Abstract
We prove a characterization of Hardy’s inequality in Sobolev–Slobodeckiĭ spaces in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation. This extends previous results by Ancona Kinnunen & Korte for standard Sobolev spaces. The proof is based on variational methods.
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1 Introduction
1.1 Main result
The present note deals with the fractional Hardy inequality in Sobolev–Slobodeckiĭ spaces. For \(1<p<\infty \), \(0<s<1\) and \(\Omega \subsetneq {\mathbb {R}}^N\) an open set, this takes the form
where
This note can be seen as a companion paper of [2], where the main result here presented is applied. We refer the reader to [2] for more details on (1.1). The paper [2] deals with the problem of determining the sharp constant in (1.1). This is the quantity defined by
In order to present the main result, we need to introduce the equation
where \(\lambda \ge 0\). The symbol \((-\Delta _p)^s\) stands for the fractional \(p-\) Laplacian of order s, defined in weak form by the first variation of the convex functional
see Sect. 2 for more details.
Actually, There is a tight connection between positive supersolutions of (1.2) and the constant \({\mathfrak {h}}_{s,p}(\Omega )\). This connection is encoded in the following result, which is the main goal of the present note. We refer to the comments below, for a comparison with known results.
Theorem 1.1
Let \(1<p<\infty \), \(0<s<1\) and let \(\Omega \subsetneq {\mathbb {R}}^N\) be an open set. Then we have
The proof of this result is a direct consequence of the following equivalence (see Lemmas 4.1 and 4.2 below)
Such an equivalence is quite well-known among experts, at least in the local case, i.e. for standard Sobolev spaces. The case \(p=2\) is contained for example in the classical paper [1, Appendix] by Alano Ancona and it has been generalized to \(p\not =2\) by Juha Kinnunen and Riika Korte, see [10, Theorem 5.1]. We should notice that the case \(p=2\) and \(0<s<1\) can be obtained from [8, Theorem 1.9] by Patrick Joseph Fitzsimmons, which is concerned with the more general framework of Dirichlet forms associated to symmetric Markov processes. We point out that in [8] the author uses a probabilistic approach, which can not be applied to the case \(p\not =2\).
In all these references, the proof of the implication “\(\Longrightarrow \)” is based on the Lax-Milgram Theorem for coercive bilinear forms and its non-Hilbertian variants.Footnote 1 Our proof of this fact is slightly different: more precisely, we use a purely variational approach (see Lemma 4.2 below) in order to show existence of a supersolution. This is quite delicate, since in Theorem 1.1 we do not have any assumption on the open set \(\Omega \), apart from the fact that it admits a Hardy inequality. Thus, when employing the Direct Method in the Calculus of Variations, some non-trivial compactness issues arise. A careful study of a suitable weighted Sobolev–Slobodeckiĭ space is needed at that point (see Sect. 3). This also gives us the opportunity to make some precisions on the correct functional analytic setting which is needed for this result (see Remark 3.7).
1.2 Plan of the paper
In Sect. 2 we introduce the main notation and definitions. Section 3 is devoted to discuss in detail a weighted Sobolev–Slobodeckiĭ space. This in an essential tool in the proof of Theorem 1.1, which can be found in Sect. 4.
2 Notation and definitions
For every \(1<p<\infty \), we indicate by \(J_p:{\mathbb {R}}\rightarrow {\mathbb {R}}\) the monotone increasing continuous function defined by
For \(x_0\in {\mathbb {R}}^N\) and \(R>0\), we will set
and \(\omega _N:=|B_1(x_0)|\). For an open set \(\Omega \subsetneq {\mathbb {R}}^N\), we denote by
the distance function from the boundary. Whenever such a distance is bounded, we will set
This will be called inradius of the set \(\Omega \). For two open sets \(\Omega '\subset \Omega \subset {\mathbb {R}}^N\), we will write \(\Omega '\Subset \Omega \) to indicate that the closure \(\overline{\Omega '}\) is a compact set contained in \(\Omega \).
