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1
Content available remote

On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

100%
Kałamajska A.
Colloquium Mathematicum
|
2001
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tom 89
|
nr 1
43-59
EN
We study the functional $I_{f}(u)=∫_{Ω} f(u(x))dx$, where u=(u₁, ..., uₘ) and each $u_{j}$ is constant along some subspace $W_{j}$ of ℝⁿ. We show that if intersections of the $W_{j}$'s satisfy a certain condition then $I_{f}$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_{j}}_{j=1,...,m}$ to have the equivalence: $I_{f}$ is weakly continuous if and only if f is Λ-affine.
2
Content available remote

Coercive inequalities on weighted Sobolev spaces

100%
Kałamajska A.
Colloquium Mathematicum
|
1993
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tom 66
|
nr 2
309-318
3
Content available remote

On lower semicontinuity of multiple integrals

100%
Kałamajska A.
Colloquium Mathematicum
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1997
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tom 74
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nr 1
71-78
EN
We give a new short proof of the Morrey-Acerbi-Fusco-Marcellini Theorem on lower semicontinuity of the variational functional $\int_{Ω} F(x,u,∇u)dx$. The proofs are based on arguments from the theory of Young measures.
4
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Gagliardo-Nirenberg inequalities in logarithmic spaces

64%
Kałamajska A., Pietruska-Pałuba K.
Colloquium Mathematicum
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2006
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tom 106
|
nr 1
93-107
EN
We obtain interpolation inequalities for derivatives: $∫ M_{q,α}(|∇f(x)|)dx ≤ C[∫M_{p,β}(Φ₁(x,|f|,|∇^{(2)}f|))dx + ∫M_{r,γ}(Φ₂(x,|f|,|∇^{(2)}f|))dx]$, and their counterparts expressed in Orlicz norms: ||∇f||²_{(q,α)} ≤ C||Φ₁(x,|f|,|∇^{(2)}f|)||_{(p,β)} ||Φ₂(x,|f|,|∇^{(2)}f|)||_{(r,γ)}$, where $||·||_{(s,κ)}$ is the Orlicz norm relative to the function $M_{s,κ}(t) = t^{s}(ln(2+t))^{κ}$. The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher order gradients are also considered.
5
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Gagliardo-Nirenberg inequalities in weighted Orlicz spaces

64%
Kałamajska A., Pietruska-Pałuba K.
Studia Mathematica
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2006
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tom 173
|
nr 1
49-71
EN
We derive inequalities of Gagliardo-Nirenberg type in weighted Orlicz spaces on ℝⁿ, for maximal functions of derivatives and for the derivatives themselves. This is done by an application of pointwise interpolation inequalities obtained previously by the first author and of Muckenhoupt-Bloom-Kerman-type theorems for maximal functions.
6
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On a Variant of the Gagliardo-Nirenberg Inequality Deduced from the Hardy Inequality

64%
Kałamajska A., Pietruska-Pałuba K.
Bulletin of the Polish Academy of Sciences. Mathematics
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2011
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tom 59
|
nr 2
133-149
EN
We obtain new variants of weighted Gagliardo-Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.
7
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On a variant of the Hardy inequality between weighted Orlicz spaces

64%
Kałamajska A., Pietruska-Pałuba K.
Studia Mathematica
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2009
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tom 193
|
nr 1
1-28
EN
Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities $∫_{ℝ₊} M(ω(x)|u(x)|) exp(-φ(x)) dx ≤ C ∫_{ℝ₊} M(|u'(x)|) exp(-φ(x)) dx$, where u belongs to some set 𝓡 of locally absolutely continuous functions containing $C₀^{∞}(ℝ₊)$. We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set 𝓡. The set 𝓡 may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.
8
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New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

64%
Kałamajska A., Pietruska-Pałuba K.
Open Mathematics
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2012
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tom 10
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nr 6
2033-2050
EN
We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.
9
Content available remote

Polynomial asymptotics and approximation of Sobolev functions

64%
Hajłasz P., Kałamajska A.
Studia Mathematica
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1995
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tom 113
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nr 1
55-64
EN
We prove several results concerning density of $C_{0}^{∞}$, behaviour at infinity and integral representations for elements of the space $L^{m,p} = {⨍ | ∇^{m}⨍ ∈ L^p}$.
10
Content available remote

Pointwise multiplicative inequalities and Nirenberg type estimates in weighted Sobolev spaces

50%
Kałamajska A.
Studia Mathematica
|
1994
|
tom 108
|
nr 3
275-290
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