Factorizations and Hardy-Rellich-Type Inequalities

0.21 MB·

by  Fritz Gesztesy

FACTORIZATIONS AND HARDY–RELLICH-TYPE INEQUALITIES
FRITZGESZTESYANDLANCELITTLEJOHN
7
1 Dedicated with great pleasure toHelge Holden on the occasion of his60th birthday.
0
2
Abstract. Theprincipalaimofthisnoteistoillustratehowfactorizationsofsingular,
n
even-orderpartialdifferentialoperatorsyieldanelementaryapproachtoclassicalinequal-
a
itiesofHardy–Rellich-type. Moreprecisly,introducingthetwo-parametern-dimensional
J homogeneousscalardifferentialexpressionsTα,β :=−∆+α|x|−2x·∇+β|x|−2,α,β∈R,
1 x∈Rn\{0}, n∈N, n≥2, and itsformaladjoint, denoted byT+ ,we show that non-
3 negativityofTα+,βTα,β onC0∞(Rn\{0})impliesthefundamentalαi,nβequality,

P] [(∆f)(x)]2dnx≥[(n−4)α−2β] |x|−2|(∇f)(x)|2dnx
ZRn ZRn
A
−α(α−4) |x|−4|x·(∇f)(x)|2dnx (0.1)
h. ZRn
at +β[(n−4)(α−2)−β]ZRn|x|−4|f(x)|2dnx, f ∈C0∞(Rn\{0}).
m
A particular choice of values for α and β in (0.1) yields known Hardy–Rellich-type
[ inequalities,includingtheclassicalRellichinequalityandaninequalityduetoSchmincke.
Bylocality,theseinequalitiesextendtothesituationwhereRnisreplacedbyanarbitrary
1 opensetΩ⊆Rn forfunctionsf ∈C∞(Ω\{0}).
0
v Perhaps more importantly, we will indicate that our method, in addition to being
9 elementary,isquiteflexiblewhenitcomestoavarietyofgeneralizedsituationsinvolving
2 theinclusionofremaindertermsandhigher-ordersituations.
9
8
0
.
1
1. Introduction
0
7 We dedicate this note with great pleasure to Helge Holden, whose wide range of contri-
1
butions to a remarkable variety of areas in mathematical physics, stochastics, partial differ-
:
v ential equations, and integrable systems, whose exemplary involvement with students, and
i whose tireless efforts on behalf of the mathematical community, deserve our utmost respect
X
and admiration. Happy Birthday, Helge, we hope our modest contribution toHardy–Rellich-
r
a type inequalities will give some joy.
The celebrated (multi-dimensional) Hardy inequality,
|(∇f)(x)|2dnx≥[(n−2)/2]2 |x|−2|f(x)|2dnx, f ∈C∞(Rn\{0}), n∈N, n≥3,
ZRn ZRn 0
(1.1)
and Rellich’s inequality,
|(∆f)(x)|2dnx≥[n(n−4)/4]2 |x|−4|f(x)|2dnx, f ∈C∞(Rn\{0}), n∈N, n≥5,
ZRn ZRn 0
(1.2)
Date:February1,2017.
2010 Mathematics Subject Classification. Primary: 35A23,35J30; Secondary: 47A63,47F05.
Keywordsandphrases. Hardyinequality,Rellich-typeinequality,factorizationsofdifferentialoperators.
1
2 F.GESZTESYANDL.L.LITTLEJOHN
the first two inequalities in an infinite sequence of higher-order Hardy-type inequalities,
received enormous attention in the literature due to their ubiquity in self-adjointness and
spectraltheoryproblemsassociatedwithsecondandfourth-orderdifferentialoperatorswith
strongly singular coefficients, respectively (see, e.g., [2], [3], [6], [10], [15, Sect. 1.5], [16,
Ch.5],[28],[31],[32],[35],[38]–[42],[54,Ch.II],[57]). WerefertoRemark2.11foraselection
of Rellich inequality references and some pertinent monographs on Hardy’s inequality.
