) Let \(\alpha \in (0,1)\), \(p\in [1,+\infty ]\) and assume that \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\). We have the following cases:
-
(i)
if \(p\in \left[ 1,\frac{n}{1-\alpha }\right) \), then \(|D^\alpha f|\ll {\mathscr {H}}^{{n-1}}\);
-
(ii)
if \(p\in \left[ \frac{n}{1-\alpha },+\infty \right] \), then \(|D^\alpha f|\ll {\mathscr {H}}^{{n-\alpha -\frac{n}{p}}}\).
As shown by Theorem 1, the fractional variation in the subcritical regime \(p<\frac{n}{1-\alpha }\) is comparable with the Hausdorff measure of dimension \(n-1\), in accordance with (1.8). In fact, we can actually prove a deeper property, in analogy with the relation (1.7). Precisely, the Riesz potential operator
is continuous whenever \(p<\frac{n}{1-\alpha }\) (see Proposition 4(i) below for the detailed statement), from which item (i) in Theorem 1 immediately follows. Here and in the following, for any \(p\in [1,+\infty ]\), we let
be the space of \(L^p\) functions having finite variation, extending the definition in (1.1) to the integer case \(\alpha =1\).
In the supercritical regime \(p\ge \frac{n}{1-\alpha }\) instead, the fractional variation is comparable with the Hausdorff measure of dimension \(n-\alpha -\frac{n}{p}\), thus recovering (1.6) in the case \(p=+\infty \). The proof of item (ii) of Theorem 1 is more delicate and requires a finer analysis. The overall idea is to adapt the strategy developed in [7, Section 5] for sets with (locally) finite fractional Caccioppoli perimeter to the present more general \(L^p\) framework. The key role in this approach is played by the following decay estimate for the fractional variation of a function \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\) with \(p\ge \frac{n}{n-\alpha }\),
valid for \(|D^\alpha f|\)-a.e. \(x\in {\mathbb {R}}^n\) and all \(r>0\) sufficiently small, where \(c_{n,\alpha ,p}>0\) is a constant depending on n, \(\alpha \), and p only (see Theorem 10 below for the precise statement). The validity of (1.9) is suggested by the following heuristic argument, valid for all \(f \in BV^{\alpha , p}({\mathbb {R}}^n)\) such that
If \(\varphi \in C^{\infty }_{c}(B_2)\) is such that \(\varphi \ge 0\) and \(\varphi \equiv 1\) on \(B_1\), then
thanks to (1.3) and the \(\alpha \)-homogeneity of the fractional gradient ([30, Theorem 4.3]), so that
which gives (1.9). Without (1.10), the decay estimate (1.9) is a consequence of some new integrability properties in Lorentz spaces of the fractional gradient and of an integration-by-parts formula of \(BV^{\alpha ,p}\) functions on balls which may be of some independent interest (see Theorems 8 and 9, respectively).
We note that Theorem 1 still holds even in the limit as \(\alpha \rightarrow 1^-\). Indeed, for all \(p \in [1, +\infty ]\) and \(f \in BV^{1,p}({\mathbb {R}}^n)\) we get that \(|D f|\ll {\mathscr {H}}^{{n-1}}\), since point (i) now applies to all \(p \in [1, +\infty )\), while point (ii) refers only to \(p=+\infty \), for which we have \(n-1-\frac{n}{p} = n - 1\). This is in fact a well-known result for functions in \(BV_{\mathrm{loc}} ({\mathbb {R}}^n)\), see [3, Lemma 3.76] for instance. On the contrary, Theorem 1 is not optimal in the limit as \(\alpha \rightarrow 0^+\). Indeed, in virtue of [6, Theorem 3.3 and Remark A.3], if \(p \in [1, +\infty )\) and \(f \in BV^{0,p}({\mathbb {R}}^n)\), then \(|D^0 f| \ll {\mathscr {L}}^{{n}}\) (where the space \(BV^{0,p}({\mathbb {R}}^n)\) is defined as in (1.1) with \(\alpha = 0\), see [6] for a more detailed presentation).
1.3 The precise representative of a \(BV^{\alpha ,p}\) function
Formulas (1.4) and (1.5) suggest that the set of discontinuity points (in the measure-theoretical sense) of a BV function should have Hausdorff dimension \(n-1\). In more precise terms, if \(f\in BV({\mathbb {R}}^n)\), then the limit
exists for \({\mathscr {H}}^{{n-1}}\)-a.e. \(x\in {\mathbb {R}}^n\). In fact, the limit in (1.11) can be strengthened as
for \({\mathscr {H}}^{{n-1}}\)-a.e. \(x\in {\mathbb {R}}^n\setminus J_f\), see [11, Section 5.9] for example, where \(J_f\subset {\mathbb {R}}^n\) is the so-called jump set of the function \(f\in BV({\mathbb {R}}^n)\) (if \(f\in W^{1,1}({\mathbb {R}}^n)\), then \(J_f\) is empty).
The function \(f^{\star }\) defined by (1.11) is the so-called precise representative of the function f (by convention, we set \(f^*(x)=0\) if the limit in (1.11) does not exist). The well-posedness of the precise representative (1.11) of a \(BV^{\alpha ,p}\) function is not known at the present moment. Our second main result moves in this direction and can be briefly stated as follows (for a more precise statement, we refer the reader to Corollary 5 below).
Theorem 2
(The precise representative of a \(BV^{\alpha ,p}\) function) Let \(\alpha \in (0,1)\), \(p\in [1,+\infty ]\) and \(\varepsilon >0\). If \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\), then the limit \(f^{\star }(x)\) exists for \({\mathscr {H}}^{{n - \alpha + \varepsilon }}\)-a.e. \(x\in {\mathbb {R}}\). Moreover, for any such point \(x\in {\mathbb {R}}^n\), it holds that
for any \(q\in \left[ 1,\bar{q}_\varepsilon \right] \), where \(\bar{q}_\varepsilon \in \left[ 1,\frac{n}{n-\alpha }\right) \) is such that \(\lim \limits _{\varepsilon \rightarrow 0^+}\bar{q}_\varepsilon =\frac{n}{n-\alpha }\).
The idea behind the proof of Theorem 2 is very simple and relies on three ingredients naturally arising from our general investigation of the \(BV^{\alpha ,p}({\mathbb {R}}^n)\) space. First, we show that \(C^\infty _c\) functions are dense in energy in \(BV^{\alpha ,p}({\mathbb {R}}^n)\) provided that \(p\in \left[ 1,\frac{n}{n-\alpha }\right) \), extending the approximation [7, Theorem 3.8] already proved by the first and the third author in the geometric regime \(p=1\). Second, by combining this approximation with an optimal embedding inequality [32] due to the second author, we establish a fractional analogue of the Gagliardo–Nirenberg–Sobolev inequality, that is, \(BV^{\alpha ,p}({\mathbb {R}}^n)\subset L^{\frac{n}{n-\alpha }}({\mathbb {R}}^n)\) with continuous inclusion. Third, we exploit this fractional embedding inequality to prove the continuous inclusion of \(BV^{\alpha ,p}({\mathbb {R}}^n)\) into some Bessel potential space of suitable fractional order. At this point, the existence of the precise representative of a \(BV^{\alpha ,p}\) function for \(p<\frac{n}{n-\alpha }\) can be inferred from the known theory of Bessel potential spaces, see [1, Section 6.1] for example. The remaining exponents \(p\ge \frac{n}{n-\alpha }\) can be recovered from the previous analysis by a simple cut-off argument that may be of some separate interest (see Lemma 1 for the detailed statement).
1.4 Future developments
Generally speaking, the precise representative of a function turns out to be the correct object when dealing with the product between the function itself and a sufficiently well-behaved measure.
For example, the precise representative allows to state the general Leibniz rule for the product of two BV functions. Precisely, if \(f,g\in BV({\mathbb {R}}^n)\cap L^\infty ({\mathbb {R}}^n)\), then \(fg\in BV({\mathbb {R}}^n)\) with
Note that the two products appearing in right-hand side of (1.12) are well posed thanks to the combination of the absolute continuity property of the variation (1.4) and the existence of the precise representative (1.11).
With Theorems 1 and 2 at hand, the analysis developed in the present work naturally leads to study the interactions between the fractional variation measure and the precise representative of \(BV^{\alpha ,p}\) functions, aiming at a more general formulation of the Leibniz rule and of the Gauss–Green formula in this fractional setting. These results are the main topic of the subsequent paper [9].
1.5 Organization of the paper
The paper is organized as follows.
In Sect. 2, we quickly set up the notation used throughout the entire work and recall the elementary features of the fractional operators involved.
In Sect. 3, we carry out the general analysis of the \(BV^{\alpha ,p}({\mathbb {R}}^n)\) space. On the one side, we deal with the approximation in energy by smooth functions and the consequent embedding theorems in Lebesgue and Bessel potential spaces, preparing the ground for the proof of Theorem 2. On the other side, we treat some integration-by-parts formulas of \(BV^{\alpha ,p}\) functions against rough test vector-fields and on balls, developing the tools needed for the proof of the decay estimate (1.9) and thus of Theorem 1.
In Sect. 4, we prove our first main result Theorem 1. We divide the proof into two parts, dealing with the subcritical regime (i) and the supercritical regime (ii) separately, see Proposition 4(i) and Corollary 3 respectively. At the end of this section, we provide two examples to show the sharpness of our result in the one-dimensional case \(n=1\).
In Sect. 5, after having recalled some known properties of the fractional capacity in Bessel potential spaces and having proved a localization lemma for \(BV^{\alpha ,p}\) functions, we end our paper with the proof of our second main result Theorem 2.
2 Preliminaries
2.1 General notation
We start with a brief description of the main notation used in this paper. In order to keep the exposition the most reader-friendly as possible, we retain the same notation adopted in the previous works [6,7,8].
Given an open set \(\varOmega \subset {\mathbb {R}}^n\), we say that a set E is compactly contained in \(\varOmega \), and we write \(E\Subset \varOmega \), if the \(\overline{E}\) is compact and contained in \(\varOmega \). We let \({\mathscr {L}}^{{n}}\) and \({\mathscr {H}}^{{\alpha }}\) be the n-dimensional Lebesgue measure and the \(\alpha \)-dimensional Hausdorff measure on \({\mathbb {R}}^n\), respectively, with \(\alpha \in [0,n]\). Unless otherwise stated, a measurable set is a \({\mathscr {L}}^{{n}}\)-measurable set. We also use the notation \(|E|={\mathscr {L}}^{{n}}(E)\). All functions we consider in this paper are Lebesgue measurable, unless otherwise stated. We denote by \(B_r(x)\) the standard open Euclidean ball with center \(x\in {\mathbb {R}}^n\) and radius \(r>0\). We let \(B_r=B_r(0)\). For all \(\beta > 0\), we set \(\omega _{\beta } =\pi ^{\frac{\beta }{2}}/\varGamma \left( \frac{\beta +2}{2}\right) \), where \(\varGamma \) is Euler’s Gamma function, and we recall that \(|B_1| = \omega _{n}\) and \({\mathscr {H}}^{{n-1}}(\partial B_{1}) = n \omega _n\).
