Upper bounds on second order operators, acting on metric function
We prove upper bounds on the general second order operator acting on metric function. The suggested approach does not use traditional formulas for deviations of geodesics and Jacobi fields construction and leads to the manifolds generalization of the classical coercitivity and dissipativity conditio...
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| Дата: | 2007 |
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Інститут прикладної математики і механіки НАН України
2007
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| Цитувати: | Upper bounds on second order operators, acting on metric function / A.V. Antoniouk // Український математичний вісник. — 2007. — Т. 4, № 2. — С. 163-172. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859618845933699072 |
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| author | Antoniouk, A.V. |
| author_facet | Antoniouk, A.V. |
| citation_txt | Upper bounds on second order operators, acting on metric function / A.V. Antoniouk // Український математичний вісник. — 2007. — Т. 4, № 2. — С. 163-172. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Український математичний вісник |
| description | We prove upper bounds on the general second order operator acting on metric function. The suggested approach does not use traditional formulas for deviations of geodesics and Jacobi fields construction and leads to the manifolds generalization of the classical coercitivity and dissipativity conditions for diffusion equations.
|
| first_indexed | 2025-11-29T00:10:58Z |
| format | Article |
| fulltext |
Український математичний вiсник
Том 4 (2007), № 2, 163 – 172
Upper bounds on second order operators,
acting on metric function
Alexander Val. Antoniouk
(Presented by S. Ya. Makhno)
Abstract. We prove upper bounds on the general second order op-
erator acting on metric function. The suggested approach does not use
traditional formulas for deviations of geodesics and Jacobi fields con-
struction and leads to the manifolds generalization of the classical co-
ercitivity and dissipativity conditions for diffusion equations.
2000 MSC. 35A15, 53C21, 58E35.
Key words and phrases. Second order operators, metric distance
function, geodesic deviations, upper bounds.
In this paper we turn to the geometric problem of upper estimates
on the general second order operators, acting on metric function. We
consider operators of form
Lf =
1
2
d∑
σ=1
Aσ(Aσf) + A0f, (1)
where A0, Aα represent C∞ smooth globally defined vector fields on the
oriented smooth complete connected Riemannian manifold M without
boundary.
The known approaches of differential geometry were mostly invented
for the case of Laplace-Beltrami ∆ or similar operators [5, 6], because
it was hard to find the implicit representations for arbitrary differential
operators on metric function, defined as a minimum of length functional
ρ2(x, y) = inf
{ 1∫
0
|γ̇(ℓ)|2dℓ, γ(0) = x, γ(1) = y
}
. (2)
Received 20.04.2004
The research was supported by A.von Humboldt Foundation (Germany).
ISSN 1810 – 3200. c© Iнститут математики НАН України
164 Upper bounds on second order operators...
Corresponding techniques were related with the use of geodesic deviations
formulas and related Jacobi fields, with the study of the global geometry
of manifold, e.g. [5, 6], survey [7] and references therein.
However, for upper bounds one does not need the precise represen-
tations for differential operators on metric (!). Below we develop such
estimates and demonstrate, that the traditional approach of geodesic de-
viations is a little advanced for such simple problem.
The found conditions on coefficients of operator L generalize the clas-
sical dissipativity and coercitivity conditions from the linear base space
to manifold. They relate the coefficients of operator with the geometric
properties of manifold, without traditional separation of geometry:
• coercitivity: ∃ o ∈ M such that ∀C ∈ R+ ∃KC ∈ R
1 such that
∀x ∈ M
〈Ã0(x),∇xρ2(x, o)〉 + C
d∑
σ=1
‖Aσ(x)‖2 ≤ KC(1 + ρ2(x, o)); (3)
• dissipativity: ∀C, C ′ ∈ R+ ∃KC ∈ R
1 such that ∀x ∈ M , ∀h ∈
TxM
〈∇Ã0(x)[h], h〉 + C
d∑
σ=1
‖∇Aσ(x)[h]‖2
− C ′
d∑
σ=1
〈Rx(Aσ(x), h)Aσ(x), h〉 ≤ KC‖h‖
2, (4)
where Ã0 = A0 + 1
2
∑d
σ=1 ∇Aσ
Aσ and [R(A, h)A]m = R m
p ℓqA
pAℓhq
denotes the curvature operator, related with (1,3)-curvature tensor
with components
R 2
1 34 =
∂Γ 2
1 3
∂x4
−
∂Γ 2
1 4
∂x3
+ Γ j
1 3Γ
2
j 4 − Γ j
1 4Γ
2
j 3. (5)
For simplicity of further calculations we only point the positions of
corresponding indexes.