As one of our principal results we will derive the following two-parameter family of in-
equalities (a special case of inequality (1.5) below): If either α ≤ 0 or α ≥ 4, and β ∈ R,
then,
|(∆f)(x)|2dnx≥[α(n−α)−2β] |x|−2|(∇f)(x)|2dnx
ZRn ZRn
+β[(n−4)(α−2)−β] |x|−4|f(x)|2dnx, (1.3)
ZRn
f ∈C∞(Rn\{0}), n∈N, n≥2.
0
Aswillbeshown,(1.3)containsRellich’sinequality(1.2),andSchmincke’sone-parameter
family of inequalities,
|(∆f)(x)|2dnx≥−s |x|−2|(∇f)(x)|2dnx
ZRn ZRn
+[(n−4)/4]2 4s+n2 |x|−4|f(x)|2dnx, (1.4)
(cid:0) (cid:1)ZRn
s∈ −2−1n(n−4),∞ ,
(cid:2) (cid:1)
as special cases. By locality, the inequalities (1.1)–(1.4) naturally extend to the case where
Rn is replaced by an arbitrary open set Ω ⊂ Rn for functions f ∈ C∞(Ω\{0}) (without
0
changing the constants in these inequalities).
Our approach is based on factorizing even-order differential equations. More precisely,
focusingonthe 4th-ordercaseforsimplicity,weintroducethe two-parametern-dimensional
homogeneous scalar differential expressions T := −∆+α|x|−2x·∇+β|x|−2, α,β ∈ R,
α,β
x ∈ Rn\{0}, n ∈ N, n ≥ 2, and its formal adjoint, denoted by T+ . Nonnegativity of
α,β
T+ T on C∞(Rn\{0}) then implies the fundamental inequality,
α,β α,β 0
[(∆f)(x)]2dnx≥[(n−4)α−2β] |x|−2|(∇f)(x)|2dnx
ZRn ZRn
−α(α−4) |x|−4|x·(∇f)(x)|2dnx (1.5)
ZRn
+β[(n−4)(α−2)−β] |x|−4|f(x)|2dnx, f ∈C∞(Rn\{0}),
ZRn 0
which in turn contains inequality (1.3) as a special case.
We conclude our note with a series of remarks putting our approachinto proper context
by indicating that our method is elementary and very flexible in handling a variety of
generalized situations involving the inclusion of remainder terms and higher even-order
differential operators.
2. Factorizations and Hardy–Rellich-type Inequalities
The principal inequality to be proven in this note is of the following form:
HARDY–RELLICH-TYPE INEQUALITIES 3
Theorem 2.1. Let α,β ∈R, and f ∈C∞(Rn\{0}), n∈N, n≥2. Then,
0
[(∆f)(x)]2dnx≥[(n−4)α−2β] |x|−2|(∇f)(x)|2dnx
ZRn ZRn
−α(α−4) |x|−4|x·(∇f)(x)|2dnx (2.1)
ZRn
+β[(n−4)(α−2)−β] |x|−4|f(x)|2dnx.
ZRn
In addition, if either α≤0 or α≥4, then,
|(∆f)(x)|2dnx≥[α(n−α)−2β] |x|−2|(∇f)(x)|2dnx
ZRn ZRn
+β[(n−4)(α−2)−β] |x|−4|f(x)|2dnx. (2.2)
ZRn
Proof. Given α,β ∈ R and n ∈ N, n ≥ 2, we introduce the two-parameter n-dimensional
homogeneous scalar differential expressions
T :=−∆+α|x|−2x·∇+β|x|−2, x∈Rn\{0}, (2.3)
α,β
and its formal adjoint, denoted by T+ ,
α,β
T+ :=−∆−α|x|−2x·∇+[β−α(n−2)]|x|−2, x∈Rn\{0}. (2.4)
α,β
Assumingf ∈C∞(Rn\{0})throughoutthisproof,employingelementarymulti-variabledif-
0
ferentialcalculus,weproceedtothecomputationofT+ T (which,whileentirelystraight-