For \(k \in {\mathbb {N}}_{0} \cup \left\{ {+ \infty }\right\} \) and \(m \in {\mathbb {N}}\), we let \(C^{k}_{c}(\varOmega ; {\mathbb {R}}^{m})\) and \({{\,\mathrm{Lip}\,}}_c(\varOmega ; {\mathbb {R}}^{m})\) be the spaces of \(C^{k}\)-regular and, respectively, Lipschitz-regular, m-vector-valued functions defined on \({\mathbb {R}}^n\) with compact support in the open set \(\varOmega \subset {\mathbb {R}}^n\).
For \(m\in {\mathbb {N}}\), the total variation on \(\varOmega \) of the m-vector-valued Radon measure \(\mu \) is defined as
We thus let \({\mathscr {M}}(\varOmega ;{\mathbb {R}}^m)\) be the space of m-vector-valued Radon measure with finite total variation on \(\varOmega \). We say that \((\mu _k)_{k\in {\mathbb {N}}}\subset {\mathscr {M}}(\varOmega ;{\mathbb {R}}^m)\) weakly converges to \(\mu \in {\mathscr {M}}(\varOmega ;{\mathbb {R}}^m)\), and we write \(\mu _k\rightharpoonup \mu \) in \({\mathscr {M}}(\varOmega ;{\mathbb {R}}^m)\) as \(k\rightarrow +\infty \), if
for all \(\varphi \in C_c^0(\varOmega ;{\mathbb {R}}^m)\). Note that we make a little abuse of terminology, since the limit in (2.1) actually defines the weak*-convergence in \({\mathscr {M}}(\varOmega ;{\mathbb {R}}^m)\).
For any exponent \(p\in [1,+\infty ]\), we let \(L^p(\varOmega ;{\mathbb {R}}^m)\) be the space of m-vector-valued Lebesgue p-integrable functions on \(\varOmega \).
We let
be the space of m-vector-valued Sobolev functions on \(\varOmega \), see for instance [18, Chapter 11] for its precise definition and main properties. We also let
We let
be the space of m-vector-valued functions of bounded variation on \(\varOmega \), see for instance [3, Chapter 3] or [11, Chapter 5] for its precise definition and main properties. We also let
For \(\alpha \in (0,1)\) and \(p\in [1,+\infty )\), we let
where
be the space of m-vector-valued fractional Sobolev functions on \(\varOmega \), see [10] for its precise definition and main properties. We also let
For \(\alpha \in (0,1)\) and \(p=+\infty \), we simply let
so that \(W^{\alpha ,\infty }(\varOmega ;{\mathbb {R}}^m)=C^{0,\alpha }_b (\varOmega ;{\mathbb {R}}^m)\), the space of m-vector-valued bounded \(\alpha \)-Hölder continuous functions on \(\varOmega \).
In order to avoid heavy notation, if the elements of a function space \(F(\varOmega ;{\mathbb {R}}^m)\) are real-valued (i.e., \(m=1\)), then we will drop the target space and simply write \(F(\varOmega )\).
Given \(\alpha \in (0,n)\), we let
be the Riesz potential of order \(\alpha \) of \(f\in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^m)\). We recall that, if \(\alpha ,\beta \in (0,n)\) satisfy \(\alpha +\beta <n\), then we have the following semigroup property
for all \(f\in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^m)\). In addition, if \(1<p<q<+\infty \) satisfy \(\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}\), then there exists a constant \(C_{n,\alpha ,p}>0\) such that the operator in (2.2) satisfies
for all \(f\in C^\infty _c({\mathbb {R}}^n;\,{\mathbb {R}}^m)\). As a consequence, the operator in (2.2) extends to a linear continuous operator from \(L^p({\mathbb {R}}^n;{\mathbb {R}}^m)\) to \(L^q({\mathbb {R}}^n;{\mathbb {R}}^m)\), for which we retain the same notation. For a proof of (2.3) and (2.4), we refer the reader to [34, Chapter V, Section 1] and to [14, Section 1.2.1].
Given \(\alpha \in (0,1)\), we also let
be the fractional Laplacian (of order \(\alpha \)) of \(f \in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^{n};{\mathbb {R}}^m)\), where
Finally, we let
be the (vector-valued) Riesz transform of a (sufficiently regular) function f. We refer the reader to [14, Sections 2.1 and 2.4.4], [34, Chapter III, Section 1] and [35, Chapter III] for a more detailed exposition. We warn the reader that the definition in (2.6) agrees with the one in [35] and differs from the one in [14, 34] for a minus sign, so that \(R=\nabla I_1\) on \(C^\infty _c({\mathbb {R}}^n)\) in particular. The Riesz transform (2.6) is a singular integral of convolution type, thus in particular it defines a continuous operator \(R:L^p({\mathbb {R}}^n)\rightarrow L^p({\mathbb {R}}^n;{\mathbb {R}}^{n})\) for any given \(p\in (1,+\infty )\), see [13, Corollary 5.2.8]. We also recall that its components \(R_i\) satisfy
see [13, Proposition 5.1.16].
2.2 The operators \(\nabla ^\alpha \) and \(\mathrm{div}^\alpha \)
We briefly recall the definitions and the essential features of the non-local operators \(\nabla ^\alpha \) and \({{\,\mathrm{div}\,}}^\alpha \), see [6,7,8, 31] and [24, Section 15.2].
Let \(\alpha \in (0,1)\) and set
We let
be the fractional \(\alpha \)-gradient of \(f\in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\) and, similarly, we let
be the fractional \(\alpha \)-divergence of \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\). The non-local operators \(\nabla ^\alpha \) and \({{\,\mathrm{div}\,}}^\alpha \) are well defined in the sense that the involved integrals converge and the limits exist. Moreover, since
it is immediate to check that \(\nabla ^{\alpha }c=0\) for all \(c\in {\mathbb {R}}\) and
for all \(f\in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\). Analogously, we have
for all \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\). From the above expressions, it is not difficult to recognize that, given \(f\in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\) and \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\), it holds that
for all \(p\in [1,+\infty ]\), see [7, Corollary 2.3]. Finally, the fractional operators \(\nabla ^\alpha \) and \(\mathrm{div}^\alpha \) are dual, in the sense that
for all \(f\in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\) and \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\), as proved in [30, Section 6] and [7, Lemma 2.5].
With a slight abuse of notation, in the following we let \(\nabla ^1\) and \(\mathrm{div}^1\) be the usual (local) gradient and divergence. Note that this notation is coherent with the asymptotic behavior of the fractional operators \(\nabla ^\alpha \) and \(\mathrm{div}^\alpha \) when \(\alpha \rightarrow 1^-\) for sufficiently regular functions, see the analysis made in [8].
3 The \(BV^{\alpha ,p}({\mathbb {R}}^n)\) space
In this section we study the main properties of the \(BV^{\alpha ,p}\) functions, following the strategy adopted in [7, Section 3].
3.1 Definition of \(BV^{\alpha ,p}({\mathbb {R}}^n)\)
Let \(\alpha \in (0,1]\) and \(p\in [1,+\infty ]\). We say that a function \(f\in L^p({\mathbb {R}}^n)\) belongs to the space \(BV^{\alpha ,p}({\mathbb {R}}^n)\) if \(|D^\alpha f|({\mathbb {R}}^n)<+\infty \), where
see [7, Section 3] for the case \(p=1\) and the discussion in [8, Section 3.3] for the case \(p\in (1,+\infty ]\). In the case \(p=1\), we simply write \(BV^{\alpha ,1}({\mathbb {R}}^n)=BV^\alpha ({\mathbb {R}}^n)\). The resulting linear space
endowed with the norm
is a Banach space and that the fractional variation defined in (3.1) is lower semicontinuous with respect to the \(L^p\)-convergence. Similarly as it was proved in the case \(p=1\) in [7, Theorem 3.2], it is possible to show the following result relating non-local distributional gradients of \(BV^{\alpha ,p}\) functions to vector valued Radon measures.
Theorem 3
(Structure Theorem for \(BV^{\alpha ,p}\) functions) Let \(\alpha \in (0,1)\), \(p \in [1, +\infty ]\) and \(f \in L^{p}({\mathbb {R}}^{n})\). Then, \(f \in BV^{\alpha ,p}({\mathbb {R}}^n)\) if and only if there exists a finite vector valued Radon measure \(D^{\alpha } f \in {\mathscr {M}}({\mathbb {R}}^n; {\mathbb {R}}^{n})\) such that
for all \(\varphi \in C^{\infty }_{c}({\mathbb {R}}^n; {\mathbb {R}}^{n})\). In addition, for any open set \(U\subset {\mathbb {R}}^n\) it holds
3.2 Approximation by smooth functions
Here and in the rest of the paper, we let \((\varrho _\varepsilon )\subset C^\infty _c({\mathbb {R}}^n)\) be a family of standard mollifiers as in [7, Section 3.3]. The following approximation theorem is the extension to \(BV^{\alpha ,p}\) functions of [7, Lemma 3.5 and Theorem 3.7]. We leave its proof to the interested reader.
Theorem 4
(Approximation by \(C^\infty \cap BV^{\alpha ,p}\) functions) Let \(\alpha \in (0,1]\) and \(p \in [1, +\infty ]\). Let \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\) and define \(f_{\varepsilon }= f*\varrho _{\varepsilon }\) for all \(\varepsilon >0\). Then \((f_{\varepsilon })_{\varepsilon >0}\subset BV^{\alpha ,p}({\mathbb {R}}^n) \cap C^{\infty }({\mathbb {R}}^n)\) with \(D^{\alpha } f_{\varepsilon } = (\varrho _{\varepsilon } *D^{\alpha } f) {\mathscr {L}}^{{n}}\) for all \(\varepsilon >0\). Moreover, the following properties hold.
-
(i)
If \(p<+\infty \), then \(f_{\varepsilon } \rightarrow f\) in \(L^p({\mathbb {R}}^n)\) as \(\varepsilon \rightarrow 0^+\); if \(p=+\infty \), then \(f_{\varepsilon } \rightarrow f\) in \(L^q_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) as \(\varepsilon \rightarrow 0^+\) for all \(q\in [1,+\infty )\);
-
(ii)
\(D^{\alpha } f_{\varepsilon } \rightharpoonup D^{\alpha } f\) in \({\mathscr {M}}({\mathbb {R}}^n; {\mathbb {R}}^{n})\) and \(|D^{\alpha } f_{\varepsilon }|({\mathbb {R}}^n) \rightarrow |D^{\alpha } f|({\mathbb {R}}^n)\) as \(\varepsilon \rightarrow 0^+\).