Notation ∇H[h] means the directional covariant derivative, defined
by
(∇H(x)[h])i = ∇jH
i(x) · hj . (6)
Main result of article provides
A. V. Antoniouk 165
Theorem 1. Suppose that conditions (3)–(4) hold.
Then there is constant K such that at the points of C2-regularity of
metric distance
{
AI
0 + AII
0 +
1
2
d∑
σ=1
(AI
σ + AII
σ )2
}
ρ2(x, y) ≤ Kρ2(x, y). (7)
Notations AI , AII mean vector fields, acting on the first and second vari-
ables x and y of function ρ2(x.y) correspondingly, for example AIIρ2(x, y)
= 〈A(y),∇y〉ρ
2(x, y).
Similarly ∀C ∃KC such that
LIρ2(x, o) + C
d∑
σ=1
(AI
σρ2(x, o))2
ρ2(x, o)
≤ K(1 + ρ2(x, o)). (8)
Proof. Step 1. First note, that for smooth vector field X in a vicinity
of some point z of manifold N and smooth function f on N there are
following representations
Xf(z) = lim
ε→0
1
ε
ε∫
0
Xf(zs) ds = lim
ε→0
1
ε
ε∫
0
d
ds
f(zs) ds = lim
ε→0
f(zε) − f(z)
ε
,
X(Xf)(z) = lim
ε→0
1
ε2
ε∫
0
ds
s∫
−s
X(Xf)(zℓ) dℓ
= lim
ε→0
1
ε2
ε∫
0
ds
s∫
−s
d
dℓ
(Xf)(zℓ) dℓ
= lim
ε→0
1
ε2
ε∫
0
{(Xf)(zs)−(Xf)(z−s)} ds = lim
ε→0
1
ε2
ε∫
0
d
ds
{f(zs)+f(z−s)} ds
= lim
ε→0
f(zε) + f(z−ε) − 2f(z)
ε2
. (9)
Here we used notation zε for the differential flow along field X: zε =
z +
∫ ε
0 X(zs) ds.
Step 2. In the vicinity of geodesic γ(ℓ), ℓ ∈ [0, 1] from γ(0) = x to
γ(1) = y that minimizes (2) consider smooth vector field H. Introduce
a family of paths
[0, 1] × (−δ, δ) ∋ (ℓ, s) → γ(ℓ, s) ∈ M
166 Upper bounds on second order operators...
such that at s = 0 path γ(ℓ, s)
∣∣
s=0
= γ(ℓ) gives geodesic γ above and
parameter s appears as a result of evolution along H:
∂
∂s
γ(ℓ, s) = H(γ(ℓ, s)). (10)
Note that for s 6= 0 each path γ(ℓ, s)s-fixed must not be geodesic, unlike
in formulas for geodesic deviations.
H
]1,0[Îthe only
geodesic
is here
field H
Figure 1: Field H (white vectors) in a vicinity of geodesic from x to y deter-
mines a set of paths, parameterized by s. The resulting surface is parameterized
by (ℓ, s) ∈ [0, 1]× (−δ, δ). Note that for s 6= 0 each path γ(ℓ, s)s-fixed should not
be geodesic
Step 3. Now let’s apply (9) with N = M × M , X = HI ⊗ HII and
function f(z) = ρ(x, y) for z = (x, y). Using the minimal property of
geodesic, i.e. that the path γ(ℓ, s) is longer than geodesic from γ(0, s)
to γ(1, s), we can estimate terms with ±ε in (9) from above and obtain
(point at which we get rid of implicit representations, see also (16))
(HI + HII)ρ2(x, y) = lim
ε→0
ρ2(γ(1, ε), γ(0, ε)) − ρ2(x, y)
ε
≤ lim
ε→0
∫ 1
0 | ∂
∂ℓ
γ(ℓ, ε)|2dℓ −
∫ 1
0 | ∂
∂ℓ
γ(ℓ, 0)|2dℓ
ε
=
1∫
0
∂
∂s
∣∣∣
s=0
∣∣∣ ∂
∂ℓ
γ(ℓ, s)
∣∣∣
2
dℓ.