α,β α,β
forward, may well produce some tears in the process),
(T+ T f)(x)=(∆2f)(x)+[(n−4)α−2β]|x|−2(∆f)(x)
α,β α,β
n
+α(4−α)|x|−4 x x f (x)
j k xj,xk
jX,k=1
+ −(n−3)α2+2(n−2)α+4β |x|−4x·(∇f)(x)
+(cid:2)β2+2(n−4)β−(n−4)αβ |x(cid:3)|−4f(x). (2.5)
(cid:2) (cid:3)
Thus, choosingf ∈C∞(Rn\{0})real-valuedfromthis point on andintegrating by parts
0
(observing the support properties of f, which results in vanishing surface terms) implies
0≤ [(T f)(x)]2dnx= f(x)(T+ T f)(x)dnx
ZRn α,β ZRn α,β α,β
= [(∆f)(x)]2dnx+[(n−4)α−2β] |x|−2f(x)(∆f)(x)dnx
ZRn ZRnZRn
n
+α(α−4) |x|−4f(x)x x f (x)dnx
jX,k=1ZRn j k xj,xk
+ −(n−3)α2+2(n−2)α+4β |x|−4f(x)[x·(∇f)(x)]dnx
(cid:2) (cid:3)ZRn
+ β2+2(n−4)β−(n−4)αβ |x|−4f(x)2dnx. (2.6)
(cid:2) (cid:3)ZRn
4 F.GESZTESYANDL.L.LITTLEJOHN
To simplify and exploit expression (2.6), we make two observations. First, a standard
integration by parts (again observing the support properties of f) yields
|x|−2f(x)(∆f)(x)dnx=2 |x|−4f(x)(x·(∇f)(x)dnx
ZRn ZRn (2.7)
− |x|−2|(∇f)(x)|2dnx.
ZRn
Similarly, one confirms that
n
x x f(x)f (x)=−(n−3) |x|−4f(x)[x·(∇f)(x)]dnx
jX,k=1ZRn j k xj,xk ZRn (2.8)
− |x|−4[x·(∇f)(x)]2dnx.
ZRn
Combining (2.6)–(2.8) then yields (2.1).
Since by Cauchy’s inequality,
− |x|−4[x·(∇f)(x)]2dnx≥− |x|−2|(∇f)(x)|2dnx, (2.9)
ZRn ZRn
one concludes that as long as α(α−4) ≥ 0, that is, as long as either α ≤ 0 or α ≥ 4, one
can further estimate (2.1) from below and thus arrive at inequality (2.2). (cid:3)
As a special case of (2.2) one obtains Rellich’s classical inequality in its original form as
follows:
Corollary 2.2. Let n∈N, n≥5, and f ∈C∞(Rn\{0}). Then,
0
|(∆f)(x)|2dnx≥[n(n−4)/4]2 |x|−4|f(x)|2dnx. (2.10)
ZRn ZRn
Proof. Choosing β =α(n−α)/2 in (2.2) results in
|(∆f)(x)|2dnx≥G (α) |x|−4|f(x)|2dnx, (2.11)
n
ZRn ZRn
with
G (α)=α(n−α){(n−4)(α−2)−[α(n−α)/2]}/2. (2.12)
n
Maximizing G (α) with respect to α (it is advantageous to introduce the new variable
n
a=α−2) yields maxima at
α =2± n2/2 −2n+4 1/2, (2.13)
±
(cid:2)(cid:0) (cid:1) (cid:3)
and taking the constraints α≤0 or α≥4 into account results in n≥5. The fact
G (α )=[n(n−4)/4]2, (2.14)
n ±
then yields the Rellich inequality (2.10). (cid:3)
Inequality (2.1) also implies the following result:
Corollary 2.3. Let n∈N and f ∈C∞(Rn\{0}). Then,
0
|(∆f)(x)|2dnx≥ n2/4 |x|−2|(∇f)(x)|2dnx, n≥8, (2.15)
ZRn (cid:0) (cid:1)ZRn
and
|(∆f)(x)|2dnx≥4(n−4) |x|−2|(∇f)(x)|2dnx, 5≤n≤7. (2.16)
ZRn ZRn
HARDY–RELLICH-TYPE INEQUALITIES 5
In addition,
|(∆f)(x)|2dnx≥ n2/4 |x|−4|x·(∇f)(x)|2dnx, n≥2. (2.17)
ZRn (cid:0) (cid:1)ZRn
Proof. Again, we chose f ∈ C∞(Rn\{0}) real-valued for simplicity throughout this proof.