The following result extends the approximation by test functions given in [7, Theorem 3.8] to functions in \(BV^{\alpha ,p}({\mathbb {R}}^n)\) for \(\alpha \in (0,1)\) and all exponents \(p\in \left[ 1,\frac{n}{n-\alpha }\right) \). In the proof below and in the following, we let
for any \(f\in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^m)\), \(m\in {\mathbb {N}}\). Note that \(|\nabla ^\alpha f(x)| \le \mu _{n,\alpha } {\mathcal {D}}^\alpha f(x)\) for all \(x\in {\mathbb {R}}^n\) and that \({\mathcal {D}}^\alpha f\in L^p({\mathbb {R}}^n)\) for all \(p\in [1,+\infty ]\).
Theorem 5
(Approximation by \(C^\infty _c\) functions) Let \(\alpha \in (0,1)\) and \(p\in \left[ 1,\frac{n}{n-\alpha }\right) \). If \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\), then there exists \((f_k)_{k\in {\mathbb {N}}}\subset C^\infty _c({\mathbb {R}}^n)\) such that
-
(i)
\(f_k\rightarrow f\) in \(L^p({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \);
-
(ii)
\(|D^\alpha f_k|({\mathbb {R}}^n) \rightarrow |D^\alpha f|({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \).
Proof
Let \((\eta _R)_{R>0}\subset C^\infty _c({\mathbb {R}}^n)\) be a family of cut-off functions such that
We can also assume that \(\eta _R(x)=\eta _1(\frac{x}{R})\) for all \(x\in {\mathbb {R}}^n\) and \(R>0\). The proof now goes as the one of [7, Theorem 3.8] with minor modifications. We simply have to check that
Indeed, by Hölder’s inequality, we have
where \(\frac{1}{p}+\frac{1}{q}=1\), and a simple change of variables shows that
for all \(R>0\). The claim in (3.4) thus follows provided that \(\frac{n}{q}-\alpha <0\), which is equivalent to \(p\in \left[ 1,\frac{n}{n-\alpha }\right) \), and the proof is complete. \(\square \)
For the sake of completeness, we also treat the case \(\alpha =1\) of the previous result.
Proposition 1
Let \(n\in {\mathbb {N}}\) and \(p\in [1,+\infty )\) be such that \(p\le \frac{n}{n-1}\) for \(n\ge 2\). If \(f\in BV^{1,p}({\mathbb {R}}^n)\), then there exists \((f_k)_{k\in {\mathbb {N}}}\subset C^\infty _c({\mathbb {R}}^n)\) such that
-
(i)
\(f_k\rightarrow f\) in \(L^p({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \);
-
(ii)
\(|D f_k|({\mathbb {R}}^n) \rightarrow |D f|({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \).
Proof
Thanks to Theorem 4, we can assume \(f \in C^{\infty }({\mathbb {R}}^n) \cap BV^{1,p}({\mathbb {R}}^n)\) without loss of generality. Now let \((\eta _R)_{R>0}\subset C^\infty _c({\mathbb {R}}^n)\) be a family of cut-off functions as in the proof of Theorem 5. Clearly, \(\eta _R f \rightarrow f\) in \(L^p({\mathbb {R}}^n)\) as \(R \rightarrow + \infty \). Moreover, since \(\nabla (\eta _R f) = \eta _R \nabla f + f \nabla \eta _R\), we thus just need to check that \(\Vert f\, \nabla \eta _R\Vert _{L^1({\mathbb {R}}^n;\,{\mathbb {R}}^n)} \rightarrow 0^+\) as \(R \rightarrow + \infty \). Indeed, by Hölder’s inequality, we can estimate
and the conclusion immediately follows. \(\square \)
3.3 Gagliardo–Nirenberg–Sobolev inequality
Thanks to the approximation by test functions given by Theorem 5, we can extend [7, Theorem 3.9] and prove the analogue of the Gagliardo–Nirenberg–Sobolev inequality for the space \(BV^{\alpha ,p}({\mathbb {R}}^n)\) whenever \(p\in \left[ 1,\frac{n}{n-\alpha }\right) \).
Theorem 6
(Gagliardo–Nirenberg–Sobolev inequality) Let \(\alpha \in (0,1)\) and let \(p\in \left[ 1,\frac{n}{n-\alpha }\right) \). There exists a constant \(c_{n,\alpha }>0\), depending on n and \(\alpha \) only, such that
for all \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\), where \(r=+\infty \) if \(n=1\) and \(r=1\) if \(n\ge 2\). As a consequence, \(BV^{\alpha ,p}({\mathbb {R}}^n)\subset L^q({\mathbb {R}}^n)\) continuously for all \(q\in \left[ p,\frac{n}{n-\alpha }\right) \), with also \(q=\frac{n}{n-\alpha }\) if \(n\ge 2\).
Proof
Assume that \(f\in C^\infty _c({\mathbb {R}}^n)\) to start. Arguing as in the proof of [8, Theorem 3.8], we can estimate \( |f| \le c_{n,\alpha }\, I_\alpha |\nabla ^\alpha f| \) for some constant \(c_{n,\alpha }>0\) depending only on n and \(\alpha \) (possibly varying from line to line). Thanks to the Hardy–Littlewood–Sobolev inequality, we immediately deduce that
for all \(f\in C^\infty _c({\mathbb {R}}^n)\). Moreover, if \(n\ge 2\), then we can apply [32, Theorem 1.1] to the vector field \(F=\nabla ^\alpha f\) in order to get that
for all \(f\in C^\infty _c({\mathbb {R}}^n)\). Since \(I_\alpha \nabla ^\alpha f = Rf\) for all \(f\in C^\infty _c({\mathbb {R}}^n)\) and
strongly (recall the definition in (2.6) and the properties of the Riesz transform), we immediately deduce that
for all \(f\in C^\infty _c({\mathbb {R}}^n)\), with \(n\ge 2\). The conclusion then follows by combining a standard approximation argument exploiting Theorem 5 with Fatou’s Lemma. \(\square \)
For \(\alpha =1\), the previous result can be stated as follows.
Proposition 2
(Alvino’s inequality) Let \(n\in {\mathbb {N}}\) and \(p\in [1,+\infty )\). If \(n\ge 2\) and \(p\le \frac{n}{n-1}\), then there exists a dimensional constant \(c_n>0\) such that
for all \(f\in BV^{1,p}({\mathbb {R}}^n)\). If \(n=1\), then \(\Vert f\Vert _{L^\infty ({\mathbb {R}})} \le |Df|({\mathbb {R}})\) for all \(f\in BV^{1,p}({\mathbb {R}})\).
Proof
While the case \(n=1\) is a well-known property of functions having bounded variation, the case \(n\ge 2\) follows from Alvino’s inequality [2] for functions in \(BV({\mathbb {R}}^n)\) (also see [33, Section 5]) in combination with Proposition 1. We leave the details to the interested reader. \(\square \)
3.4 The space \(S^{\alpha ,p}({\mathbb {R}}^n)\) and the embedding \(BV^{\alpha ,p}\subset S^{\beta ,q}\)
Let \(\alpha \in (0,1)\) and \(p\in [1,+\infty ]\). We define the weak fractional \(\alpha \)-gradient of a function \(f\in L^p({\mathbb {R}}^n)\) as the function \(\nabla ^\alpha f\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n;{\mathbb {R}}^n)\) satisfying
for all \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\). We hence let the linear space
endowed with the norm
be the distributional fractional Sobolev space.
As shown in [7, Proposition 3.20], (\(S^{\alpha ,p}({\mathbb {R}}^n)\),\(\Vert \cdot \Vert _{S^{\alpha ,p}({\mathbb {R}}^n)}\)) is a Banach space for all \(p\in [1,+\infty ]\). Thanks to [7, Theorem 3.23] for \(p=1\) and to [6, Theorem A.1] for \(p\in (1,+\infty )\) (we refer the reader also to [17, Theorem 2.7] for a simpler proof), the set \(C^\infty _c({\mathbb {R}}^n)\) is dense in \(S^{\alpha ,p}({\mathbb {R}}^n)\). As a consequence, for \(p\in (1,+\infty )\) it is possible to identify \(S^{\alpha ,p}({\mathbb {R}}^n)\) with the fractional Bessel potential space \(L^{\alpha ,p}({\mathbb {R}}^n)\), see [6, Corollary 2.1] and the discussion therein.
We now want to provide a rigorous formulation of the naïve intuition that
that is to say, if \(\nabla ^\alpha f\in L^p({\mathbb {R}}^n;{\mathbb {R}}^n)\) for some \(\alpha \in (0,1)\) and \(p\in [1,+\infty )\), then \(\nabla ^\beta f\in L^q({\mathbb {R}}^n;{\mathbb {R}}^n)\) for some lower fractional differentiation order \(\beta <\alpha \) and some higher integrability exponent \(q>p\) (depending on \(\beta \)).
For \(p>1\), the above principle is a simple consequence of the known embedding theorems between the fractional Bessel potential spaces, thanks to the aforementioned identification \(S^{\alpha ,p}({\mathbb {R}}^n)=L^{\alpha ,p}({\mathbb {R}}^n)\).
The more delicate case \(p=1\) is covered in Theorem 7 below. We refer the reader also to [7, Theorem 3.32] and to [8, Propositions 3.2(i), 3.3 and 3.12] for similar results in this direction.
Theorem 7
(\(BV^{\alpha ,p}\subset S^{\beta ,q}\) for \(p<\frac{n}{n-\alpha }\)) Let \(\alpha ,\beta \in (0,1]\), with \(\beta <\alpha \), and let \(p, q \in [1,+\infty ]\) be such that \(p \le q < \frac{n}{n+\beta -\alpha }\). Then \(BV^{\alpha ,p}({\mathbb {R}}^n)\subset S^{\beta ,q}({\mathbb {R}}^n)\) continuously.
Proof
Assume that \(f\in C^\infty _c({\mathbb {R}}^n)\) and let \(R>0\). Arguing as in the proof of [8, Proposition 3.12], we can estimate
for all \(x\in {\mathbb {R}}^n\). On the one side, we can write
for all \(x\in {\mathbb {R}}^n\), so that
by Young’s inequality. On the other side, arguing as in the proof of [8, Proposition 3.12], we can write
for all \(x\in {\mathbb {R}}^n\), so that
by Minkowski’s integral inequality, for some constant \(c_{n,\alpha ,\beta }>0\) depending only on n, \(\alpha \) and \(\beta \). Hence we get that
whenever \(R>0\), for some constant \(c_{n,\alpha ,\beta ,q}>0\) depending only on n, \(\alpha \), \(\beta \) and q. Choosing \(R = \left( \frac{\Vert f\Vert _{L^q({\mathbb {R}}^n)}}{\Vert \nabla ^\alpha f\Vert _{L^1({\mathbb {R}}^n; {\mathbb {R}}^n)}}\right) ^{\frac{1}{\frac{n}{q}-n+\alpha }}\), we get that
for all \(f\in C^\infty _c({\mathbb {R}}^n)\). The conclusion thus follows from Theorems 5 and 6 (Propositions 1 and 2 in the case \(\alpha =1\)) via a routine approximation argument, since clearly \(p < \frac{n}{n - \alpha }\). \(\square \)
3.5 Generalized integration-by-parts formula for \(BV^{\alpha ,p}\) functions
The following result is a generalization of the fractional integration-by-parts formula (3.2) (the case \(p=1\) was actually already analyzed in [8, Proposition 2.7]). This result will be useful for integrating by parts \(BV^{\alpha ,p}\) functions on balls, see Theorem 9 below.
