                TECHNION -- ISRAEL INSTITUTE OF TECHNOLOGY
                         DEPARTMENT OF MATHEMATICS

A homothety argument for computing the best constant in some Hardy’s inequalities for exterior and punctured domains

Shayne Waldron^{∗} (waldron@math.auckland.ac.nz)

                             Technical Report
                                 July 1997

ABSTRACT

An argument based on homothety transformations is used to prove a conjecture about multivariate Hardy’s inequalities of Matskewich and Sobolevskii [MS96]. This result and more general inequalities are seen to follow from a multivariate form of Hardy’s inequality recently given in Waldron [W97].

Key Words: Hardy’s inequality, homothety transformation, integral form of Minkowskii’s inequality, Sobolev’s embedding theorem, Taylor interpolation

AMS (MOS) Subject Classifications: primary 26D10, 41A44; secondary 41A80

∗ This work was supported by the Israel Council for Higher Education.

1. Introduction

Following the notation of Matskewich and Sobolevskii [MS96], for Ω an open subset of IR^{n}, define the ‘generalised Hardy’s functional’

l_{p}(u, Ω, m) := ∥D^{m}u∥_{L_{p}(Ω)} ; ∥u/(ρ_{∂Ω})^{m}∥_{L_{p}(Ω)} , 1 ≤ p ≤ ∞, u ∈ C_{0}^{∞}(Ω), u ≠ 0, m = 1, 2, …,

where

∥D^{m}u∥_{L_{p}(Ω)} := the L_{p}(Ω)–norm of |D^{m}u|,

with

|D^{m}u|(x) := sup_{∥ξ_{i}∥=1} |D_{ξ_{1}}D_{ξ_{2}} ⋯ D_{ξ_{m}}u(x)|,

and ρ_{∂Ω}(x) := the distance of x from ∂Ω, the boundary of the domain Ω.

When m = 1 this reduces to l_{p}(u, Ω) as in [MS96]. Here ∥ · ∥ is the Euclidean norm, and D_{ξ}f is the directional derivative of f in the (possibly nonunit) direction ξ, defined by

D_{ξ}f(x) := lim_{t→0} f(x + tξ) − f(x) ; t.

The main result of this paper is the computation for certain Ω ⊂ IR^{n} of

inf_{{u∈C^{∞}_{0}(Ω) \atop u≠0}} l_{p}(u, Ω, m) = C, (p ≥ n),

which for C > 0 is equivalent to the existence of the sharp Hardy–type inequality

∥u/(ρ_{∂Ω})^{m}∥_{L_{p}(Ω)} ≤ 1 ; C ∥D^{m}u∥_{L_{p}(Ω)},  ∀u ∈ C_{0}^{∞}(Ω).  (1.1)

We say that Ω is punctured (at a) if Ω = U \ {a}, a ∈ U, where U is an open set, and is an exterior domain if Ω = IR^{n} \ D, where D is a nonempty compact set. The precise form of our result is:

Theorem 1.2. Suppose that Ω ⊂ IR^{n} is a punctured open set, or an exterior domain, i.e., Ω := IR^{n} \ D, where D ⊂ IR^{n} is a nonempty compact set. Then, for p ≥ n, m = 1, 2, …,

inf_{{u∈C^{∞}_{0}(Ω) \atop u≠0}} l_{p}(u, Ω, m) = (1 − n ; p)(2 − n ; p) ⋯ (m − n ; p).  (1.3)

From this follows:

Corollary 1.4 ([MS96:Th.1]). Let Ω := IR^{3} \ D, where D ⊂ IR^{3} is a compact set whose boundary is a C^{2}–smooth, 2–dimensional manifold. For p ≥ 3,

inf_{{u∈C^{∞}_{0}(Ω) \atop u≠0}} l_{p}(u, Ω) = inf_{{u∈C^{∞}_{0}(Ω) \atop u≠0}} l_{p}(u, Ω, 1) = 1 − 3 ; p.

The result analogous to Corollary 1.4 for IR^{2} (n = 2, m = 1) was also proved by Matskewich and Sobolevskii (unpublished).

In Section 3, some other Hardy–type inequalities similar to those of Theorem 1.2 are discussed. In particular, (1.3) can be extended to (3.5).

Fig. 1.1. A punctured open set and an exterior domain

2. Proof of Theorem 1.2

By using homothety transformations (dilations)

x ↦ λx, λ > 0,

we reduce the proof of Theorem 1.2 to the case when Ω = IR^{n} \ {0}.