0
The choice β =(n−4)(α−2) in (2.1) then results in
|(∆f)(x)|2dnx≥(n−4)(4−α) |x|−2|(∇f)(x)|2dnx
ZRn ZRn (2.18)
−α(α−4) |x|−4|x·(∇f)(x)|2dnx.
ZRn
If in addition α < 0, then applying Cauchy’s inequality to the 2nd term on the right-hand
side of (2.18) yields
|(∆f)(x)|2dnx≥H (α) |x|−2|(∇f)(x)|2dnx, (2.19)
n
ZRn ZRn
whereH (α)=(n−4+α)(4−α). MaximizingH withrespecttoαsubjecttotheconstraint
n n
α<0yieldsamaximumatα =(8−n)/2,withH ((8−n)/2)=n2/4,implying inequality
1 n
(2.15) for n≥9. On the other hand, choosing α=0 in (2.18) yields
|(∆f)(x)|2dnx≥4(n−4) |x|−2|(∇f)(x)|2dnx. (2.20)
ZRn ZRn
Since 4(n−4)=n2/4 for n=8, this proves (2.15). Actually, one can arriveat (2.15) much
quicker,butsincewewillsubsequentlyuse(2.18),wekepttheaboveargumentinthisproof:
Indeed, choosing β =0 in (2.2) yields
|(∆f)(x)|2dnx≥α(n−α) |x|−2|(∇f)(x)|2dnx. (2.21)
ZRn ZRn
Maximizing F (α) = α(n − α) with respect to α yields a maximum at α = n/2, and
n 1
subjecting it to the constraint α≥4 proves (2.15).
Choosing α=4, β =0 in (2.1) yields (2.16).
For n ≥ 2 and (4−n) < α < 4, applying Cauchy’s inequality to the 1st term on the
right-hand side of (2.18) now yields
|(∆f)(x)|2dnx≥K (α) |x|−4|x·(∇f)(x)|2dnx, (2.22)
n
ZRn ZRn
whereK (α)=−(α+n−4)(α−4). MaximizingK subjecttotheconstraint(4−n)<α<4
n n
yields a maximum at α =(8−n)/2, with K ((8−n)/2)=n2/4, implying (2.17). (cid:3)
1 n
We conclude with a series of remarks that put our approach into proper context and
point out natural continuations into various other directions.
Remark2.4. (i)Theconstantininequality(2.10)isknowntobeoptimal,see,forinstance,
[6, p. 222], [17], [47], [50], [58], [61].
(ii)Asequenceofextensionsof (2.15),validforn≥5,andforboundeddomainscontaining
0, was derived by Tertikas and Zographopoulos [58, Theorem 1.7]. Moreover, an extension
of inequality (2.15) valid for n=4 and for bounded open domains containing 0 was proved
by [1, Theorem 2.1(b)]. An alternative inequality whose special cases also imply Rellich’s
inequality (2.10) and inequality (2.15) appeared in [14]. Thus, while the constant n2/4 in
(2.15) is known to be optimal (cf. [58] for n≥5), the constant4(n−4) in (2.16) is not, the
sharp constant being known to be n2/4 (also for n=4, cf. [1]). ⋄
Next, we comment on a special case of inequality (2.2) originally due to Schmincke [56]:
6 F.GESZTESYANDL.L.LITTLEJOHN
Remark 2.5. The choice β =2−1(n−4)[α−2−4−1(n−4)], and the introduction of the
new variable
s=s(α)=α2−4α−2−1n(n−4), (2.23)
renders the two-parameter inequality (2.2) into Schmincke’s one-parameter inequality
|(∆f)(x)|2dnx≥−s |x|−2|(∇f)(x)|2dnx
ZRn ZRn
+[(n−4)/4]2 4s+n2 |x|−4|f(x)|2dnx, (2.24)
(cid:0) (cid:1)ZRn
s∈ −2−1n(n−4),∞ .