For \(1<p<\infty \) and \(0<s<1\), we consider the fractional Sobolev space
where
This is a reflexive Banach space, when endowed with the natural norm
see for example [11, Proposition 17.6 and Theorem 17.41]. For an open set \(\Omega \subset {\mathbb {R}}^N\), we indicate by \({\widetilde{W}}^{s,p}_0(\Omega )\) the closure of \(C^\infty _0(\Omega )\) in \(W^{s,p}({\mathbb {R}}^N)\). It is intended that functions in \(C^\infty _0(\Omega )\) are considered as elements of \(C^\infty _0({\mathbb {R}}^N)\), by extending them to be zero outside \(\Omega \). Occasionally, for an open set \(\Omega \subset {\mathbb {R}}^N\), we will need the fractional Sobolev space defined by
where
Finally, \(W^{s,p}_{\textrm{loc}}(\Omega )\) is the space of functions \(u\in L^p_{\textrm{loc}}(\Omega )\) such that \(u\in W^{s,p}(\Omega ')\) for every \(\Omega '\Subset \Omega \).
For \(0<\beta <\infty \), we also denote by \(L^\beta _{s\,p}({\mathbb {R}}^N)\) the following weighted Lebesgue space
We observe that this is a Banach space for \(\beta \ge 1\), when endowed with the natural norm. Moreover, it is not difficult to see that
It is sufficient to use that
and then apply Jensen’s inequality.
Definition 2.1
We say that \(u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) is a
-
local weak supersolution of (1.2) if
$$\begin{aligned} \iint _{{\mathbb {R}}^N\times {\mathbb {R}}^N} \frac{J_p(u(x)-u(y))\,(\varphi (x)-\varphi (y))}{|x-y|^{N+s\,p}}\,dx\,dy \ge \lambda \,\int _\Omega \frac{|u(x)|^{p-2}\,u(x)}{d_\Omega (x)^{s\,p}}\,\varphi (x)\,dx,\nonumber \\ \end{aligned}$$(2.3)for every non-negative \(\varphi \in W^{s,p}({\mathbb {R}}^N)\) with compact support in \(\Omega \);
-
local weak solution of (1.2) if (2.3) holds as an equality, for every \(\varphi \in W^{s,p}({\mathbb {R}}^N)\) with compact support in \(\Omega \).
Remark 2.2
Under the assumptions taken on u and the test functions, the previous definition is well-posed, i.e.
We also observe that if a local weak solution u belongs to \({{\widetilde{W}}}^{s,p}_0(\Omega )\), then by a density argument we can take \(\varphi =u\) itself as a test function in the weak formulation.
3 A weighted fractional Sobolev space
In the proof of Theorem 1.1, we will crucially exploit a suitable weighted fractional Sobolev space, whose definition is inspired by [1, Appendix].
Definition 3.1
Let \(1<p<\infty \), \(0<s<1\) and let \(\Omega \subsetneq {\mathbb {R}}^N\) be an open set, we define
endowed with the norm
Then we define \({\mathcal {X}}^{s,p}_0(\Omega ; d_{\Omega })\) as the closure of \(C^{\infty }_0(\Omega )\) in \({\mathcal {X}}^{s,p}(\Omega ;d_\Omega )\).
Remark 3.2
We observe that if the open set \(\Omega \subsetneq {\mathbb {R}}^N\) is such that \({\mathfrak {h}}_{s,p}(\Omega )>0\), then by a simple density argument we can assure that Hardy’s inequality holds in \({\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\), as well. That is, we have
Accordingly, this implies that in this case
defines an equivalent norm on \({\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\).
Proposition 3.3
Let \(1<p<\infty \) and \(0<s<1\). Let \(\Omega \subsetneq {\mathbb {R}}^N\) be an open set. Then
and we have the estimate
Moreover, \({\mathcal {X}}^{s,p}(\Omega ;d_\Omega )\) and \({\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\) are Banach spaces.
Proof
The first fact is straightforward, by also taking into account (2.2).
We prove the estimate (3.1). We take a ball \(B_{R}(x_0)\Subset \Omega \) such that \(B_{2\,R}(x_0)\Subset \Omega \), as well. We then write
thanks to Minkowski’s inequality. By observing that
we have
This implies that we have
for a constant \(C=C(N,s,p,\Omega ,B_R(x_0))>0\). We now use that
together with the fact that
By using these in (3.2), we get
possibly with a different constant \(C=C(N,s,p,\Omega ,B_R(x_0))>0\). The proof of estimate (3.1) is almost over: it is now sufficient to add on both sides of the previous estimate the term
By using that
for some \(C=C(s,p,\Omega ,B_{2\,R}(x_0))>0\), we eventually get the desired conclusion.