Lemma 2.1. Suppose that Ω ⊂ IR^{n} is a punctured open set, or an exterior domain Then, for 1 ≤ p ≤ ∞, m = 1, 2, …,

inf_{{u∈C^{∞}_{0}(Ω) \atop u≠0}} l_{p}(u, Ω, m) = inf_{{u∈C^{∞}_{0}(IR^{n}\\{0\}) \atop u≠0}} l_{p}(u, IR^{n} \ \{0\}, m) =: I(p, m, n).

Proof: In the argument below, we find it convenient to view C_{0}^{∞}(Ω) as the subspace of C_{0}^{∞}(IR^{n}) consisting of those functions with compact support contained in Ω.

Without loss of generality assume that Ω is punctured at 0, or is an exterior domain for which 0 ∉ Ω. Then, each u ∈ C_{0}^{∞}(Ω) has compact support not containing 0, and so C_{0}^{∞}(Ω) ⊂ C_{0}^{∞}(IR^{n} \ {0}), giving

I(p, m, n) := inf_{{u∈C^{∞}_{0}(IR^{n}\\{0\}) \atop u≠0}} l_{p}(u, IR^{n} \ \{0\}, m) ≤ inf_{{u∈C^{∞}_{0}(Ω) \atop u≠0}} l_{p}(u, Ω, m).

Conversely, suppose that u ∈ C_{0}^{∞}(IR^{n} \ {0}). The change of variables formula shows that the generalised Hardy’s functional tranforms under the homothety transformation x ↦ λx as follows

l_{p}(u, Ω, m) = l_{p}(u(·/λ), λΩ, m),  ∀u ∈ C_{0}^{∞}(Ω).  (2.2)

If Ω is punctured at 0, then u(·/λ) ∈ C_{0}^{∞}(Ω) for sufficiently small λ > 0, and if Ω is an exterior domain, then u(·/λ) ∈ C_{0}^{∞}(Ω) for sufficiently large λ > 0. Hence in either case, by (2.2), we conclude

inf_{{u∈C^{∞}_{0}(Ω) \atop u≠0}} l_{p}(u, Ω, m) ≤ inf_{{u∈C^{∞}_{0}(IR^{n}\\{0\}) \atop u≠0}} l_{p}(u, IR^{n} \ \{0\}, m) =: I(p, m, n).

Thus, it is sufficient to compute I(p, m, n). To do this it is convenient to use the divided difference functional

f ↦ ∫_{[^{0,...,0}_{m},x]} f := 1 ; (m − 1)! ∫_{0}^{1}(1 − t)^{m−1}f(tx) dt.  (2.3)

(cf equation (2.1.9) of Waldron [W97]). This allows the (univariate) error formula for T_{<m} the operator of Taylor interpolation at 0 by n–variate polynomials of degree < m to be expressed as

u(x) − T_{<m}u(x) = ∫_{[^{0,...,0}_{m},x]} D_{x}^{m}u,  ∀u ∈ C^{m}(IR^{n}), ∀x ∈ IR^{n}.  (2.4)

The multivariate Hardy’s inequality that for Ω starshaped with respect to 0, and p > n

∥ x ↦ ∫_{[^{0,...,0}_{m},x]} f ∥_{L_{p}(Ω)} ≤ 1 ; (1 − n/p) ⋯ (m − n/p)∥f∥_{L_{p}(Ω)}  (2.5)

will be used. This is proved by applying the integral form of Minkowski’s inequality to (2.3), and (for p < ∞) making the change of variables y = tx, dy = t^{n}dx, as follows:

∥ x ↦ ∫_{[^{0,...,0}_{m},x]} f ∥_{L_{p}(Ω)} ≤ 1 ; (m − 1)! ∫_{0}^{1}(1 − t)^{m−1}∥ x ↦ f(tx) ∥_{L_{p}(Ω)} dt

= 1 ; (m − 1)! ∫_{0}^{1}(1 − t)^{m−1} (∫_{Ω} |f(tx)|^{p} dx)^{1/p} dt

≤ 1 ; (m − 1)! ∫_{0}^{1}(1 − t)^{m−1} (∫_{Ω} |f(y)|^{p} ; t^{n} dy)^{1/p} dt

= 1 ; (m − 1)!(∫_{0}^{1}(1 − t)^{m−1}t^{−n/p} dt)∥f∥_{L_{p}(Ω)}

= 1 ; (m − 1)! Γ(m)Γ(1 − n/p) ; Γ(m + 1 − n/p)∥f∥_{L_{p}(Ω)}

= 1 ; (1 − n/p) ⋯ (m − n/p)∥f∥_{L_{p}(Ω)}.

More general forms of this multivariate Hardy’s inequality are given in Waldron [W97].