(11)
A. V. Antoniouk 167
To find derivative ∂
∂s
∣∣
s=0
in the above expression let us use that
by continuity arguments, for any ℓ and sufficiently small δ(ℓ) the path
{γ(ℓ, z)}z∈(−δ(ℓ),δ(ℓ)) completely lies in some coordinate vicinity (xi). In
this coordinate system relation (10) has integral form
γi(ℓ, s) = γi(ℓ) +
s∫
0
H i(γ(ℓ, z)) dz (12)
with point γ(ℓ) on initial geodesic. Therefore
γ̇i(ℓ, s) = γ̇i(ℓ) +
s∫
0
∂kH
i(γ(ℓ, z))γ̇k(ℓ, z) dz
and
∂
∂s
γ̇i(ℓ, s) = ∂kH
i(γ(ℓ, s))γ̇k(ℓ, s) = (∇kH
i − Γ i
k h)γ̇k(ℓ, s), (13)
where we changed to the covariant derivatives and introduced notation
γ̇ = ∂
∂ℓ
γ. In particular, from above formula and (10) we conclude com-
mutation
∂
∂s
∂
∂ℓ
γi(ℓ, s) =
∂
∂ℓ
∂
∂s
γi(ℓ, s).
Relation (13) and autoparallel property of Riemannian connection
∂kgmn(x) = ghnΓ h
k m + gmhΓ h
k n (14)
lead to
∂
∂s
|γ̇(ℓ, s)|2 =
∂
∂s
[gij(γ(ℓ, s))γ̇i(ℓ, s)γ̇j(ℓ, s)]
= ∂kgij
∂
∂s
γk · γ̇iγ̇j + 2gij γ̇
i ∂
∂s
γ̇j
= 2gij γ̇
i(∇kH
j)γ̇k = 2〈γ̇,∇H[γ̇]〉. (15)
Therefore estimate (11) transforms to
(HI + HII)ρ2(x, y) ≤ 2
1∫
0
〈∇H[γ̇], γ̇〉 dℓ.
168 Upper bounds on second order operators...
Step 4. In a similar to (11) way,
(HI + HII)(HI + HII)ρ2(x, y)
= lim
ε→0
ρ2(γ(1, ε), γ(0, ε)) + ρ2(γ(1,−ε), γ(0, ε)) − 2ρ2(x, y)
ε2
≤ lim
ε→0
∫ 1
0 | ∂
∂ℓ
γ(ℓ, ε)|2dℓ +
∫ 1
0 | ∂
∂ℓ
γ(ℓ,−ε)|2dℓ − 2
∫ 1
0 | ∂
∂ℓ
γ(ℓ, 0)|2dℓ
ε2
=
1∫
0
∂2
∂s2
∣∣∣
s=0
∣∣∣ ∂
∂ℓ
γ(ℓ, s)
∣∣∣
2
dℓ. (16)
Using relation (15) we find
1
2
∂2
∂s2
∣∣∣ ∂
∂ℓ
γ(ℓ, ε)
∣∣∣
2
=
∂
∂s
〈γ̇(ℓ, s),∇H[γ̇(ℓ, s)]〉
=
∂
∂s
{ gij(γ)γ̇i[∇kH
j(γ)]γ̇k } = ∂mgij(γ)Hmγ̇i[∇kH
j(γ)]γ̇k
+ gij{ (∇mH i − Γ i
m h)γ̇m }[∇kH
j(γ)]γ̇k
+ gij γ̇
i[∂m∇kH
j(γ) · Hm(γ)]γ̇k
+ gij(γ)γ̇i[∇kH
j(γ)]{ (∇mHk − Γ k
m h)γ̇m },
where, after the differentiation of product, we substituted relations (10)
and (13).