We prove the second part of the statement. We first observe that it is sufficient to prove that \({\mathcal {X}}^{s,p}(\Omega ;d_\Omega )\) is a Banach space. We take \(\{u_n\}_{n\in {\mathbb {N}}}\subset {\mathcal {X}}^{s,p}(\Omega ;d_\Omega )\) to be a Cauchy sequence. Then we get that this is a Cauchy sequence in the Banach space \(L^p(\Omega ;d_\Omega ^{-s\,p})\) and that
is a Cauchy sequence in \(L^p({\mathbb {R}}^N\times {\mathbb {R}}^N)\). This follows from the fact that
Moreover, according to (3.1), the sequence \(\{u_n\}_{n\in {\mathbb {N}}}\) is also a Cauchy sequence in the Banach space \(L^p_{s\,p}({\mathbb {R}}^N)\). The last fact implies that there exists \(u\in L^p_{s\,p}({\mathbb {R}}^N)\) such that
In particular, up to a subsequence, we can suppose that \(u_n\) converges to u almost everywhere in \({\mathbb {R}}^N\). By using the completeness of \(L^p(\Omega ;d_\Omega ^{-s\,p})\), we get similarly the existence of \({\widetilde{u}}\in L^p(\Omega ;d_\Omega ^{-s\,p})\) such that
By uniqueness of the limit, we must have \(u={{\widetilde{u}}}\) almost everywhere in \(\Omega \). Finally, by using that \(L^p({\mathbb {R}}^N\times {\mathbb {R}}^N)\) is a Banach space, we get that there exists \(\phi \in L^p({\mathbb {R}}^N\times {\mathbb {R}}^N)\) such that
This in particular would imply that
up to a subsequence. On the other hand, by using the almost everywhere convergence of \(u_n\) previously inferred, we also obtain that
By using the uniqueness of the limit, we get at the same time that
This concludes the proof. \(\square \)
In the next technical lemma, we show that the summability of a negative power of the distance implies certain geometric properties of the open set.
Lemma 3.4
Let \(N\ge 1\) and let \(\Omega \subsetneq {\mathbb {R}}^N\) be an open set such that
for some \(\alpha >0\). Then we must have \(\alpha <N\). Moreover, we have the estimates
where \(r_\Omega \) is defined in (2.1).
Proof
We take \(x_0\in \Omega \) and consider the open ball \(B_r(x_0)\) with radius \(r=d_\Omega (x_0)\). This implies that
Let us call \({\widetilde{x}}_0\) a point contained in this intersection. By observing that
we get
By using spherical coordinates, we see that the last integral diverges for \(\alpha \ge N\). Thus we get the first statement.
In order to get the claimed estimates, we go on by estimating from below the last integral as follows
Since \(\alpha <N\) from the first part of the proof, we can take the supremum on \(x_0\in \Omega \) and get that the distance function is actually bounded. Moreover, we obtain the first estimate in (3.3), thus in particular the inradius is finite. In turn, by using this fact we get
which shows that the volume is finite, as well, together with the second estimate in (3.3). This concludes the proof. \(\square \)
As a straightforward consequence of Lemma 3.4, we get the following
Lemma 3.5
Let \(1<p<\infty \), \(0<s<1\) and let \(\Omega \subsetneq {\mathbb {R}}^N\) be an open set. Then for \(s\,p\ge N\) the unique constant function contained in \({\mathcal {X}}^{s,p}(\Omega ;d_\Omega )\) is the null one.
The same conclusion holds also for \(s\,p<N\), if we additionally suppose that \(|\Omega |=+\infty \).
In the next result we compare the two spaces \({\widetilde{W}}^{s,p}_0(\Omega )\) and \({\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\).
Proposition 3.6
Let \(1<p<\infty \) and \(0<s<1\). Let \(\Omega \subsetneq {\mathbb {R}}^N\) be an open set such that \({\mathfrak {h}}_{s,p}(\Omega )>0\). Then we have
and the inclusion is continuous. Moreover, if we assume that \(r_\Omega <+\infty \), then
and
is an equivalent norm on this space. Finally, if we further require that \(|\Omega |<+\infty \), then we have the continuous embedding
and this is compact, as well.
Proof
By recalling Remark 3.2, we know that (3.5) is an equivalent norm on \({\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\). Since we trivially have
the continuous inclusion (3.4) easily follows.
