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1
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An Extension of Kotzig’s Theorem

100%
Aksenov V. A., Borodin O. V., Ivanova A. O.
Discussiones Mathematicae Graph Theory
|
2016
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tom 36
|
nr 4
889-897
EN
In 1955, Kotzig proved that every 3-connected planar graph has an edge with the degree sum of its end vertices at most 13, which is tight. An edge uv is of type (i, j) if d(u) ≤ i and d(v) ≤ j. Borodin (1991) proved that every normal plane map contains an edge of one of the types (3, 10), (4, 7), or (5, 6), which is tight. Cole, Kowalik, and Škrekovski (2007) deduced from this result by Borodin that Kotzig’s bound of 13 is valid for all planar graphs with minimum degree δ at least 2 in which every d-vertex, d ≥ 12, has at most d − 11 neighbors of degree 2. We give a common extension of the three above results by proving for any integer t ≥ 1 that every plane graph with δ ≥ 2 and no d-vertex, d ≥ 11+t, having more than d − 11 neighbors of degree 2 has an edge of one of the following types: (2, 10+t), (3, 10), (4, 7), or (5, 6), where all parameters are tight.
2
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Two-weight norm inequalities for maximal functions on homogeneous spaces and boundary estimates

100%
Luís Zani S.
Studia Mathematica
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1997
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tom 126
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nr 1
67-94
EN
Let D be an open subset of a homogeneous space(X,d,μ). Consider the maximal function $M_φ f(x) = sup1/φ(B) ʃ_{B∩∂D} |f|dν$, x∈ D, where the supremum is taken over all balls of the form B = B(a(x),r) with r > t(x) = d(x,∂D), a(x)∈ ∂D is such that d(a(x),x) < 3/2 t(x)$ and φ is a nonnegative set function defined for all Borel sets of X satisfying the quasi-monotonicity and doubling properties. We give a necessary and sufficient condition on the weights w and v for the weighted norm inequality (0.1) $(ʃ_D [M_φ(f)]^q wdμ)^{1/q} ≤ c(ʃ_{∂D} |f|^p vdν)^{1/p}$ to hold when 1 < p < q < ∞, $σdν = v^{1-p'}dν$ is a doubling weight, and dν is a doubling measure, and give a sufficient condition for (0.1) when 1 < p ≤ q < ∞ without assuming that σ is a doubling weight but with an extra assumption on φ. Another characterization for (0.1) is also provided for 1 < p ≤ q < ∞ and D of the form Y×(0,∞), where Y is a homogeneous space with group structure. These results generalize some known theorems in the case when $M_φ$ is the fractional maximal function in $ℝ^{n+1}_+$, that is, when $M_φ f(x,t) = M_γ f(x,t) = sup_{r>t} 1/(ν(B(x,r))^{1-γ}) ʃ_{B(x,r)} |f|dν$, where $(x,t) ∈ ℝ^{n+1}_+$, 0 < γ < 1, and ν is a doubling measure in $ℝ^n$.
3
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Pointwise multipliers for reverse Holder spaces

100%
M. Buckley S.
Studia Mathematica
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1994
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tom 109
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nr 1
23-39
EN
We classify weights which map reverse Hölder weight spaces to other reverse Hölder weight spaces under pointwise multiplication. We also give some fairly general examples of weights satisfying weak reverse Hölder conditions.
4
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Weak type estimates for operators of potential type

100%
Wheeden R. L., Zhao S.
Studia Mathematica
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1996
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tom 119
|
nr 2
149-160
EN
We derive two-weight weak type estimates for operators of potential type in homogeneous spaces. The conditions imposed on the weights are testing conditions of the kind first studied by E. T. Sawyer [4]. We also give some applications to strong type estimates as well as to operators on half-spaces.
5
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Minimal vertex degree sum of a 3-path in plane maps

88%
Borodin O.
Discussiones Mathematicae Graph Theory
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1997
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tom 17
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nr 2
279-284
EN
Let wₖ be the minimum degree sum of a path on k vertices in a graph. We prove for normal plane maps that: (1) if w₂ = 6, then w₃ may be arbitrarily big, (2) if w₂ < 6, then either w₃ ≤ 18 or there is a ≤ 15-vertex adjacent to two 3-vertices, and (3) if w₂ < 7, then w₃ ≤ 17.
6
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5-Stars of Low Weight in Normal Plane Maps with Minimum Degree 5

88%
Borodin O. V., Ivanova A. O., Jensen T. R.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 3
539-546
EN
It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48
7
Content available remote

$L^p$-spaces and quantum dynamical semigroups

63%
Goldstein S., Martin Lindsay J.
Banach Center Publications
|
1998
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tom 43
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nr 1
211-216
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