Proposition 3
(\(W^{1,q}\cap C_b\)-regular test) Let \(\alpha \in (0, 1)\) and let \(p,q \in [1,+\infty ]\) be such that \(\frac{1}{p}+\frac{1}{q}=1\). If \(f \in BV^{\alpha , p}({\mathbb {R}}^n)\), then
for all \(\varphi \in W^{1,q}({\mathbb {R}}^n;{\mathbb {R}}^n)\cap C_b({\mathbb {R}}^n;{\mathbb {R}}^n)\) if \(q > 1\), and for all \(\varphi \in BV({\mathbb {R}}^n;{\mathbb {R}}^n)\cap C_b({\mathbb {R}}^n;{\mathbb {R}}^n)\) if \(q=1\).
Proof
We divide the proof into two steps and adopt the same strategy of [7, Theorem 3.8] and [8, Proposition 2.7] with minor modifications.
Step 1. Assume \(\varphi \in W^{1,q}({\mathbb {R}}^n;{\mathbb {R}}^n)\cap {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\cap C^\infty ({\mathbb {R}}^n; {\mathbb {R}}^n)\) and let \((\eta _R)_{R>0}\subset C^\infty _c({\mathbb {R}}^n)\) be a family of cut-off functions as in [7, Section 3.3]. On the one hand, since
for all \(R>0\), by Lebesgue’s Dominated Convergence Theorem we have
On the other hand, by [8, Lemmas 2.2 and 2.5] we can write
for all \(R>0\). Since \(\varphi \eta _R\in C^\infty _c({\mathbb {R}}^n; {\mathbb {R}}^n)\), (3.5) implies that
for all \(R>0\). Since
for all \(R>0\), by Lebesgue’s Dominated Convergence Theorem (with respect to the finite measure \(|D^\alpha f|\)) we have
Finally, on the one side we can estimate
for all \(R>0\), while, on the other side,
for all \(R>0\). We claim that
Indeed, \(f\varphi \in L^1({\mathbb {R}}^n;{\mathbb {R}}^n)\) and (3.6) follows by Lebesgue’s Dominated Convergence Theorem. We also claim that
Indeed, since \(\varphi \in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\) and \(\Vert \eta _R\Vert _{L^\infty ({\mathbb {R}}^n)}\le 1\) for all \(R>0\), by Lebesgue’s Dominated Convergence Theorem we get that
for all \(x\in {\mathbb {R}}^n\). Moreover, for a.e. \(x\in {\mathbb {R}}^n\) we have
Therefore, combining (3.8) and (3.9), again by Lebesgue’s Dominated Convergence Theorem we get (3.7). Thus (3.5) is proved whenever \(\varphi \in W^{1,q}({\mathbb {R}}^n;{\mathbb {R}}^n)\cap {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\cap C^\infty ({\mathbb {R}}^n; {\mathbb {R}}^n)\).
Step 2. Now assume \(\varphi \in W^{1,q}({\mathbb {R}}^n;{\mathbb {R}}^n)\cap C_b({\mathbb {R}}^n;{\mathbb {R}}^n)\) (if \(q=1\), then we instead take \(\varphi \in BV({\mathbb {R}}^n;{\mathbb {R}}^n)\cap C_b({\mathbb {R}}^n;{\mathbb {R}}^n)\)) and let \((\varrho _\varepsilon )_{\varepsilon >0}\subset C^\infty _c({\mathbb {R}}^n)\) be a family of standard mollifiers as in [7, Section 3.3]. Then \(\varphi _\varepsilon =\varrho _\varepsilon *\varphi \in W^{1,q}({\mathbb {R}}^n;{\mathbb {R}}^n)\cap {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\cap C^\infty ({\mathbb {R}}^n; {\mathbb {R}}^n)\) and so, by Step 1, we can write
for all \(\varepsilon >0\). On the one hand, it is not difficult to see that \(\mathrm{div}^\alpha \varphi _\varepsilon =\varrho _\varepsilon *\mathrm{div}^\alpha \varphi \) for all \(\varepsilon >0\), so that
On the other hand, since \(\varphi \in C_b({\mathbb {R}}^n;{\mathbb {R}}^n)\), we get
by Lebesgue’s Dominated Convergence Theorem (with respect to the finite measure \(|D^\alpha f|\)). This concludes the proof of (3.5). \(\square \)
3.6 Integrability of \({\mathcal {D}}^\alpha \) in Lorentz space for \(BV^{1,p}\) functions
We now need to focus on the weak integrability properties of the operator \({\mathcal {D}}^\alpha \) defined in (3.3) when applied to functions belonging to \(BV^{1,p}({\mathbb {R}}^n)\). This analysis will be useful for studying the integrability properties of the fractional gradient of the characteristic function of a ball, see Corollary 1 below. This result, in turn, will be useful in the proof of the integration-by-parts formula of \(BV^{\alpha ,p}\) functions on balls in Theorem 9.
Theorem 8
(Weak integrability of \({\mathcal {D}}^\alpha \)) Let \(\alpha \in (0,1)\), \(p\in [1,+\infty ]\) and define
The operator \({\mathcal {D}}^\alpha :BV^{1,p}({\mathbb {R}}^n)\rightarrow L^{p_\alpha ,\infty }({\mathbb {R}}^n)\) is well defined and satisfies
for all \(f\in BV^{1,p}({\mathbb {R}}^n)\), where \(c_{n,\alpha ,p}>0\) is a constant depending on n, \(\alpha \) and p only.
Proof
The case \(p=1\) is easy, since (3.10) holds in the following stronger form
for all \(f\in BV({\mathbb {R}}^n)\), whose simple proof is left to the reader (for instance, one can follow the strategy of the proof of [10, Proposition 2.2]). In the following, we thus assume that \(p>1\). We now divide the proof into three steps.
Step 1. Assume \(f\in W^{1,1}({\mathbb {R}}^n)\cap C^\infty ({\mathbb {R}}^n)\cap {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n)\). By [32, Lemma 3.2], there exists a constant \(C_{n, \alpha } > 0\) such that
for all \(x\in {\mathbb {R}}^n\). Since \(f\in L^p({\mathbb {R}}^n)\), by Hölder’s inequality in Lorentz spaces (see [23, Theorem 3.4] and [33, Theorem 5.1]) and the well-known continuity properties of the maximal function, we get that
where \(c_{n,\alpha ,p}>0\) is a constant depending only on n, \(\alpha \) and p.
Step 2. Now assume \(f\in BV({\mathbb {R}}^n)\cap L^p({\mathbb {R}}^n)\) and define \(f_\varepsilon =f*\varrho _\varepsilon \) for all \(\varepsilon >0\). Then \(f_\varepsilon \in W^{1,1}({\mathbb {R}}^n)\cap C^\infty ({\mathbb {R}}^n)\cap {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n)\) with \(\Vert f_\varepsilon \Vert _{L^p({\mathbb {R}}^n)}\le \Vert f\Vert _{L^p({\mathbb {R}}^n)}\) for all \(\varepsilon >0\) and \(|Df_\varepsilon |({\mathbb {R}}^n)\rightarrow |Df|({\mathbb {R}}^n)\) as \(\varepsilon \rightarrow 0^+\). Moreover, by the Fatou Lemma, we have
for a.e. \(x\in {\mathbb {R}}^n\). Hence, by [13, Exercise 1.1.1(b)], we get
thanks to Step 1.
Step 3. Finally, assume \(f\in BV^{1,p}({\mathbb {R}}^n)\). Let \((\eta _R)_{R>0}\subset C^\infty _c({\mathbb {R}}^n)\) be a family of cut-off functions as in [7, Section 3.3] and define \(f_R=f\eta _R\) for all \(R>0\). Then \(f_R\in BV({\mathbb {R}}^n)\cap L^p({\mathbb {R}}^n)\) with \(\Vert f_R\Vert _{L^p({\mathbb {R}}^n)}\le \Vert f\Vert _{L^p({\mathbb {R}}^n)}\) for all \(R>0\) and \(|Df_R|({\mathbb {R}}^n)\rightarrow |Df|({\mathbb {R}}^n)\) as \(R\rightarrow +\infty \). Moreover, by the Fatou Lemma, we have
for a.e. \(x\in {\mathbb {R}}^n\). Inequality (3.10) thus follows again by [13, Exercise 1.1.1(b)], thanks to Step 2. This concludes the proof. \(\square \)
From Theorem 8, we immediately deduce the following integrability properties of the fractional gradient of the indicator function of a ball.
Corollary 1
(Integrability of \(\nabla ^\alpha \chi _{B_r(x)}\)) Let \(\alpha \in (0,1)\) and let \(p \in \left[ 1, \frac{1}{\alpha } \right) \). There exists a constant \(c_{n,\alpha ,p} > 0\), depending on n, \(\alpha \) and p only, such that
for all \(x \in {\mathbb {R}}^{n}\) and \(r > 0\).
Proof
Let \(x \in {\mathbb {R}}^{n}\) and \(r > 0\) be fixed. Since
by the rescaling property of \(\nabla ^\alpha \), we immediately get that
Since \(\nabla ^{\alpha } \chi _{B_1} \in L^{1}({\mathbb {R}}^n; {\mathbb {R}}^n) \cap L^{\frac{1}{\alpha }, \infty }({\mathbb {R}}^n;{\mathbb {R}}^n)\) by [7, Proposition 4.8] and Theorem 8, the conclusion follows by observing that \(L^1({\mathbb {R}}^n)\cap L^{\frac{1}{\alpha },\infty }({\mathbb {R}}^n)\subset L^p({\mathbb {R}}^n)\) with continuous inclusion for all \(p\in \left[ 1,\frac{1}{\alpha }\right) \). This interpolation result can be proved for instance by arguing as in the proof of [13, Proposition 1.1.14] with some minor modifications. We leave the simple details to the interested reader. \(\square \)
Remark 1
(The case \(n=1\) in Corollary 1) The estimate in (3.11) in the case \(n = 1\) can be obtained by a direct computation from the explicit formula
given by [7, Example 4.11]. In particular, we deduce that
for all \(s \in [1,+\infty )\).
From Proposition 2, we immediately deduce the following improvement of Theorem 8. We leave its simple proof to the reader.
