Optimal Sobolev embeddings
Weighted estimates for the Hilbert transform of odd functions.
Two-weight weak type maximal inequalities in Orlicz classes
Necessary and sufficient conditions are shown in order that the inequalities of the form , or hold with some positive C independent of λ > 0 and a μ-measurable function f, where (X,μ) is a space with a complete doubling measure μ, is the maximal operator with respect to μ, Φ, Ψ are arbitrary Young functions, and ϱ, σ are weights, not necessarily doubling.
Hrášek a sluníčko. O matematickém paradoxu Stefana Banacha a Alfreda Tarského
Dirichletovy šuplíčky
Článek obsahuje několik příkladů (téměř ze života), jejichž společným jmenovatelem je jednoduchý matematický princip známý jako princip Dirichletův. Hlavním úkolem uvedených příkladů je ilustrovat poněkud překvapivou šíři pole jeho aplikací.
O využití iracionality při hledání sedmého nebe
Věnujeme se otázce V. I. Arnol'da, zda nějaká mocnina dvojky začíná číslicí sedm. Uvedeme dvě různá řešení problému a zmíníme se o některých souvisejících otázkách a možnostech zobecnění.
O princezně a píd’alce
Optimal Sobolev imbedding spaces
This paper continues our study of Sobolev-type imbedding inequalities involving rearrangement-invariant Banach function norms. In it we characterize when the norms considered are optimal. Explicit expressions are given for the optimal partners corresponding to a given domain or range norm.
Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities
We study imbeddings of the Sobolev space : = u: Ω → ℝ with < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, and , that are optimal with respect to the inclusions . General formulas for and are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.
Compactness of Sobolev imbeddings involving rearrangement-invariant norms
We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space be compactly imbedded into the rearrangement-invariant space , where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from into . The results are illustrated with examples in which ϱ and σ are both Orlicz norms...
Discretization and anti-discretization of rearrangement-invariant norms.
An optimal endpoint trace embedding
We find an optimal Sobolev-type space on all of whose functions admit a trace on subspaces of of given dimension. A corresponding trace embedding theorem with sharp range is established.
Compactness of Hardy Operators Involving Suprema
We study compactness properties of Hardy operators involving suprema on weighted Banach function spaces. We first characterize the compactness of abstract operators assumed to have their range in the class of non-negative monotone functions. We then define a category of pairs of weighted Banach function spaces for which a suitable Muckenhoupt-type condition implies the boundedness of Hardy operators involving suprema, and prove a criterion for the compactness of these operators between such a couple...
A note on the Dirichlet problem for the elliptic linear operator in Sobolev spaces with weight
Weak and extra-weak type inequalities for the maximal operator and the Hilbert transform
Compactness of Hardy-type integral operators in weighted Banach function spaces
We consider a generalized Hardy operator . For T to be bounded from a weighted Banach function space (X,v) into another, (Y,w), it is always necessary that the Muckenhoupt-type condition be satisfied. We say that (X,Y) belongs to the category M(T) if this Muckenhoupt condition is also sufficient. We prove a general criterion for compactness of T from X to Y when (X,Y) ∈ M(T) and give an estimate for the distance of T from the finite rank operators. We apply the results to Lorentz spaces and characterize...
Characterization of associate spaces of weighted Lorentz spaces with applications
We characterize associate spaces of weighted Lorentz spaces GΓ(p,m,w) and present some applications of this result including necessary and sufficient conditions for a Sobolev-type embedding into .
Weighted inequalities for Hardy-type operators involving suprema.
A sharp iteration principle for higher-order Sobolev embeddings
We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order involving...
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