Proof of Theorem 1.2: By Lemma 2.1, it is sufficient to consider Ω := IR^{n}\{0}. For this choice ρ_{∂Ω}(x) = ∥x∥, and each u ∈ C_{0}^{∞}(Ω) can be extended to a u ∈ C_{0}^{∞}(IR^{n}) by letting u(0) := 0. Since u is zero in a neighbourhood of 0, it follows from (2.4) that

u(x) ; (ρ_{∂Ω}(x))^{m} = u(x) ; ∥x∥^{m} = ∫_{[^{0,...,0}_{m},x]} Dm_{x ; ∥x∥} u,  (2.6)

giving

| u(x) ; (ρ_{∂Ω}(x))^{m} | ≤ ∫_{[^{0,...,0}_{m},x]} |D^{m}u|.  (2.7)

To this apply the multivariate Hardy’s inequality (2.5), with f = |D^{m}u|, to obtain that for 1 − n/p > 0 (p > n)

\begin{aligned}∥u/(ρ_{∂Ω})^{m}∥_{L_{p}(Ω)} & ≤ 1 ; (1 − n/p) ⋯ (m − n/p)∥ |D^{m}u| ∥_{L_{p}(Ω)} \\ & = 1 ; (1 − n/p) ⋯ (m − n/p)∥D^{m}u∥_{L_{p}(Ω)}, \end{aligned}  (2.8)

which, cf (1.1), proves

I(p, m, n) ≥ (1 − n ; p)(2 − n ; p) ⋯ (m − n ; p),  p > n.  (2.9)

The reverse inequality of (2.9) (for p = n also) follows since the sharpness of the multivariate Hardy’s inequality as used in (2.8) is for functions of the form

f = |D^{m}u| = ∥ · ∥^{−n/p+ε}χ_{B} as ε → 0

(with the relevant feature of f being its behaviour at 0). These functions do not belong to C_{0}^{∞}(Ω), and so we must use some mollification argument. Here are the details.

Since the map u ↦ l_{p}(u, Ω, m), u ≠ 0, is continuous when C_{0}^{∞}(Ω) is given the norm

∥u∥^{∗} := ∥u/(ρ_{∂Ω})^{m}∥_{L_{p}(Ω)} + ∥D^{m}u∥_{L_{p}(Ω)},

it is sufficient to consider functions u ≠ 0 from the Sobolev–type space

S_{p}^{m}(Ω) := ∥ · ∥^{∗}–closure of C_{0}^{∞}(Ω)

(cf Maz’ja [M85:Ch.1]), for which ∥D^{m}u∥_{L_{p}(Ω)}, ∥u/(ρ_{∂Ω})^{m}∥_{L_{p}(Ω)}, and l_{p}(u, Ω, m) are defined (and coincide with the extensions given by the density of C_{0}^{∞}(Ω) in S_{p}^{m}(Ω)).

Let u := u_{ε}, ε > 0 be the radial function

u(x) := \{ \begin{matrix}∥x∥^{m−n/p+ε}, & ∥x∥ ≤ 1; \\ g_{ε}(∥x∥), & ∥x∥ > 1, \end{matrix}

where g_{ε} is chosen so that u is C^{m} on Ω, i.e., | · |^{m−n/p+ε} and g_{ε} have a C^{m} join at 1. Then, u ∈ S_{p}^{m}(Ω) and

|D^{m}u|(x) = \{ \begin{matrix}(1 − n/p + ε) ⋯ (m − n/p + ε)∥x∥^{−n/p+ε}, & ∥x∥ ≤ 1; \\ |D^{m}g_{ε}|(∥x∥), & ∥x∥ > 1, \end{matrix}

giving

l_{p}(u, Ω, m) ≤ ∥D^{m}u∥_{L_{p}(B)} + ∥D^{m}u∥_{L_{p}(IR^{n}\B)} ; ∥u/(ρ_{∂Ω})^{m}∥_{L_{p}(B)}

= (1 − n/p + ε) ⋯ (m − n/p + ε)∥(∥ · ∥^{−n/p+ε})∥_{L_{p}(B)} + ∥D^{m}u∥_{L_{p}(IR^{n}\B)} ; ∥(∥ · ∥^{m−n/p+ε}/∥ · ∥^{m})∥_{L_{p}(B)}