Corollary 2
(Improved weak integrability of \({\mathcal {D}}^\alpha \)) Let \(\alpha \in (0,1)\), \(n\in {\mathbb {N}}\) and \(p\in [1,+\infty )\) be such that \(p\le \frac{n}{n-1}\) for \(n\ge 2\). If \(f\in BV^{1,p}({\mathbb {R}}^n)\), then
3.7 Integration by parts of \(BV^{\alpha ,p}\) functions on balls
We are now ready to state and prove the following integration-by-parts of \(BV^{\alpha ,p}\) functions on balls, which is a generalization of [7, Theorem 5.2] to \(BV^{\alpha ,p}\) functions for \(p\in \left( \frac{1}{1-\alpha },+\infty \right] \). This result will be the central ingredient of the proof of the decay estimates for \(BV^{\alpha ,p}\) functions in Theorem 10 below.
Theorem 9
(Integration by parts on balls) Let \(\alpha \in (0, 1)\) and \(p\in \left( \frac{1}{1-\alpha },+\infty \right] \). If \(f \in BV^{\alpha , p}({\mathbb {R}}^{n})\), \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) and \(x \in {\mathbb {R}}^n\), then
for \({\mathscr {L}}^{{1}}\)-a.e. \(r > 0\).
Proof
Let \(x \in {\mathbb {R}}^{n}\) and \(\varphi \in {{\,\mathrm{Lip}\,}}_{c}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) be fixed. We divide the proof into two parts, dealing with the cases \(p = +\infty \) and \(p\in \left( \frac{1}{1-\alpha },+\infty \right) \) separately.
Case 1: \(p = +\infty \). Let \(\varepsilon >0\) and define the function \(h_{\varepsilon , r, x}\in {{\,\mathrm{Lip}\,}}({\mathbb {R}}^n)\) by setting
for all \(y\in {\mathbb {R}}^n\). By [7, Lemma 5.1], we know that \(\nabla ^\alpha h_{\varepsilon ,r,x}\in L^1({\mathbb {R}}^n;{\mathbb {R}}^n)\) with
for \({\mathscr {L}}^{{n}}\)-a.e. \(y \in {\mathbb {R}}^{n}\). On the one hand, since \(h_{\varepsilon , r, x}\,\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\), by Proposition 3 we have
Since \(h_{\varepsilon , r, x}(y) \rightarrow \chi _{\overline{B_r(x)}}(y)\) as \(\varepsilon \rightarrow 0^+\) for all \(y \in {\mathbb {R}}^{n}\) and \(|D^{\alpha } f|(\partial B_r(x)) = 0\) for \({\mathscr {L}}^{{1}}\)-a.e. \(r > 0\), we can compute
for \({\mathscr {L}}^{{1}}\)-a.e. \(r > 0\). On the other hand, by [8, Lemma 2.5], we have
We deal with each term of the right-hand side of (3.15) separately. For the first term, since \(0\le h_{\varepsilon , r, x}\le \chi _{B_{r+1}(x)}\) for all \(\varepsilon \in (0,1)\) and \(h_{\varepsilon , r, x} \rightarrow \chi _{B_r(x)}\) in \(L^{q}({\mathbb {R}}^{n})\) as \(\varepsilon \rightarrow 0^+\) for any \(q \in [1,+\infty )\), by [7, Corollary 2.3] and Lebesgue’s Dominated Convergence Theorem we can compute
For the second term, by (3.13) we have
By Fubini’s Theorem, we can compute
By Lebesgue’s Differentiation Theorem, we have
for \({\mathscr {L}}^{{1}}\)-a.e. \(r > 0\). Therefore, by [7, Theorem 3.18, equation (3.26)], we get that
for \({\mathscr {L}}^{{1}}\)-a.e. \(r > 0\). Finally, for the third term, we note that
for all \(y\in {\mathbb {R}}^n\), so that
for \({\mathscr {L}}^{{n}}\)-a.e. \(y\in {\mathbb {R}}^n\) by Lebesgue’s Dominated Convergence Theorem. Since
again by Lebesgue’s Dominated Convergence Theorem we can compute
Combining (3.14), (3.15), (3.16), (3.17) and (3.18), we obtain (3.12).
Case 2: \(p \in \left( \frac{1}{1 - \alpha }, +\infty \right) \). Let \((\varrho _{\varepsilon })_{\varepsilon > 0}\) be a family of standard mollifiers (see [7, Section 3.3]) and define \(f_\varepsilon =f*\varrho _\varepsilon \) for all \(\varepsilon >0\). By Theorem 4, we have that
for all \(\varepsilon >0\). Hence, by Step 1, for each \(\varepsilon > 0\) we have
for \({\mathscr {L}}^{{1}}\)-a.e. \(r > 0\). We now need to study the convergence as \(\varepsilon \rightarrow 0^+\) of each term of the above equality. By Theorem 4, we know that \(D^{\alpha } f_{\varepsilon } \rightharpoonup D^{\alpha } f\) as \(\varepsilon \rightarrow 0^+\), so that
for \({\mathscr {L}}^{{1}}\)-a.e. \(r > 0\) thanks to [3, Proposition 1.62]. Since \(\mathrm{div}^{\alpha } \varphi \in L^{1}({\mathbb {R}}^{n}) \cap L^{\infty }({\mathbb {R}}^{n})\) by [7, Corollary 2.3] and \(\mathrm{div}^{\alpha }_{\mathrm{NL}}(\chi _{B_{r}(x)}, \varphi ) \in L^{1}({\mathbb {R}}^{n}) \cap L^{\infty }({\mathbb {R}}^{n})\) by [7, Lemma 2.7 and Remark 2.8], we have
and
thanks to the fact that \(f_\varepsilon \rightarrow f\) in \(L^p({\mathbb {R}}^n)\) as \(\varepsilon \rightarrow 0^+\). Finally, thanks to Corollary 1, we have \(\nabla ^{\alpha } \chi _{B_{r}(x)} \in L^{p'}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) for any \(p \in \left( \frac{1}{1 - \alpha }, + \infty \right) \) and so
again by the convergence \(f_\varepsilon \rightarrow f\) in \(L^p({\mathbb {R}}^n)\) as \(\varepsilon \rightarrow 0^+\). The conclusion thus follows. \(\square \)
4 Absolute continuity of the fractional variation
In this section, we prove our first main result Theorem 1. We divide the proof into two parts, dealing with the subcritical regime (i) and the supercritical regime (ii) separately, see Proposition 4(i) and Corollary 3 respectively. At the end of this section, we provide two examples to show the sharpness of our result in the one-dimensional case \(n=1\).
4.1 The subcritical regime \(p\in \left[ 1,\frac{n}{1-\alpha }\right) \)
Thanks to [7, Lemma 3.28], if \(f\in BV^{\alpha }({\mathbb {R}}^n)\) then \(u = I_{1 - \alpha } f\in bv({\mathbb {R}}^{n}) \cap L^{\frac{n}{n - 1 + \alpha }, \infty }({\mathbb {R}}^{n})\), with \(D u =D^\alpha f\) in \({\mathscr {M}}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\). As a consequence, we immediately deduce that \(|D^{\alpha } f| \ll {\mathscr {H}}^{{n - 1}}\) for all \(f \in BV^{\alpha }({\mathbb {R}}^{n})\).
In the following result, which is a generalization of [7, Lemma 3.28] to the present setting, we show that this phenomenon is typical of the functions belonging to \(BV^{\alpha ,p}({\mathbb {R}}^n)\) in the subcritical regime \(p\in \left[ 1,\frac{n}{1-\alpha }\right) \).
Proposition 4
(Relation between \(BV^{\alpha ,p}({\mathbb {R}}^n)\) and \(BV^{1,p}({\mathbb {R}}^n)\)) Let \(\alpha \in (0,1)\), \(p\in \left( 1,\frac{n}{1-\alpha }\right) \) and \(q = \frac{np}{n - (1-\alpha )p }\).
-
(i)
If \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\), then \(u = I_{1 - \alpha } f\in BV^{1, q}({\mathbb {R}}^n)\) with
$$\begin{aligned} \Vert u\Vert _{L^q({\mathbb {R}}^n)} \le c_{n,\alpha ,p}\,\Vert f\Vert _{L^p({\mathbb {R}}^n)} \quad \text {and}\quad D u =D^\alpha f\ \text {in}\ {\mathscr {M}}({\mathbb {R}}^{n}; {\mathbb {R}}^{n}). \end{aligned}$$As a consequence, we have \(|D^{\alpha } f| \ll {\mathscr {H}}^{{n - 1}}\) for all \(f \in BV^{\alpha ,p}({\mathbb {R}}^{n})\) and the operator \(I_{1-\alpha }:BV^{\alpha ,p}({\mathbb {R}}^n)\rightarrow BV^{1,q}({\mathbb {R}}^n)\) is continuous.
-
(ii)
If \(p\in \left( 1,\frac{n}{n-\alpha }\right) \) and \(u\in BV^{1,p}({\mathbb {R}}^n)\), then \(f=(-\varDelta )^{\frac{1-\alpha }{2}}u\in BV^{\alpha ,p}({\mathbb {R}}^n)\) with
$$\begin{aligned} \Vert f\Vert _{L^p({\mathbb {R}}^n)} \le c_{n,\alpha ,p}\Vert u\Vert _{BV^{\alpha ,p}({\mathbb {R}}^n)} \quad \text {and} \quad D^\alpha f = D u\ \text {in}\ {\mathscr {M}}({\mathbb {R}}^{n}; {\mathbb {R}}^{n}). \end{aligned}$$As a consequence, the operator \((-\varDelta )^{\frac{1-\alpha }{2}}:BV^{1,p}({\mathbb {R}}^n)\rightarrow BV^{\alpha ,p}({\mathbb {R}}^n)\) is continuous.
Proof
Let \(s=\frac{p}{p-1}\) and note that \(r = \frac{n s}{n + (1 - \alpha )s}\in \left( 1,\frac{n}{1-\alpha }\right) \). We prove the two properties separately.
Proof of (i). Let \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\). By the Hardy–Littlewood–Sobolev inequality, we immediately get that
Given \(\varphi \in C^{\infty }_{c}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\), we clearly have \(I_{1 - \alpha } |\mathrm{div}\varphi | \in L^{s}({\mathbb {R}}^{n})\), because \(|\mathrm{div}\varphi |\in L^r({\mathbb {R}}^n)\). Hence, by Fubini Theorem, we have
for all \(\varphi \in C^{\infty }_{c}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\), proving that \(D^{\alpha } f = Du\) in \({\mathscr {M}}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\). The remaining part of the statement in (i) follows easily.