Using property (14), transforming partial derivative ∂m∇kH
j to co-
variant ∇m∇kH
j and contracting the terms with connection Γ we have
1
2
∂2
∂s2
|γ̇(ℓ, ε)|2 = gij(∇mH i)γ̇m(∇kH
j)γ̇k
+ gij γ̇
i(∇m∇kH
j)Hmγ̇k + gij(γ)γ̇i(∇kH
j)(∇mHk)γ̇m.
Now we commute the covariant derivatives in the second term
∇m∇kH
j = ∇k∇mHj + R
j
h kmHh
to obtain
1
2
∂2
∂s2
|γ̇(ℓ, ε)|2 = |∇H[γ̇] |2 + gij γ̇
i(∇k∇mHj + R
j
h kmHh)Hmγ̇k
+ gij γ̇
i(∇kH
j)(∇mHk)γ̇m = |∇H[γ̇] |2 − 〈γ̇, R(H, γ̇)H〉
+ gij γ̇
i(∇k∇mHj)Hmγ̇k + gij γ̇
i(∇kH
j)(∇mHk)γ̇m
with curvature operator.
A. V. Antoniouk 169
Redenoting indexes m ↔ k in the third term we have
3rd + 4th terms = gij γ̇
i(∇m∇kH
j)Hkγ̇m
+ gij γ̇
i(∇kH
j)(∇mHk)γ̇m = gij γ̇
i(∇m{Hk∇kH
j})γ̇m
which leads to the final representation
1
2
∂2
∂s2
|γ̇(ℓ, ε)|2 = |∇H[γ̇] |2 − 〈γ̇, R(H, γ̇)H〉 + 〈γ̇,∇(∇HH)[γ̇]〉.
Taking now H = A0 for the first order estimate and H = Aα for the
second order estimate we find
{
AI
0 + AII
0 +
1
2
d∑
α=1
(AI
α + AII
α )2
}
ρ2(x, y)
≤
1∫
0
(
2〈∇Ã0[γ̇], γ̇〉 +
d∑
α=1
{|∇Aα[γ̇] |2 − 〈R(Aα, γ̇)Aα, γ̇〉}
)
dℓ. (17)
Dissipativity condition (4), in view of (2) leads to the statement (7).
Step 5. To get estimate (8), one proceeds like above with a choice
y = o and
∂
∂s
γ(ℓ, s) = c(ℓ, s)H(γ(ℓ, s))
instead of (10), taking c(ℓ, s0) = a(ℓ) and c(ℓ, sα) = b(ℓ) for the first and
second order operators A0 and (Aα)2 correspondingly. One has from (17)
{
AI
0 +
1
2
d∑
α=1
(AI
α)2
}
ρ2(x, o) ≤
1∫
0
(
2〈∇(aA0 +
1
2
d∑
α=1
∇bAα
[bAα] )[γ̇], γ̇〉
+
d∑
α=1
{|∇(bAα)[γ̇] |2 − 〈R(bAα, γ̇)bAα, γ̇〉}
)
dℓ.
Using that ∇c(ℓ)[γ̇] = ∂c(ℓ)
∂ℓ
we can further rewrite the last inequality
{
AI
0 +
1
2
d∑
α=1
(AI
α)2
}
ρ2(x, o) ≤
1∫
0
(2a〈∇A0[γ̇], γ̇〉 + 2
∂a
∂ℓ
〈A0, γ̇〉
+
d∑
α=1
{b2|∇Aα[γ̇] |2+
∂b2
∂ℓ
〈Aα,∇Aα[γ̇]〉+
(∂b
∂ℓ
)2
|Aα|
2−b2〈R(Aα, γ̇)Aα, γ̇〉
+ b2〈∇(∇Aα
Aα)[γ̇], γ̇〉 +
∂b2
∂ℓ
〈∇Aα
Aα, γ̇〉}) dℓ. (18)
170 Upper bounds on second order operators...
To get the last line we also applied ∇Aα
b(ℓ) = ∂b(ℓ)
∂s
= 0, leading to
calculation
∇(∇bAα
[bAα])[γ̇] = ∇γ̇(b2∇Aα
Aα) = b2∇(∇Aα
Aα)[γ̇] +
∂b2
∂l
∇Aα
Aα.