(cid:2) (cid:1)
Here the requirements α≤0, equivalently, α≥4, both yield the range requirementfor s in
the form s∈ −2−1n(n−4),∞ . Inequality (2.24) is precisely the content of Lemma 2 in
Schmincke [56(cid:2)], in particular, (2(cid:1).2) thus recovers Schmincke’s result. Moreover, assuming
n ≥ 5 (the case n = 4 being trivial) permits the value s = 0 and hence implies Rellich’s
inequality (2.10). If n ≥ 8, the value s = −n2/4 is permitted, yielding inequality (2.15).
Finally, for 5 ≤ n ≤ 7, s ∈ −2−1n(n−4),∞ and 4s+n2 ≥ 0 permit one to choose
s=−n(n−4)/2 and hence to(cid:2)conclude (cid:1)
|(∆f)(x)|2dnx≥2−1n(n−4) |x|−2|(∇f)(x)|2dnx
ZRn ZRn
+[(n−4)/4]2 4s+n2 |x|−4|f(x)|2dnx(2.21) (2.25)
(cid:0) (cid:1)ZRn
≥2−1n(n−4) |x|−2|(∇f)(x)|2dnx, 5≤n≤7, (2.26)
ZRn
but inequality (2.16) is strictly superior to (2.26). Hence the two-parameter version (2.2)
yields the better result (2.16), eventhough, the latter is not optimaleither as mentioned in
the previous Remark 2.4(ii). ⋄
Remark 2.6. Since all differential expressions employedare local, and only integrationby
parts was involvedin deriving (2.6) (cf., e.g., [39, Remark 4] in this context), the estimates
(2.1), (2.2), (2.10), (2.15)–(2.17), (2.24), all extend to the case where Rn is replaced by an
arbitrary open set Ω ⊂ Rn for functions f ∈ C∞(Ω\{0}) (without changing the constants
0
in these inequalities). ⋄
Remark 2.7. Since C∞(Rn\{0}) is dense in H2(Rn) if and only if n ≥ 4, equivalently,
0
−∆ is essentially self-adjoint in L2(Rn) if and only if n ≥ 4 (see, e.g., [21,
C0∞(Rn\{0})
p. 4(cid:12)12–413], see also [41], [56]), Rellich’s inequality (2.10) extends from C∞(Rn\{0}) to
(cid:12) 0
H2(Rn) for n≥5, and inequalities (2.15), and (2.17) extend from C∞(Rn\{0}) to H2(Rn)
0
for n≥4. ⋄
Remark 2.8. This factorization approach was originally employed in the context of the
classicalHardyinequality in [31](and someof its logarithmicrefinements in [28]). Without
repeatingthe analogousstepsindetailwe justmentionthatgivenn∈N, n≥3, α∈R,one
introduces the one-parameter family of homogeneous vector-valued differential expressions
T :=∇+α|x|−2x, x∈Rn\{0}, (2.27)
α
with formal adjoint, denoted by T+,
α
T+ =−div(·)+α|x|−2x·, x∈Rn\{0}, (2.28)
α
such that (e.g., on C∞(Rn\{0})-functions),
0
T+T =−∆+α(α+2−n)|x|−2. (2.29)
α α
HARDY–RELLICH-TYPE INEQUALITIES 7
Thus, for f ∈C∞(Rn\{0}),
0
0≤ |T f)(x)|2dnx= f(x)(T+T f)(x)dnx
ZRn α ZRn α α (2.30)
= |(∇f)(x)|2dnx+α(α+2−n) |x|−2|f(x)|2dnx,
ZRn ZRn
and hence,
|(∇f)(x)|2dnx≥α[(n−2)−α] |x|−2|f(x)|2dnx. (2.31)
ZRn ZRn
Maximizing α[(n−2)−α] with respect to α yields the classical Hardy inequality,
|(∇f)(x)|2dnx≥[(n−2)/2]2 |x|−2|f(x)|2dnx, f ∈C∞(Rn\{0}). (2.32)
ZRn ZRn 0
Again, it is well-known that the constant in (2.32) is optimal (cf., e.g., [61]). ⋄
Actually,ourfactorizationapproachalsoyieldsaknownimprovementofHardy’sinequal-
ity (see, e.g.,[6, Theorem1.2.5],specializing it to p=2, ε=0). Next, we briefly sketchthe
corresponding argument.
