Hardy inequalities in function spaces

Hans Triebel

Displaying similar documents to “Hardy inequalities in function spaces”

On a higher-order Hardy inequality

David Eric Edmunds, Jiří Rákosník (1999)

Mathematica Bohemica

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The Hardy inequality $\int_{Ω} {| u (x) |}^{p} d {(x)}^{- p} \ddot{x} \leq c \int_{Ω} {| \nabla u (x) |}^{p} \ddot{x}$ with $d (x) = dist (x, \partial Ω)$ holds for $u \in C_{0}^{\infty} (Ω)$ if $Ω \subset ℝ^{n}$ is an open set with a sufficiently smooth boundary and if $1 < p < \infty$ . P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for $p = 1$ .

The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations

Peter Raith (1997)

Mathematica Bohemica

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In this paper piecewise monotonic maps $T [0, 1] \to [0, 1]$ are considered. Let $Q$ be a finite union of open intervals, and consider the set $R (Q)$ of all points whose orbits omit $Q$ . The influence of small perturbations of the endpoints of the intervals in $Q$ on the dynamical system $(R (Q), T)$ is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed...

Linear integral equations in the space of regulated functions

Milan Tvrdý (1998)

Mathematica Bohemica

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n this paper we investigate systems of linear integral equations in the space $𝔾_{L}^{n}$ of $n$ -vector valued functions which are regulated on the closed interval $[0, 1]$ (i.e. such that can have only discontinuities of the first kind in $[0, 1]$ ) and left-continuous in the corresponding open interval $(0, 1) .$ In particular, we are interested in systems of the form x(t) - A(t)x(0) - 01B(t,s)[d x(s)] = f(t), where $f \in 𝔾_{L}^{n}$ , the columns of the $n \times n$ -matrix valued function $A$ belong to $𝔾_{L}^{n}$ , the entries of $B (t, .)$ have a bounded variation...

Upper and lower solutions for singularly perturbed semilinear Neumann's problem

Róbert Vrábeľ (1997)

Mathematica Bohemica

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The paper establishes sufficient conditions for the existence of solutions of Neumann’s problem for the differential equation $μ y " + k y = f (t, y)$ which tend to the solution of the reduced problem $k y = f (t, y)$ on $[0, 1]$ as $μ \to 0 .$