Taking further a(ℓ) = b2(ℓ), b(ℓ) = 1 − ℓ and using estimate
|〈∇Aα[γ̇], Aα〉| ≤
(1 − ℓ)
2
|∇Aα[γ̇] |2 +
1
2(1 − ℓ)
|Aα|
2
we find
LIρ2(x, o) ≤
1∫
0
{
(1 − ℓ)2(2〈∇Ã0[γ̇], γ̇〉
+2
d∑
α=1
|∇Aα[γ̇] |2 −
d∑
α=1
〈R(Aα, γ̇)Aα, γ̇〉) (19)
+ 4(ℓ − 1)〈Ã0(γ), γ̇〉 + 2
d∑
α=1
|Aα|
2
}
dℓ. (20)
Using that
∇γ(ℓ)ρ2(γ(ℓ), o) = 2ρ(γ(ℓ), o)∇γ(ℓ)ρ(γ(ℓ), o)
= 2(ℓ − 1)ρ(x, o)
γ̇(ℓ)
ρ(x, o)
= 2(ℓ − 1)γ̇,
the first term in (20) gives
2(ℓ − 1)〈Ã0(γ), γ̇〉 = 〈Ã0(γ),∇γ(ℓ)ρ2(γ(ℓ), o)〉.
Finally, using the coercitivity and dissipativity assumptions (3)–(4)
for lines (19) and (20) correspondingly, we conclude
LIρ2(x, o) ≤
1∫
0
{
2KC(1 − ℓ)2|γ̇|2 + KC′(1 + ρ2(γ(ℓ), o))
}
dℓ
≤ K(1 + ρ2(x, o))
where we applied (1 − ℓ) ≤ 1 for ℓ ∈ [0, 1] and that path γ(ℓ, 0) = γ(ℓ)
realizes the geodesic distance.
A. V. Antoniouk 171
Term (AI
αρ2(x,o))2
ρ2(x,o)
in (8) is treated like the first order term in (11) with
choice of coefficient c(ℓ) = b(ℓ) = 1 − ℓ. We get
AI
αρ2(x, o) ≤ 2
1∫
0
〈∇(bAα)[γ̇], γ̇〉 dℓ
=
1∫
0
{∂b
∂ℓ
〈Aα(γ), γ̇〉 + b〈∇Aα[γ̇], γ̇〉
}
dℓ.
Therefore
(AI
αρ2(x, o))2 ≤
1∫
0
{(∂b
∂ℓ
)2
‖Aα‖
2 + b2‖∇Aα[γ̇] ‖2
}
dℓ + 2
1∫
0
|γ̇|2 dℓ.
The last integral gives ρ2(x, o) by (2). Moreover, we can add the first
and second terms to line (18) to apply, like before, the coercitivity and
dissipativity conditions and finish estimate (8).
Remark 1. Let us note, that the upper bounds in (11) and (16) gave
us more freedom in a choice of paths {γ(ℓ, s)}ℓ∈[0,1] for s 6= 0. Otherwise
they all should be geodesics (compare with picture 1) and we would have
to work with curvature, arising in Jacobi equation on geodesic deviations
∇γ̇∇γ̇
∂γ(ℓ, s)
∂s
+ R
(
γ̇,
∂γ
∂s
)
γ̇ = 0. (21)
The work with Jacobi equation, as a second order Sturm-Liuville equa-
tions which depends on curvature, would require the knowledge of global
geometry instead of pointwise conditions (3)–(4), i.e., for example, more
precise information about the structure of harmonic tensors and Betti
numbers, etc, e.g. [5, 6].
Remark 2. The applications of upper bounds (7)–(8) to the smooth
properties of parabolic equations on manifolds are discussed in [1–4].
Here we show that under conditions (3)–(4) plus some additional
assumption on the behaviour of coefficients on the infinity, one has C∞–
regularity of process yx
t with respect to the initial data x and regularity
properties of corresponding diffusion semigroups. In particular, dissipa-
tivity condition (4) actually represents the coercitivity condition for the
high order variational systems for process yx
t .
Finally, in [4] we apply the technique of upper bounds on opera-
tors, acting on metric function, to state the existence and uniqueness of
solutions to the Stratonovich diffusions on noncompact manifolds with
globally non-Lipschitz coefficients.