Remark 2.9. Given n ∈ N, n ≥ 3, α ∈ R, one introduces the following modified one-
parameter family of homogeneous vector-valued differential expressions
T := |x|−1x ·∇+α|x|−1, x∈Rn\{0}, (2.33)
α
(cid:0) (cid:1)
with formal adjoint, deneoted by T +,
α
(cid:0) (cid:1)
T + =− |x|−1xe ·∇+(α−n+1)|x|−1, x∈Rn\{0}. (2.34)
α
(cid:0) (cid:1) (cid:0) (cid:1)
Exploiting the ideentities (for f ∈C∞(Rn\{0}), for simplicity),
0
n
|x|−1x·∇ |x|−1x·(∇f)(x) =|x|−2 x x f (x), x∈Rn\{0}, (2.35)
j k xj,xk
(cid:2) (cid:3)(cid:2) (cid:3) jX,k=1
x·∇ |x|−1f(x) =|x|−1[x·(∇f)(x)]−|x|−1f(x), x∈Rn\{0}, (2.36)
(cid:0) (cid:1)
one computes (e.g., on C∞(Rn\{0})-functions),
0
n
T +T =−|x|−2 x x ∂ ∂ −(n−1)|x|−2[x·(∇f)(x)]+α(α+2−n)|x|−2,
α α j k xj xk
(cid:0) (cid:1) jX,k=1
e e
x∈Rn\{0}. (2.37)
Thus, appropriate integration by parts yield
0≤ T f (x) 2dnx= f(x) T +T f (x)dnx
α α α
ZRn(cid:12)(cid:0) (cid:1) (cid:12) ZRn (cid:0)(cid:0) (cid:1) (cid:1)
(cid:12) e (cid:12)n e e
=− |x|−2 x x f(x)f (x)+(n−1)f(x)[x·(∇f)(x)]
ZRn (cid:26)jX,k=1 j k xj,xk
−α(α−n+2)|f(x)|2 dnx,
(cid:27)
= |x|−2 |[x·(∇f)(x)]|2+α(α−n+2)|f(x)|2 , f ∈C∞(Rn\{0}). (2.38)
ZRn (cid:26) (cid:27) 0
8 F.GESZTESYANDL.L.LITTLEJOHN
Here we used
n
|x|−2x x f(x)f (x)dnx=− |x|−2|[x·(∇f)(x)]|2dnx
jX,k=1ZRn j k xj,xk ZRn (2.39)
−(n−1) |x|−2f(x)[x·(∇f)(x)]dnx, f ∈C∞(Rn\{0}).
ZRn 0
Thus,
|x|−1x·∇f](x) 2dnx≥α[(n−2)−α] |x|−2|f(x)|2dnx. (2.40)
ZRn(cid:12)(cid:2) (cid:12) ZRn
Maximizing α[(n(cid:12)−2)−α] with re(cid:12)spect to α yields the improved Hardy inequality,
|x|−1x·∇f](x) 2dnx≥[(n−2)/2]2 |x|−2|f(x)|2dnx. (2.41)
ZRn(cid:12)(cid:2) (cid:12) ZRn
(cid:12) (cid:12)
(By Cauchy’s inequality, (2.41) implies the classical Hardy inequality (2.32).) Again, it is
known that the constant in (2.41) is optimal (cf., e.g., [6, Theorem 1.2.5]). ⋄
Remark 2.10. The case of Rellich (and Hardy) inequalities in the half-line case is com-
pletely analogous (and much more straightforward): Consider the differential expressions
d2 α d β d2 α d α+β
T =− + + , T+ =− − + , (2.42)
dx2 xdx x2 dx2 xdx x2
with α,β ∈R, which are formal adjoints to each other. One verifies,
d4 α−α2−2β d2 2α2−2α+4β d 3αβ+β2−6β
T+T = + + + , (2.43)
dx4 x2 dx2 x3 dx x4
and hence upon some integrations by parts,
∞ ∞
0≤ (Tf)(x)2dx= f(x)(T+Tf)(x)dx
Z Z
0 0
∞ ∞ [f′(x)]2
= [f′′(x)]2dx− α−α2−2β dx
Z Z x2
0 (cid:0) (cid:1) 0
∞ f(x)2
+β(3α+β−6) dx, f ∈C∞((0,∞)), (2.44)
Z x4 0
0
choosing f real-valued (for simplicity and w.l.o.g.). Thus, one obtains,
∞ ∞ |f′(x)|2
|f′′(x)|2dx≥ α−α2−2β dx
Z Z x2
0 (cid:0) (cid:1) 0
∞ |f(x)|2
+β(6−β−3α) dx, (2.45)
Z x4
0
f ∈C∞((0,∞)), α,β ∈R.