We now assume that \(r_\Omega <+\infty \). In conjuction with Hardy’s inequality and recalling (2.1), this yields
Thus we get that
are equivalent norms on \(C^\infty _0(\Omega )\), again thanks to Remark 3.2. Then the claimed identity of the two closures immediately follows. The last statement is now an easy consequence of the same property for the space \({\widetilde{W}}^{s,p}_0(\Omega )\), which is well-known. \(\square \)
Remark 3.7
Under the sole assumption that \({\mathfrak {h}}_{s,p}(\Omega )>0\), in general we have
contrary to what incorrectly stated in [10, Theorem 5.1], for the local case \(s=1\). As a counter-example, it is sufficient to take any open set \(\Omega \subsetneq {\mathbb {R}}^N\) such that
For example, we can take \(\Omega \) to be a half-space. In such a case, we have by construction
while
We can finally prove a compactness result for the space \({\mathcal {X}}^{s,p}_0(\Omega )\), under minimal assumptions on the open set \(\Omega \). This will be crucially exploited in the proof of Lemma 4.2.
Theorem 3.8
Let \(1<p<\infty \), \(0<s<1\) and let \(\Omega \subsetneq {\mathbb {R}}^N\) be an open set such that \({\mathfrak {h}}_{s,p}(\Omega )>0\). Let \(\{u_n\}_{n \in {\mathbb {N}}}\subset {\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\) be such that
for some \(M>0\). Then there exist a function \(u \in {\mathcal {X}}^{s,p}_0(\Omega ; d_{\Omega })\) and a subsequence \(\{u_{n_k}\}_{k\in {\mathbb {N}}}\) such that
Moreover, for every \(\Omega '\Subset \Omega \), we also have
up to a possible further subsequence.
Proof
We need two distinguish two cases: either \(|\Omega |<+\infty \) or \(|\Omega |=+\infty \).
Case 1: \(\Omega \) has finite volume. This is the easiest case: here the result plainly follows from Proposition 3.6. We also observe that the last statement actually holds in a stronger form, since we can infer convergence in \(L^p(\Omega )\).
Case 2: \(\Omega \) has infinite volume. We still use the notation \(D^s \varphi \) for a measurable function, as in Proposition 3.3. Thus, by assumption, we get that \(\{D^s u_n\}_{n\in {\mathbb {N}}}\) is a bounded sequence in \(L^p({\mathbb {R}}^N\times {\mathbb {R}}^N)\). This entails that, up to a subsequence, it is weakly converging in \(L^p({\mathbb {R}}^N\times {\mathbb {R}}^N)\). For simplicity, we do not relabel the subsequence. Let us call \(\phi \) such a limit. We may apply Mazur’s Lemma (see [12, Theorem 2.13]) and get that for every \(n\in {\mathbb {N}}\) there exists
and such that the new sequence made of convex combinations
strongly converges in \(L^p({\mathbb {R}}^N\times {\mathbb {R}}^N)\) to \(\phi \), as n goes to \(\infty \). Observe that by construction we have
and
since the latter is a vector space. This proves that \(\{D^s {\widetilde{u}}_n\}_{n\in {\mathbb {N}}}\) is a Cauchy sequence in \(L^p({\mathbb {R}}^N\times {\mathbb {R}}^N)\) and this, in turn, implies that \(\{{{\widetilde{u}}}_n\}_{n\in {\mathbb {N}}}\) is a Cauchy sequence in \({\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\), thanks to Remark 3.2. By using that \({\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\) is a Banach space, we get that \(\{{{\widetilde{u}}}_n\}_{n\in {\mathbb {N}}}\) converges in this space to a limit function \(u\in {\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\). In particular, we must have
We now want to prove that \(\{u_n\}_{n\in {\mathbb {N}}}\) converges almost everywhere on \({\mathbb {R}}^N\) to the function u, up to a subsequence. We first observe that all the elements of \({\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\) vanish almost everywhere in \({\mathbb {R}}^N\setminus \Omega \), by construction. Thus we only need to prove convergence almost everywhere in \(\Omega \).