Proof of (ii). Let \(p\in \left( 1,\frac{n}{n-\alpha }\right) \) and \(u\in BV^{1,p}({\mathbb {R}}^n)\). Since \(p<\frac{n}{n-\alpha }\), we can apply Theorem 7 to get that \(BV^{1,p}({\mathbb {R}}^n)\subset S^{1-\alpha ,p}({\mathbb {R}}^n)\) with continuous inclusion, so that \(f=(-\varDelta )^{\frac{1-\alpha }{2}}u\in L^p({\mathbb {R}}^n)\) (thanks to the identification \(S^{1-\alpha ,p}({\mathbb {R}}^n)=L^{1-\alpha ,p}({\mathbb {R}}^n)\) following from [6, Corollary 2.1], also see the discussion in [6, Section 2.1]) and thus \(I_{1-\alpha }f\in L^q({\mathbb {R}}^n)\) by the Hardy–Littlewood–Sobolev inequality. Since \(p<\frac{n}{n-\alpha }\), we also have that \(p<q<\frac{n}{n-1}\) and thus \(BV^{1,p}({\mathbb {R}}^n)\subset L^q({\mathbb {R}}^n)\) with continuous inclusion by Proposition 2. Hence \(u\in L^q({\mathbb {R}}^n)\) and we can now claim that \(I_{1-\alpha }f=u\) in \(L^q({\mathbb {R}}^n)\). Indeed, this is easily verified if \(u\in C^\infty _c({\mathbb {R}}^n)\) by applying the Fourier transform (see [19, Lemma 2.3] for instance), so that the claim follows by a plain approximation argument. Therefore, by applying Fubini Theorem again, we can write (4.1) and prove that \(D^\alpha f=Du\) in \({\mathscr {M}}({\mathbb {R}}^n;{\mathbb {R}}^n)\), reaching the conclusion. \(\square \)
Remark 2
(About Proposition 4(ii)) The validity of Proposition 4(ii) when \(p=1\) was already proved in [7, Lemma 3.28]. We also refer the reader to [17, Proposition 3.1], in which the authors prove that, if \(u\in W^{1,p}({\mathbb {R}}^n)\) for some \(p\in [1,+\infty ]\), then \(f=(-\varDelta )^{\frac{1-\alpha }{2}}u\in S^{\alpha ,p}({\mathbb {R}}^n)\) with \(\nabla ^\alpha f=\nabla u\) in \(L^p({\mathbb {R}}^n;{\mathbb {R}}^n )\).
4.2 The supercritical regime \(p\in \left[ \frac{n}{1-\alpha },+\infty \right] \)
We now focus on the absolute continuity property of the fractional variation with respect to the Hausdorff measure for functions belonging to \(BV^{\alpha ,p}({\mathbb {R}}^n)\) in the supercritical regime \(p\in \left[ \frac{n}{1-\alpha },+\infty \right] \). The crucial tool in this case is provided by the following important consequence of Theorem 9, which extends [7, Theorem 5.3] to the present setting.
Theorem 10
(Decay estimates for \(BV^{\alpha ,p}\) functions for \(p>\frac{1}{1-\alpha }\)) Let \(\alpha \in (0,1)\) and \(p\in \left( \frac{1}{1-\alpha }, +\infty \right] \). There exist two constants \(A_{n,\alpha ,p}, B_{n,\alpha ,p} > 0\), depending on n, \(\alpha \) and p only, with the following property. If \(f \in BV^{\alpha ,p}({\mathbb {R}}^n)\) then, for \(|D^{\alpha } f|\)-a.e. \(x \in {\mathbb {R}}^n\), there exists \(r_x > 0\) such that
and
for all \(r \in (0, r_x)\), where \(q\in [1,+\infty )\) is such that \(\frac{1}{p}+\frac{1}{q}=1\).
Proof
Since \(f \in BV^{\alpha ,p}({\mathbb {R}}^n)\), by the Polar Decomposition Theorem for Radon measures there exists a Borel vector valued function \(\sigma _{f}^{\alpha }:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) such that
We divide the proof into two steps, dealing with the two estimates separately.
Step 1: Proof of (4.2). Let \(\sigma _{f}^{\alpha }: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{n}\) be as in (4.4) and let \(x\in {\mathbb {R}}^n\) be such that \(|\sigma _{f}^{\alpha }(x)| = 1\). Given \(r>0\), we define the vector field \(\varphi :{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) by setting
for all \(y\in {\mathbb {R}}^n\). We clearly have that \(\varphi _{x, r} \in {{\,\mathrm{Lip}\,}}_{c}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) with \(\Vert \varphi \Vert _{L^{\infty }({\mathbb {R}}^{n}; {\mathbb {R}}^{n})} \le 1\). Thus, on the one hand, we can find \(r_x\in (0,1)\) such that
for all \(r\in (0,r_x)\). On the other hand, by (3.12) we can write
for \({\mathscr {L}}^{{1}}\)-a.e. \(r\in (0,r_x)\). We now estimate the three terms in the right-hand side of (4.7) separately. For the first term, recalling the definition of \(\varphi _{x, r}\) in (4.5), we have
After some elementary computations, we get
for some constant \(C_{n,\alpha }>0\) depending only on n and \(\alpha \). Note that the integral appearing in the right-hand side of (4.8) converges if and only if \(\alpha q < 1\), that is, \(p > \frac{1}{1 - \alpha }\). We thus get
for some constant \(C_{n,\alpha ,q}>0\) depending only on n, \(\alpha \) and q. For the second term in the right-hand side of (4.7), we have
thanks to Corollary 1, for some constant \(C_{n, \alpha ,q}>0\) depending only on n, \(\alpha \) and q. Finally, observing that \(\varphi _{x, r}(x + r y) =\varphi _{0, 1}(y)\) for all \(y\in {\mathbb {R}}^n\), a simple change of variables gives
Thus, for the third and last term in the right-hand side of (4.7), we have
where \(C_{n, \alpha , q}=\Vert \mathrm{div}_{\mathrm{NL}}^{\alpha }(\chi _{B_{1}}, \varphi )\Vert _{L^{q}({\mathbb {R}}^{n})}\) (which is finite thanks to [7, Lemma 2.7 and Remark 2.8]). Combining (4.6) with (4.7), (4.9), (4.10) and (4.12), we get (4.2) with a simple continuity argument.
Step 2: Proof of (4.3). Let \(x\in {\mathbb {R}}^n\) be such that \(|\sigma _{u}^{\alpha }(x)| = 1\). Given \(\varphi \in {{\,\mathrm{Lip}\,}}_{c}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) such that \(\Vert \varphi \Vert _{L^{\infty }({\mathbb {R}}^{n}; {\mathbb {R}}^{n})} \le 1\), from (3.12) we deduce that
for \({\mathscr {L}}^{{1}}\)-a.e. \(r\in (0,r_x)\). Exploiting (4.2), (3.11) and (4.11), we conclude that
for \({\mathscr {L}}^{{1}}\)-a.e. \(r\in (0,r_x)\), where \(C_{n, \alpha , q}>0\) is a constant depending only on n, \(\alpha \) and q. Inequality (4.3) thus follows for all \(r\in (0,r_x)\) by a simple continuity argument. \(\square \)
Thanks to Theorem 10 and extending [7, Corollary 5.4] to the present setting, we are now ready to state and prove the following absolute continuity property of the fractional variation for \(BV^{\alpha ,p}\) functions with \(p\in \left( \frac{1}{1-\alpha },+\infty \right] \). Note that the result below is truly interesting only for \(p\in \left[ \frac{n}{1-\alpha },+\infty \right] \), due to Theorem 1(i) (see also Proposition 4) and the fact that
Corollary 3
(\(|D^{\alpha } f|\ll {\mathscr {H}}^{{\frac{n}{q} -\alpha }}\) for \(p>\frac{1}{1-\alpha }\)) Let \(\alpha \in (0,1)\) and \(p\in \left( \frac{1}{1-\alpha }, +\infty \right] \). If \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\), then there exists a \(|D^\alpha f|\)-negligible set \(Z_f^{\alpha ,p}\subset {\mathbb {R}}^n\) such that
where \(A_{n,\alpha ,p}\) is as in (4.2) and \(q\in [1,+\infty )\) is such that \(\frac{1}{p}+\frac{1}{q}=1\).
Proof
By Theorem 10, there exists a set \(Z_{f}^{\alpha ,p}\subset {\mathbb {R}}^n\) such that \(|D^{\alpha }f|(Z_{f}^{\alpha ,p}) = 0\) and (4.2) holds for any \(x \notin Z_{f}^{\alpha ,p}\). Thanks to the Borel regularity of the Radon measure \(|D^{\alpha } f|\), we can assume that \(Z_{f}^{\alpha ,p}\) is a Borel set without loss of generality. Hence, for all \(x\in {\mathbb {R}}^{n}\setminus Z_{f}^{\alpha ,p}\), we have
Inequality (4.13) thus follows from [3, Theorem 2.56]. \(\square \)
Remark 3
(The case \(n=1\) and \(p=\frac{1}{1-\alpha }\)) Note that Corollary 3 covers the supercritical regime \(p\in \left[ \frac{n}{1-\alpha },+\infty \right] \) for \(n\ge 2\), while for \(n=1\) the boundary case \(p=\frac{1}{1-\alpha }\) is missing. However, if \(n=1\) and \(p=\frac{1}{1-\alpha }\), then \(q=\frac{p}{p-1}=\frac{1}{\alpha }\) and so \({\mathscr {H}}^{{\frac{n}{q}-\alpha }}={\mathscr {H}}^{{0}}\), so that \(|D^\alpha f|\ll {\mathscr {H}}^{{0}}\) for all \(f\in BV^{\alpha ,\frac{1}{1-\alpha }}({\mathbb {R}})\) trivially. We do not know if this result is sharp.
Remark 4
(The limit as \(\alpha \rightarrow 1^-\)) It is somewhat interesting to observe that Corollary 3 still holds true if we send \(\alpha \rightarrow 1^-\). Indeed, such a limit case would apply only to functions \(f \in BV^{1, \infty }({\mathbb {R}}^n)\), for which it is well known (see [3, Theorem 3.77, Theorem 3.78 and equation (3.90)], for instance) that
where \(J_f\) is the jump set, so that \(Z_f^{1, \infty }\) could be any |Df|-negligible subset of \({\mathbb {R}}^n \setminus J_f\).
4.3 Two examples in one dimension
We conclude this section by discussing the optimality of the absolute continuity properties of the fractional variation stated in Theorem 1 in the one-dimensional case \(n=1\).
We begin with the following example, which is borrowed from [7, Theorem 3.26].
Example 1
(Proposition 4(i) is sharp for \(n=1\)) Let \(\alpha \in (0, 1)\) and consider
By [7, Theorem 3.26], we have \(f_{\alpha } \in BV^{\alpha }({\mathbb {R}})\) with \(D^{\alpha } f_{\alpha } = \delta _{0} -\delta _{1}\). Moreover, by [8, Theorem 3.8 and Remark 3.9], we have \(f_{\alpha } \in L^{\frac{1}{1 - \alpha }, \infty }({\mathbb {R}}) \setminus L^{\frac{1}{1 - \alpha }, q}({\mathbb {R}})\) for all \(q \ge 1\). In particular, since \(f_{\alpha } \in L^{1}({\mathbb {R}}) \cap L^{\frac{1}{1 - \alpha }, \infty }({\mathbb {R}})\), by interpolation we get that \(f_{\alpha } \in L^{p}({\mathbb {R}})\) for all \(p \in \left[ 1, \frac{1}{1 - \alpha } \right) \). Hence \(f_\alpha \in BV^{\alpha ,p}({\mathbb {R}})\) for all \(p \in \left[ 1, \frac{1}{1 - \alpha } \right) \) with \(|D^\alpha f_\alpha |\not \ll {\mathscr {H}}^{{\varepsilon }}\) for all \(\varepsilon > 0\). This proves that the absolute continuity property of the fractional variation stated in Theorem 1(i) is sharp for \(n=1\).