0
Choosing β = α−α2 /2 yields the Rellich-type inequality
∞ (cid:0) (cid:1) ∞ |f(x)|2
|f′′(x)|2dx≥ 3α−(19/4)α2+2α3−(1/4)α4 dx,
Z Z x4
0 (cid:2) (cid:3) 0
f ∈C∞((0,∞)). (2.46)
0
Introducing F(α) = 3α−(19/4)α2 +2α3 −(1/4)α4 = −(1/4)(α−4)(α−3)(α−1)α,
α∈R, one verifies that F(2+γ)=F(2−γ), γ ∈R, and factors its derivative as
F′(α)=3−(19/2)α+6α2−α3
(2.47)
=−(α−2) α−2+(5/2)1/2 α−2−(5/2)1/2 .
(cid:0) (cid:1)(cid:0) (cid:1)
HARDY–RELLICH-TYPE INEQUALITIES 9
One notes that α = 2 yields a local minimum with F(2) = −1, α = 2−(5/2)1/2 and
1 2
α = 2+(5/2)1/2 both yield local maxima of equal value, that is, F(α ) = F(α ) = 9 .
3 2 3 16
Thus, one obtains Rellich’s inequality for the half-line in the form
∞ 9 ∞ |f(x)|2
|f′′(x)|2dx≥ dx, f ∈C∞((0,∞)). (2.48)
Z 16Z x4 0
0 0
We refertoBirman[13,p.46](see alsoGlazman[34,p.83–84]),who presentsasequenceof
higher-orderHardy-typeinequalities on(0,∞)whose secondmember coincides with (2.48).
For a variantof (2.48)on the interval(0,1)we refer to [16, p.114];the case of higher-order
Hardy-typeinequalitiesforgeneralintervalisalsoconsideredin[52]. Wewillreconsiderthis
sequence of higher-order Hardy-type inequalities in [30].
In addition, choosing β =0 or β =6−3α and subsequently maximizing with respect to
α yields in either case
∞ 1 ∞ |f′(x)|2
|f′′(x)|2dx≥ dx, f ∈C∞((0,∞)), (2.49)
Z 4Z x2 0
0 0
however, this is just Hardy’s inequality [36], [37], with f replaced by f′. ⋄
Remark 2.11. WhilewebasicallyfocusedoninequalitiesinL2(Rn)(see,however,Remark
2.6), muchof the recentworkonRellich andhigher-orderHardyinequalities aims at Lp(Ω)
for open sets Ω ⊂ Rn (frequently, Ω is bounded with 0 ∈ Ω), p ∈ [1,∞), appropriate
remainderterms(the latteroftenassociatedwithlogarithmicrefinementsorwithboundary
terms), higher-order Hardy–Rellich inequalities, and the inclusion of magnetic fields. The
enormous number of references on this subject, especially, in the context of Hardy-type
inequalities, makes it impossible to achieve any reasonable level of completeness in such a
short note as the underlying one. Hence we felt we had to restrict ourselves basically to
Rellich and higher-order Hardy inequality references only and thus we refer, for instance,
to [1], [2], [4], [5], [6, Ch. 6], [7], [8], [9], [11], [12], [17], [18], [19], [20], [22], [23], [24],
[25], [26], [27], [33], [40], [45], [46], [48], [49], [52], [53], [55], [58], [59], [60], andthe extensive
literaturecitedtherein. ForthecaseofHardy-typeinequalitiesweonlyrefertothestandard
monographs such as, [6], [43], [44], and [51].