We denote by \(\{\Omega _k\}_{k \in {\mathbb {N}}}\) an exhausting sequence for \(\Omega \), made of bounded open subsets with smooth boundary: in other words
see [6, Proposition 8.2.1]. We preliminary observe that, thanks to the assumption \({\mathfrak {h}}_{s,p}(\Omega )>0\), we have for every \(k,n\in {\mathbb {N}}\)
which entails that \(\{u_n\}_{n \in {\mathbb {N}}}\) is a bounded sequence in each \(W^{s,p}(\Omega _k)\). By using the compactness of the embedding \(W^{s,p}({\Omega _k}) \hookrightarrow L^p({\Omega _k})\) (see for example [7, Theorem 7.1]) and a diagonal argument, we can obtain existence of a function \(U\in W^{s,p}_{\textrm{loc}}(\Omega )\) and of a subsequence \(\{u_{n_k}\}_{k\in {\mathbb {N}}}\) such that
We then extend U to be 0 outside \(\Omega \): by using Fatou’s Lemma and the almost everywhere convergence, we get
By further using Hardy’s inequality and (3.1), we also get
and
This shows that
We now observe that from the first part of the proof, by uniqueness of the weak limit we must have
By recalling the definition of \(D^s\), this in turn implies that there exists a constant \(c\in {\mathbb {R}}\) such that
By using that \({\mathcal {X}}^{s,p}(\Omega ;d_\Omega )\) is a vector space, the function constantly equal to c must belong to \({\mathcal {X}}^{s,p}(\Omega ;d_\Omega )\). In light of Lemma 3.5, we get that \(c=0\) and thus the desired conclusion holds. \(\square \)
4 Proof of Theorem 1.1
Lemma 4.1
Let \(1<p<\infty \), \(0<s<1\) and let \(\Omega \subsetneq {\mathbb {R}}^N\) be an open set. Then:
-
(i) if there exists \(\lambda \ge 0\) such that the Eq. (1.2) admits a positive local weak supersolution u, then \(\lambda \le {\mathfrak {h}}_{s,p}(\Omega )\);
-
(ii) in particular, if u is a positive weak solution in \( {\widetilde{W}}^{s,p}_0(\Omega )\), then \(\lambda ={\mathfrak {h}}_{s,p}(\Omega )\) and u is a minimizer for \({\mathfrak {h}}_{s,p}(\Omega )\).
Proof
In order to prove (i), for every \(\eta \in C^\infty _0(\Omega )\), we test the weak formulation with
where \(\varepsilon >0\). We observe that this is a feasible test function, thanks to [2, Lemma 2.7]. By using the discrete Picone inequality (see [9, Lemma 2.6] or [3, Proposition 4.2]), we obtain
In the last inequality we used that
and the inequality is strict, unless \(\eta \) has constant sign almost everywhere (see the proof of [2, Lemma 3.2]). By taking the limit as \(\varepsilon \) goes to 0 on the left-hand side, using Fatou’s lemma, the positivity of u on \(\Omega \) and the arbitrariness of \(\eta \in C^\infty _0(\Omega )\), this finally gives that \(\lambda \le {\mathfrak {h}}_{s,p}(\Omega )\), as desired.
In order to prove point (ii), we observe that if \(u\in {\widetilde{W}}^{s,p}_0(\Omega )\), we can test the weak formulation of the equation with the solution itself. This yields
On the other hand, by definition of \({\mathfrak {h}}_{s,p}(\Omega )\), we know that
This shows that \({\mathfrak {h}}_{s,p}(\Omega )\le \lambda \). Since the reverse inequality holds from (i), we conclude that it must result \(\lambda ={\mathfrak {h}}_{s,p}(\Omega )\). \(\square \)
In the next Lemma, we will use the weighted space \({\mathcal {X}}^{s,p}_0(\Omega ; d_{\Omega })\) studied in Sect. 3.
Lemma 4.2
Let \(1<p<\infty \), \(0<s<1\) and let \(\Omega \subsetneq {\mathbb {R}}^N\) be an open set such that
Then for every \(0\le \lambda <{\mathfrak {h}}_{s,p}(\Omega )\) there exists a positive local weak supersolution \(u_\lambda \in {\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\) of the Eq. (1.2). More precisely, the function \(u_\lambda \) is a weak solution of the equation
where \(B \Subset \Omega \) is a fixed ball.
Proof
We first observe that, for every \(\varphi \in {\mathcal {X}}^{s,p}_0(\Omega , d_{\Omega })\), we have
thanks to Hölder’s inequality, the definition of \({\mathfrak {h}}_{s,p}(\Omega )\) and the fact that Hardy’s inequality holds in \({\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\), as well (see Remark 3.2). This shows that we have the continuous embedding \({\mathcal {X}}^{s,p}_0(\Omega ; d_{\Omega })\hookrightarrow L^1(B)\), for every \(B \Subset \Omega \) as in the statement.
Let \(0\le \lambda <{\mathfrak {h}}_{s,p}(\Omega )\), we consider the functional
We will construct the desired supersolution as a minimizer of the following problem
We first notice that by Hardy’s inequality we have, for every \(\varphi \in {\mathcal {X}}^{s,p}_0(\Omega ; d_{\Omega })\)
with \(\varepsilon >0\), where we also used Young’s inequality. In particular, by choosing
we can infer that
where \(c_1>0\) and \(C_1>0\) do not depend on \(\varphi \). This in particular shows that \(m(\lambda )>-\infty \).