We now prove the following result, which combines the properties of the function \(f_\alpha \) introduced in Example 1 with the decay properties of a finite Radon measure.
Proposition 5
(The function \(u_\alpha =f_\alpha *\nu \)) Let \(\alpha \in (0,1)\), let \(f_\alpha \) be as in Example 1, and let \(\nu \in \mathscr {M}({\mathbb {R}})\). Then we have
with
where \(\tau _{x}(y) = y + x\) for all \(x,y\in {\mathbb {R}}\). In addition, if there exist \(C, \varepsilon > 0\) such that
then
Proof
We divide the proof into two steps.
Step 1. Let \(\nu \in \mathscr {M}({\mathbb {R}})\). We start by showing that \(u_{\alpha }\in BV^{\alpha , p}({\mathbb {R}})\) for all \(p \in \left[ 1, \frac{1}{1 - \alpha } \right) \) and that it satisfies (4.14). Indeed, by Young’s inequality (for Radon measures) we can estimate
Moreover, thanks to the translation invariance of \(\mathrm{div}^{\alpha }\) and exploiting the explicit expression of \(f_\alpha \) given in Example 1, we can write
for all \(\varphi \in C^\infty _c({\mathbb {R}})\). Thus \(u_{\alpha } \in BV^{\alpha , 1}({\mathbb {R}})\) with \(D^{\alpha } u_{\alpha } = \nu - (\tau _{1})_{\#} \nu \). In addition, by Jensen’s inequality and Tonelli’s Theorem we can estimate
for all \(p\in \left[ 1,\frac{1}{1-\alpha }\right) \), thanks to the integrability properties of \(f_\alpha \) given in Example 1.
Step 2. We prove that (4.15) implies (4.16). To this aim, let \(\delta > 0\) and \(q=\frac{p}{p-1}\). Since \(|f_{\alpha }| = |f_{\alpha }|^{\frac{\delta }{q}} |f_{\alpha }|^{1 - \frac{\delta }{q}}\), by Hölder’s inequality we get
for \(x\in {\mathbb {R}}\). We now recall the explicit expression of \(f_\alpha \) in Example 1 and write
where we have set
for all \(x\in {\mathbb {R}}\), \(r\in (0,1)\) and \(j\in {\mathbb {N}}\) for brevity. Now, on the one hand, if \(y \in \left( - \infty , x - \frac{3}{2} \right) \cup \left( x + \frac{1}{2}, \infty \right) \), then \(x - y \in \left( - \infty , - \frac{1}{2} \right) \cup \left( \frac{3}{2}, \infty \right) \), so that
for all \(y \in \left( - \infty , x - \frac{3}{2} \right) \cup \left( x + \frac{1}{2}, \infty \right) \). Therefore, we can estimate
for all \(x\in {\mathbb {R}}\). On the other hand, for all \(x \in {\mathbb {R}}\) and \(j \in {\mathbb {N}}\), we have
Reasoning analogously, we obtain
for all \(x \in {\mathbb {R}}\) and \(j \in {\mathbb {N}}\). Therefore, inserting (4.18), (4.20) and (4.21) in (4.17), we conclude that
for all \(x\in {\mathbb {R}}\), where \(C_{\alpha , \varepsilon , \delta }>0\) is constant depending on \(\alpha \), \(\varepsilon \), and \(\delta \) which is finite provided that we choose \(\delta <\frac{\varepsilon }{1-\alpha }\), as we are assuming from now on. We thus get
Now, recalling Example 1, we immediately see that
Since one easily recognizes that
the second condition on p in (4.23) can be dropped. As for the first condition on p in (4.23), it is readily seen that
and, similarly,
Finally, in the case \(\varepsilon \in (1 - \alpha , 1]\), we exploit (4.22) for \(\delta = 1\) in order to conclude that
for all \(x\in {\mathbb {R}}\), which implies that \(u_{\alpha } \in L^{\infty }({\mathbb {R}})\). The conclusion thus follows. \(\square \)
Thanks to Proposition 5, we can now give the following example.
Example 2
(Corollary 3is sharp for \(n=1\)) Let \(\alpha \in (0,1)\) and let \(\nu \) and \(u_\alpha \) be as in Proposition 5. By [12, Corollary 4.12], there exists a compact set \(K\subset {\mathbb {R}}\) such that 
, so that \(D^{\alpha } u_{\alpha }\not \ll {\mathscr {H}}^{{s}}\) for all \(s > \varepsilon \). Now we observe that, by (4.16), we have the following situations:
-
if \(\varepsilon \in (0, 1 - \alpha )\), then \(p< \frac{1 - \varepsilon }{1 - \alpha - \varepsilon } < \frac{1}{1 - \alpha - \varepsilon }\) and thus \(\varepsilon > \frac{1}{q} - \alpha \);
-
if \(\varepsilon = 1 - \alpha \), then \(p \in [1, +\infty )\) and thus \(\varepsilon > \frac{1}{q} - \alpha \);
-
if \(\varepsilon \in (1 - \alpha , 1]\), then \(p \in [1, +\infty ]\) and so, for \(p=+\infty \), if \(s > 1 - \alpha \) then we can take \(\varepsilon \in (1 - \alpha , s)\).
Therefore, the absolute continuity property of the fractional variation stated in Theorem 1(ii) is sharp for \(n=1\).
5 Fractional capacity and precise representative
In this last section, we study the fractional capacity and the existence of the precise representatives of \(BV^{\alpha ,p}\) functions.
5.1 The \((\alpha ,p)\)-capacity
We begin with the definition of fractional capacity, see [1, Chapter 2]. For the classical integer case \(\alpha =1\), we also refer the reader to [11, Sections 4.7 and 5.6.3], [15, Chapter 2], [21, Section 2.1] and [22, Section 2.2]. Here and in the following, we repeatedly use the identification \(S^{\alpha ,p}({\mathbb {R}}^n)=L^{\alpha ,p}({\mathbb {R}}^n)\) for \(\alpha \in (0,1)\) and \(p\in (1,+\infty )\) proved in [6, Corollary 2.1].
Definition 1
(The \((\alpha ,p)\)-capacity) Let \(\alpha \in (0,1)\) and \(p\in [1,+\infty )\). We let
be the \((\alpha ,p)\)-capacity of the compact set \(K\subset {\mathbb {R}}^n\).
The mapping \({{\,\mathrm{Cap}\,}}_{\alpha ,p}\) can be extended to more general sets via the following standard routine. If \(A\subset {\mathbb {R}}^n\) is an open set, then we set
and so, given any set \(E\subset {\mathbb {R}}^n\), we let
We now recall the notion of \((\alpha ,p)\)-quasievery point, see [1, Definition 2.2.5].
Definition 2
(\((\alpha ,p)\)-quasievery point) Let \(\alpha \in (0,1)\) and \(p\in [1,+\infty )\). We say that a property \(\mathscr {P}(x)\) is true for \((\alpha ,p)\)-quasievery \(x\in {\mathbb {R}}^n\) if
Recall that, if \(\alpha \in (0,1)\) and \(p\in \left( \frac{n}{\alpha },+\infty \right) \), then \(S^{\alpha ,p}({\mathbb {R}}^n)\subset C_b({\mathbb {R}}^n)\) continuously by the fractional Sobolev Embedding Theorem, see [1, Theorem 1.2.4(c)] for instance. For this reason, the notion of \((\alpha ,p)\)-capacity becomes interesting only when \(\alpha p\le n\) (see the discussion below [1, Proposition 2.6.1]). Precisely, if \(\alpha \in (0,1)\) and \(p\in \left( 1,\frac{n}{\alpha }\right] \), then \({\mathscr {H}}^{{n-\alpha p+\varepsilon }} \ll {{\,\mathrm{Cap}\,}}_{\alpha ,p} \) for all \(\varepsilon >0\), see [1, Theorem 5.1.13 and Corollary 5.1.14].
5.2 The precise representative
We now study the precise representatives of \(BV^{\alpha ,p}\) functions by combining the embedding proved in Theorem 7 with the results already known in the literature for the precise representatives of functions in fractional Bessel potential spaces.
We begin by recalling the definition of quasicontinuity. For the integer case \(\alpha =1\), we refer the reader to [1, Definition 6.1.1] and [11, Definition 4.11].
Definition 3
(\((\alpha ,p)\)-quasicontinuity) We say that a function \(f:{\mathbb {R}}^n\rightarrow [-\infty ,+\infty ]\) defined \((\alpha ,p)\)-quasieverywhere is \((\alpha ,p)\)-quasicontinuous if, for each \(\varepsilon >0\), there exists an open set \(A_\varepsilon \subset {\mathbb {R}}^n\) such that \({{\,\mathrm{Cap}\,}}_{\alpha ,p}(A_\varepsilon )<\varepsilon \) and \(f|_{{\mathbb {R}}^n\setminus A_\varepsilon }\) is continuous.
Here and in the following, the precise representative of a function \(u\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n;{\mathbb {R}}^m)\) is defined as
if the limit exists, otherwise \(u^{\star }(x)=0\) by convention. The following result provides a precise description of the continuity properties of the precise representative of a function in \(S^{\alpha ,p}({\mathbb {R}}^n)\) for \(p\in \left( 1,\frac{n}{\alpha }\right] \). We refer the reader to [1, Theorem 6.2.1] for the proof.
Theorem 11
(Quasicontinuity of \(S^{\alpha ,p}\) functions) Let \(\alpha \in (0,1)\) and \(p\in \left( 1,\frac{n}{\alpha }\right] \). If \(f\in S^{\alpha ,p}({\mathbb {R}}^n)\), then \(f^{\star }\) is an \((\alpha ,p)\)-quasicontinuous representative of f and
for \((\alpha ,p)\)-quasievery \(x\in {\mathbb {R}}^n\), where
Thanks to the embedding proved in Theorem 7, we immediately deduce the following result concerning the quasicontinuity of the functions in \(BV^{\alpha ,p}({\mathbb {R}}^n)\).
Corollary 4
(Quasicontinuity of \(BV^{\alpha ,p}\) functions for \(p<\frac{n}{n-\alpha }\)) Let \(\alpha ,\beta \in (0,1)\), with \(\beta <\alpha \), and let \(p, q \in [1,+\infty ]\) be such that \(p\le q <\frac{n}{n+\beta -\alpha }\), with \(q>1\). If \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\), then \(f^{\star }\) is a \((\beta ,q)\)-quasicontinuous representative of f and
for \((\beta ,q)\)-quasievery \(x\in {\mathbb {R}}^n\) and for all \(t\in \left[ 1,\frac{nq}{n-\beta q}\right] \).