172 Upper bounds on second order operators...
Author is grateful to referees for their comments about previous ver-
sion of the article.
References
[1] A. Val. Antoniouk, Nonlinear Symmetries of Variational Calculus and Regularity
Properties of Differential Flows on Non-Compact Manifolds, In Proceedings of 5th
Intern. Conf. “Symmetry in Nonlinear Mathematical Physics”, 2003, 1228–1235.
[2] A. Val. Antoniouk, A. Vict. Antoniouk, Regularity of Nonlinear Flows on Non-
compact Riemannian Manifolds: Differential vs. Stochastic Geometry or What
kind of Variations are Natural? // Ukrainian Math. Journal, 58 (2006), N 8,
1011–1034.
[3] A. Val. Antoniouk, A. Vict. Antoniouk, Nonlinear Calculus of variations for dif-
ferential flows on manifolds: geomentrically correct generalization of covariant
and stochastic variations // Ukrainean Math. Bulletin, 1 (2004), N 4, 449–484.
[4] A. Val. Antoniouk, A. Vict. Antoniouk, Non-explosion and solvability of nonlin-
ear diffusion equations on noncompact manifolds, to appear in Ukrainian Math.
Journal, 16 pp.
[5] A. L. Besse, Manifolds all of whose geodesics are closed, Springer–Verlag, 1978.
[6] J. Cheeger, D. G. Ebin, Comparison theorems in Riemannian geometry, North-
Holland Publ.Co., 1975.
[7] A. Grigor’yan, Analytic and geometric background of recurrence and non-
explosion of the Brownian motion on Riemannian manifolds // Bull. Amer. Math.
Soc., 36 (1999), N 2, 135–249.
[8] E. P. Hsu, Stochastic Analysis on Manifolds, Graduate studies in Mathematics,
38, Providence, Rhode Island: American Math. Soc., 2002.
[9] N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes,
Dordrecht: North-Holland publishing, 1981.
[10] N. V. Krylov, B. L. Rozovskii, On the evolutionary stochastic equations, Ser.
“Contemporary problems of Mathematics”, VINITI, Moscow, 14 (1979), 71–146.
[11] H. Kunita, Stochastic flows and stochastic differential equations, Cambridge Uni.
Pess, 1990.
[12] E. Pardoux, Stochastic partial differential equations and filtering of diffusion pro-
cesses // Stochastics, 3 (1979), 127–167.
Contact information
Alexander Val.
Antoniouk
Department of Nonlinear Analysis,
Institute of Mathematics NAS Ukraine,
Tereschenkivska 3,
01601 MSP Kiev-4,
Ukraine
E-Mail: antoniouk@imath.kiev.ua
|
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| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-11-29T00:10:58Z |
| publishDate | 2007 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Antoniouk, A.V. 2017-09-28T13:36:56Z 2017-09-28T13:36:56Z 2007 Upper bounds on second order operators, acting on metric function / A.V. Antoniouk // Український математичний вісник. — 2007. — Т. 4, № 2. — С. 163-172. — Бібліогр.: 12 назв. — англ. 1810-3200 2000 MSC. 35A15, 53C21, 58E35. https://nasplib.isofts.kiev.ua/handle/123456789/124513 We prove upper bounds on the general second order operator acting on metric function. The suggested approach does not use traditional formulas for deviations of geodesics and Jacobi fields construction and leads to the manifolds generalization of the classical coercitivity and dissipativity conditions for diffusion equations. Author is grateful to referees for their comments about previous version of the article. en Інститут прикладної математики і механіки НАН України Український математичний вісник Upper bounds on second order operators, acting on metric function Article published earlier |
| spellingShingle | Upper bounds on second order operators, acting on metric function Antoniouk, A.V. |
| title | Upper bounds on second order operators, acting on metric function |
| title_full | Upper bounds on second order operators, acting on metric function |
| title_fullStr | Upper bounds on second order operators, acting on metric function |
| title_full_unstemmed | Upper bounds on second order operators, acting on metric function |
| title_short | Upper bounds on second order operators, acting on metric function |
| title_sort | upper bounds on second order operators, acting on metric function |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/124513 |
| work_keys_str_mv | AT antonioukav upperboundsonsecondorderoperatorsactingonmetricfunction |