Let us now take a minimizing sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset {\mathcal {X}}^{s,p}_0(\Omega ; d_{\Omega })\) such that
By appealing to (4.3), we get in particular that there exists a constant \(M>0\) such that
By applying Theorem 3.8, we can infer existence of \(u \in {\mathcal {X}}^{s,p}_0(\Omega ; d_{\Omega })\) such that the sequence converges almost everywhere in \({\mathbb {R}}^N\) and such that
up to a subsequence. Observe that by construction we have
which in particular implies that
By applying the Brézis-Lieb Lemma (see [5, Theorem 1] and also [4, Lemma 2.2]), we get
and
By inserting these informations in (4.4), we obtain
We can now use Hardy’s inequality for the function \(u_n-u\in {\mathcal {X}}^{s,p}_0(\Omega ;d_\Omega )\). Thanks to the choice of \(\lambda \), it holds that
and by taking the limit as n goes to \(\infty \), we finally get that u is a minimizer.
By minimality, we get that u must be non-negative. Indeed, by using (4.1) and observing that
we have
Moreover, the inequality sign in the latter is strict, unless u has constant sign almost everywhere. By virtue of the inequality for the integral on B, we get that such a sign must be non-negative, i.e. we must have \(u\ge 0\) almost everywhere in \(\Omega \), as claimed.
Additionally, by minimality u is a weak solution of the Euler-Lagrange equation (4.2). This in particular proves that \(u\not \equiv 0\), thanks to the presence of the term \(1_B\). Observe that (see Proposition 3.3)
thus u is a local weak supersolution, in the sense of Definition 2.1. Finally, by using the minimum principle, we get that u is positive on \(\Omega \) (one can proceed as in the proof of [2, Lemma 3.2], for example). \(\square \)
By joining the previous two technical results, we finally get the characterization of the sharp fractional \((s,p)-\)Hardy constant stated in Theorem 1.1.
Proof of Theorem 1.1
We first observe that the set of admissible \(\lambda \) is non-empty: indeed, it always contains \(\lambda =0\). To see this, it is sufficient to observe that any positive constant function is a local weak solution of
which is (1.2) for \(\lambda =0\).
In order to prove the claimed identity, we first consider the case \({\mathfrak {h}}_{s,p}(\Omega )=0\). Then, the previous discussion and Lemma 4.1 imply that the set of admissible \(\lambda \) is actually given by the singleton \(\{0\}\). Thus the conclusion holds.
In the case \({\mathfrak {h}}_{s,p}(\Omega )>0\), again by Lemma 4.1, we have that \({\mathfrak {h}}_{s,p}(\Omega )\ge \lambda \) for every \(\lambda \) such that (1.2) admits a positive local weak supersolution. On the other hand, from Lemma 4.2 we have that for every \(\varepsilon >0\) if we take
then (1.2) admits a positive local weak supersolution. This concludes the proof. \(\square \)
Notes
As an historical curiosity, we notice that the prototype of this kind of result can be traced back to a paper by Giuseppe Tomaselli, dealing with weighted Hardy inequalities for standard Sobolev spaces, in the one-dimensional case. We refer to [13, Lemma 2, point (ii)] for more details.
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Acknowledgements
We wish to thank Pierre Bousquet, Juha Kinnunen and Michele Miranda for some useful discussions on the topics of this paper. Part of this paper has been written during a staying of L. B. at the Institute of Applied Mathematics and Mechanics of the University of Warsaw, in the context of the Thematic Research Programme: “Anisotropic and Inhomogeneous Phenomena”, in July 2022. Iwona Chlebicka and Anna Zatorska-Goldstein are gratefully acknowledged for their kind invitation and the nice working atmosphere provided during the whole staying. F. B. and A. C. Z. are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). F. B., L. B. and A. C. Z. gratefully acknowledge the financial support of the projects FAR 2019 and FAR 2020 of the University of Ferrara.
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Bianchi, F., Brasco, L., Sk, F. et al. A note on the supersolution method for Hardy’s inequality. Rev Mat Complut 37, 323–340 (2024). https://doi.org/10.1007/s13163-023-00460-7
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DOI: https://doi.org/10.1007/s13163-023-00460-7