In order to provide an extension of Corollary 4 also for all exponents \(p\in [1,+\infty ]\), we need the following localization result for \(BV^{\alpha ,p}\) functions.
Lemma 1
(Localization for \(BV^{\alpha ,p}\) functions for \(p\in {[}1,+\infty {]}\)) Let \(\alpha \in (0,1)\) and let \(p\in [1,+\infty ]\). If \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\) and \(\eta \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\), then \(f\eta \in BV^{\alpha ,q}({\mathbb {R}}^n)\) for all \(q\in [1,p]\), with
Proof
By Hölder’s inequality, we clearly have that \(f\eta \in L^q({\mathbb {R}}^n)\) for all \(q\in [1,p]\), so that we only need to prove (5.1). First of all, note that the right-hand side of (5.1) is well posed because \(\eta \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\). In particular, we have \(\nabla ^\alpha \eta \in L^1({\mathbb {R}}^n;{\mathbb {R}}^n)\cap L^\infty ({\mathbb {R}}^n;{\mathbb {R}}^n)\) by [7, Corollary 2.3] and \(\nabla ^\alpha _{\mathrm{NL}}(f,\eta )\in L^1({\mathbb {R}}^n)\), since by Minkowski’s and Hölder’s (generalized) inequalities we can estimate
(for the validity of the last inequality, see [18, Theorem 17.33] for instance). Now let \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\) be given. By [7, Lemma 2.7], we can write
so that
In addition, since \(\eta \varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\) and \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\), we immediately see that
Finally, let \((f_\varepsilon )_{\varepsilon >0}\subset BV^{\alpha ,p}({\mathbb {R}}^n)\cap C^\infty ({\mathbb {R}}^n)\) be such that \(f_\varepsilon =f*\varrho _\varepsilon \) for all \(\varepsilon >0\) as in Theorem 4. Note that \(f_\varepsilon \in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\), so that
for all \(\varepsilon >0\) by [8, Lemmas 2.4 and 2.5]. Now, arguing as in the proof of (5.2), we can infer that
so that we can pass to the limit as \(\varepsilon \rightarrow 0^+\) in (5.3) to get that
In conclusion, we have that
for any given \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\) and the proof is complete. \(\square \)
Corollary 5
(Quasicontinuity of \(BV^{\alpha ,p}\) functions for \(p\in {[}1,+\infty {]}\)) Let \(\alpha ,\beta \in (0,1)\) be such that \(\beta <\alpha \) and let \(p\in \left[ 1, + \infty \right] \) and \(q\in \left[ 1,\frac{n}{n+\beta -\alpha }\right) \). If \(f\in BV^{\alpha ,p}({\mathbb {R}}^n)\), then \(f^{\star }\) is a \((\beta ,q)\)-quasicontinuous representative of f and
for \((\beta ,q)\)-quasievery \(x\in {\mathbb {R}}^n\) and for all \(t\in \left[ 1,\frac{nq}{n-\beta q}\right] \). In particular, the precise representative of f is well defined \({\mathscr {H}}^{{n - \alpha +\varepsilon }}\)-a.e. for all \(\varepsilon >0\).
Proof
By Lemma 1, we know that \(f\eta \in BV^{\alpha , 1}({\mathbb {R}}^n)\) for all \(\eta \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\). Hence, Corollary 4 implies the existence of a \((\beta ,q)\)-quasicontinuous representative of \(f\eta \) for all \(\beta \in (0, \alpha )\) and \(q\in \left( 1,\frac{n}{n+\beta -\alpha }\right) \). In particular, if \(\eta (x)=1\) for all \(x\in B_R\) for some given \(R>0\), then we get the existence of \((f\eta )^{\star }(x) =f^{\star }(x)\) for \((\beta ,q)\)-quasievery \(x\in {\mathbb {R}}^n\), together with (5.4). Since \(R > 0\) is arbitrary, \(f^{\star }(x)\) must exist for \((\beta ,q)\)-quasievery \(x\in {\mathbb {R}}^n\). Finally, since \(q < \frac{n}{\beta }\), we have \( {\mathscr {H}}^{{n-\beta q+\delta }} \ll {{\,\mathrm{Cap}\,}}_{\beta ,q} \) for all \(\delta >0\). Thus, by optimizing in \(\beta \in (0, \alpha )\) and in \(q\in \left( 1,\frac{n}{n+\beta -\alpha }\right) \), the existence of \(f^{\star }(x)\) follows for \({\mathscr {H}}^{{n - \alpha + \varepsilon }}\)-a.e. \(x \in {\mathbb {R}}^n\) and the proof is complete. \(\square \)
References
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin (1996)
Alvino, A.: Sulla diseguaglianza di Sobolev in spazi di Lorentz. Boll. Un. Mat. Ital. A (5) 14(1), 148–156 (1977)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)
Bellido, J.C., Cueto, J., Mora-Corral, C.: Fractional Piola identity and polyconvexity in fractional spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 37(4), 955–981 (2020)
Bellido, J.C., Cueto, J., Mora-Corral, C.: \(\Gamma \)-convergence of polyconvex functionals involving \(s\)-fractional gradients to their local counterparts. Calc. Var. Partial Differential Equations 60(1), 1–29 (2021)
Bruè, E., Calzi, M., Comi, G.E., Stefani, G.: A distributional approach to fractional Sobolev spaces and fractional variation: Asymptotics II. C. R. Math. (to appear) (2021)
Comi, G.E., Stefani, G.: A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up. J. Funct. Anal. 277(10), 3373–3435 (2019)
Comi, G.E., Stefani, G.: A distributional approach to fractional Sobolev spaces and fractional variation: Asymptotics I. ArXiv preprint arXiv:1910.13419 (2019)
Comi, G.E., Stefani, G.: Leibniz rules and Gauss–Green formulas in distributional fractional spaces. ArXiv preprint arXiv:2111.13942 (2021)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Revised Textbooks in Mathematics, CRC Press, Boca Raton, FL (2015)
Falconer, K.: Fractal Geometry, 3rd edn. Mathematical Foundations and Applications. John Wiley & Sons Ltd, Chichester (2014)
Grafakos, L.: Classical Fourier analysis. Third Edition. Graduate Texts in Mathematics, vol. 250, Springer, New York (2014)
Grafakos, L.: Modern Fourier analysis. Third Edition. Graduate Texts in Mathematics, vol. 250, Springer, New York (2014)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover Publications Inc, Mineola, NY (2006)
Horváth, J.: On some composition formulas. Proc. Amer. Math. Soc. 10, 433–437 (1959). https://doi.org/10.2307/2032862
Kreisbeck, C., Schönberger, H.: Quasiconvexity in the fractional calculus of variations: Characterization of lower semicontinuity and relaxation. Nonlinear Anal. 215, 112625 (2022). https://doi.org/10.1016/j.na.2021.112625
Leoni, G.: A First Course in Sobolev Spaces. Second Edition. Graduate Studies in Mathematics, vol. 181, American Mathematical Society, Providence, RI (2017)
Liu, L., Xiao, J.: Fractional Hardy-Sobolev \(L^1\)-embedding per capacity-duality. Appl. Comput. Harmon. Anal. 51, 17–55 (2021). https://doi.org/10.1016/j.acha.2020.10.001
Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems. Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/CBO9781139108133
Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs, vol. 51, American Mathematical Society, Providence, RI (1997). https://doi.org/10.1090/surv/051
Maz’ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-15564-2
O’Neil, R.: Convolution operators and \(L(p,\, q)\) spaces. Duke Math. J. 30, 129–142 (1963)
Ponce, A.C.: Elliptic PDEs, Measures and Capacities. EMS Tracts in Mathematics, vol. 23, European Mathematical Society (EMS), Zürich (2016). https://doi.org/10.4171/140
Schikorra, A., Shieh, T.-T., Spector, D.: \(L^p\) theory for fractional gradient PDE with \(VMO\) coefficients. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl 26(4), 433–443 (2015). https://doi.org/10.4171/RLM/714
Schikorra, A., Shieh, T.-T., Spector, D.E.: Regularity for a fractional \(p\)-Laplace equation. Commun. Contemp. Math. 20(1), 1750003 (2018). https://doi.org/10.1142/S0219199717500031
Schikorra, A., Spector, D., Van Schaftingen, J.: An \(L^1\)-type estimate for Riesz potentials. Rev. Mat. Iberoam. 33(1), 291–303 (2017). https://doi.org/10.4171/RMI/937
Shieh, T.-T., Spector, D.E.: On a new class of fractional partial differential equations. Adv. Calc. Var. 8(4), 321–336 (2015). https://doi.org/10.1515/acv-2014-0009
Shieh, T.-T., Spector, D.E.: On a new class of fractional partial differential equations II. Adv. Calc. Var. 11(3), 289–307 (2018). https://doi.org/10.1515/acv-2016-0056
Šilhavý, M.: Fractional vector analysis based on invariance requirements (Critique of coordinate approaches). M. Continuum Mech. Thermodyn. 32(1), 207–228 (2020)
Spector, D.: A noninequality for the fractional gradient. Port. Math. 76(2), 153–168 (2019). https://doi.org/10.4171/pm/2031
Spector, D.: An optimal Sobolev embedding for \(L^1\). J. Funct. Anal. 279(3), Art. 108559 (2020). https://doi.org/10.1016/j.jfa.2020.108559
Spector, D.: New directions in harmonic analysis on \(L^1\). Nonlinear Anal. 192, Art. 111685 (2020). https://doi.org/10.1016/j.na.2019.111685
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, N.J. (1970)
Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ (1993)
Acknowledgements
The first author is a member of INdAM–GNAMPA and is partially supported by the PRIN 2017 Project Variational methods for stationary and evolution problems with singularities and interfaces (n. prot. 2017BTM7SN). The second author is supported by the Taiwan Ministry of Science and Technology under research grant number 110-2115-M-003-020-MY3 and the Taiwan Ministry of Education under the Yushan Fellow Program. Part of this work was undertaken while the second author was visiting the National Center for Theoretical Sciences in Taiwan. He would like to thank the NCTS for its support and warm hospitality during the visit. The third author is a member of INdAM–GNAMPA and is partially supported by the ERC Starting Grant 676675 FLIRT – Fluid Flows and Irregular Transport and by the INdAM–GNAMPA Project 2020 Problemi isoperimetrici con anisotropie (n. prot. U-UFMBAZ-2020-000798 15-04-2020).
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Comi, G.E., Spector, D. & Stefani, G. The fractional variation and the precise representative of \(BV^{\alpha ,p}\) functions. Fract Calc Appl Anal 25, 520–558 (2022). https://doi.org/10.1007/s13540-022-00036-0
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DOI: https://doi.org/10.1007/s13540-022-00036-0
Keywords
- Fractional gradient
- Fractional divergence
- Fractional variation
- Hausdorff measure
- Fractional capacity
- Precise representative