Inthiscontextweemphasizeonceagainthatthefactorizationmethodisentirelyindepen-
dent of the choice of domain Ω. Indeed, factorizations in the context of Hardy’s inequality
in balls with optimal constants and logarithmic correction terms were already studied in
[28], [31], based on prior work in [38], [41], and [42], although this appears to have gone
unnoticed in the recent literature on this subject. For instance, one can introduce iterated
logarithms of the form for γ >0, x∈Rn, |x|<γ,
(−ln(|x|/γ)) =1,
0
(−ln(|x|/γ)) =(−ln(|x|/γ)),
1
(−ln(|x|/γ)) =ln((−ln(|x|/γ)) ), k ∈N, (2.50)
k+1 k
10 F.GESZTESYANDL.L.LITTLEJOHN
and replace ∇ by T , where
αm,y
m j
T =∇+2−1|x−y|−2 (n−2)+ [(−ln(|x−y|/γ)) ]−1
αm,y (cid:26) k
Xj=1kY=1
m
−α [(−ln(|x−y|/γ)) ]−1 (x−y), (2.51)
m k
(cid:27)
kY=1
0<|x|<r, r<γ, α ≥0, m∈N,
m
T =∇+2−1(n−2−α )|x−y|−2(x−y), 0<|x|<r, α ≥0, m=0. (2.52)
α0,y 0 0
Then with T+ the formal adjoint of T , one obtains for f ∈C∞(B (y;r)\{y})
αm,y αm,y 0 n
(T+ T f)(x)=(−∆f)(x)−4−1|x−y|−2 (n−2)2 (2.53)
αm,y αm,y (cid:26)
m j m
+ [(−ln(|x−y|/γ)) ]−2f(x)−α2 [(−ln(|x−y|/γ)) ]−2 f(x), m∈N,
k m k (cid:27)
Xj=1Xk=1 kY=1
(T+ T f)(x)=(−∆f)(x)−4−1 (n−2)2−α2 |x−y|−2f(x), m=0. (2.54)
α0,y α0,y 0
(Here B (x ;r ) denotes the open bal(cid:2)l in Rn with c(cid:3)enter x ∈ Rn and radius r > 0.) In
n 0 0 0 0
particular, letting r ↓0 and r ↑r in [28, Lemma 1] implies
0 1
0≤ |(T f)(x)|2 = |(∇f)(x)|2−4−1|x−y|−2 (n−2)2
Z αm,y Z (cid:26) (cid:20)
B(y;r) B(y;r)
m j m
+ [(−ln(|x−y|/γ)) ]−2−α2 [(−ln(|x−y|/γ)) ]−2 |f(x)|2 dnx, (2.55)
k m k (cid:21) (cid:27)
Xj=1Xk=1 kY=1
0<r <γ, f ∈C∞(B(y;r)\{y}), m∈N∪{0},
0
and hence (with α =0),
m
|(∇f)(x)|2dnx≥4−1 |x−y|−2 (n−2)2
Z Z (cid:26)
B(y;r) B(y;r)
m j
+ [(−ln(|x−y|/γ)) ]−2 |f(x)|2dnx, (2.56)
k
(cid:27)
Xj=1kX=1
0<r <γ, f ∈C∞(B(y;r)\{y}), m∈N∪{0}.
0
(Following standard practice, a product, resp., sum over an empty index set is defined to
equal 1, resp., 0.) In analogy to Remark 2.6, inequality (2.56) extends to arbitrary open
bounded sets Ω⊂Rn as longas γ is chosensufficiently large(e.g., largerthan the diameter
of Ω). The constants in (2.55) are best possible as it is well-known that the operators
(2.53), (2.54) are nonnegative if and only if α2 ≥ 0, m ∈ N∪{0} (cf., e.g., [28, p. 99] or
m
[32, Theorem 2.2]). (They are unbounded from below for α2 <0, m∈N∪{0}, permitting
m
temporarily a continuation to negative values of α2 on the right-hand sides of (2.53) and
(2.54).)
For the special half-line case we also refer to [32].
Higher-order logarithmic refinements of the multi-dimensional Hardy–Rellich-type in-
equality appeared in [1, Theorem 2.1], and a sequence of such multi-dimensional Hardy–
Rellich-type inequalities, with additional generalizations, appeared in [58, Theorems 1.8–
1.10]. ⋄