= (1 − n/p + ε) ⋯ (m − n/p + ε) + ∥D^{m}u∥_{L_{p}(IR^{n}\B)} ; ∥(∥ · ∥^{−n/p+ε})∥_{L_{p}(B)} ,

where B is the unit ball in IR^{n}. The quantity

∥D^{m}u∥_{L_{p}(IR^{n}\B)} ; ∥(∥ · ∥^{−n/p+ε})∥_{L_{p}(B)} = ∥ | · |^{(n−1)/p}D^{m}g_{ε} ∥_{L_{p}(1,∞)} ; ∥ | · |^{−n/p+ε+(n−1)/p} ∥_{L_{p}(0,1)} = (εp)^{1/p}∥ | · |^{(n−1)/p}D^{m}g_{ε} ∥_{L_{p}(1,∞)}

can be made arbitrarily small by appropriate choice of g_{ε}, and so

\begin{aligned}I(p, m, n) & ≤ inf\{(1 − n/p + ε) ⋯ (m − n/p + ε) : ε > 0\} \\ & = (1 − n ; p)(2 − n ; p) ⋯ (m − n ; p),  p ≥ n. \end{aligned}  (2.10)

This completes the proof of Theorem 1.2.

3. Additional Hardy–type inequalities

The classical Hardy’s inequality is the sharp inequality that for p > 1

∥ x ↦ 1 ; x ∫_{0}^{x} f ∥_{L_{p}(0,∞)} ≤ p ; p − 1∥f∥_{L_{p}(0,∞)},  ∀f ∈ L_{p}(0, ∞),  (3.1)

or equivalently, the ‘differential version’

∥ u/| · | ∥_{L_{p}(0,∞)} ≤ p ; p − 1∥Du∥_{L_{p}(0,∞)},  ∀u ∈ C_{0}^{∞}(0, ∞),  (3.2)

and its variations.

The multivariate Hardy’s inequality of [W97] is a generalisation of the form (3.1), while the Theorem 1.2 proved from it is a generalisation of the form (3.2). There are many other multivariate ‘differential versions’ of Hardy’s inequality that can be proved using variations of this proof, e.g., Theorem 4.3.1 of [W97]. Let us give another example, which follows from almost exactly the same proof as that for Theorem 1.2.

Similarly to (2.4), for j = 0, 1, ⋯, m − 1, the j–th order derivatives of the error in Taylor interpolation can be expressed as

D_{ξ_{1}} ⋯ D_{ξ_{j}}u(x) − D_{ξ_{1}} ⋯ D_{ξ_{j}}(T_{<m}u)(x) = ∫_{[^{0,...,0}_{m−j},x]} D_{x}^{m−j}D_{ξ_{1}} ⋯ D_{ξ_{j}}u.

If all derivatives of u of order < m vanish at 0, i.e., T_{<m}u = 0, then this allows (2.7) to be generalised to

|D^{j}u|(x) ; ∥x∥^{m−j} ≤ ∫_{[^{0,...,0}_{m−j},x]} |D^{m}u|.  (3.3)

By applying the multivariate Hardy’s inequality (2.5) to (3.3), as in (2.8), we then obtain:

Theorem 3.4. Let Ω be an open subset of IR^{n} which is starshaped with respect to 0. Then, for p > n, j = 0, … m − 1, m = 1, 2, …, there is the sharp inequality that for ∀u ∈ W_{p}^{m}(Ω) with derivatives of order < m vanishing at 0, i.e., T_{<m}u = 0,

∥ |D^{j}u|/∥ · ∥^{m−j} ∥_{L_{p}(Ω)} ≤ 1 ; (1 − n/p) ⋯ (m − j − n/p)∥D^{m}u∥_{L_{p}(Ω)}.

Or, with the same conditions as in Theorem 1.2, and j = 0, … m − 1, we can extend (1.3) to:

inf_{{u∈C^{∞}_{0}(Ω) \atop u≠0}} ∥D^{m}u∥_{L_{p}(Ω)} ; ∥ |D^{j}u|/(ρ_{∂Ω})^{m−j} ∥_{L_{p}(Ω)} = (1 − n ; p)(2 − n ; p) ⋯ (m − j − n ; p).  (3.5)

Note, by Sobolev’s embedding theorem, the condition p > n in Theorem 3.4 ensures that u ∈ W_{p}^{m}(Ω) is C^{m−1} at 0, and so T_{<m}u is defined.

Acknowledgement

I would like to thank Simeon Reich for his encouragement to consider this problem.

References

[M85] V. G. Maz’ja (1985): Sobolev spaces. Berlin: Springer–Verlag.

[MS96] T. Matskewich, P. E. Sobolevskii (1996): The sharp constant in the Hardy’s inequality for complement of bounded domain. Preprint

[W97] S. Waldron (1997): A multivariate form of Hardy’s inequality and L_{p}–error bounds for multivariate Lagrange interpolation schemes. SIAM J. Math. Anal., 28(1):233– 258.