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 $μ \to 0 .$

On the set of solutions of the system $x_{1} + x_{2} + x_{3} = 1, x_{1} x_{2} x_{3} = 1$

Miloslav Hlaváček (1998)

Mathematica Bohemica

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A proof is given that the system in the title has infinitely many solutions of the form $a_{1} + a_{2}$ , where $a_{1}$ and $a_{2}$ are rational numbers.

On pointwise interpolation inequalities for derivatives

Vladimir G. Maz&#039;ya, Tatjana Olegovna Shaposhnikova (1999)

Mathematica Bohemica

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Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where $\nabla_{k}$ is the gradient of order $k$ , $ℳ$ is the Hardy-Littlewood maximal operator, and $I_{z}$ is the Riesz potential of order $z$ , are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space $M (W_{p}^{m} (ℝ^{n}) \to W_{p}^{l} (ℝ^{n}))$ is described.

Displaying similar documents to “Hardy inequalities in function spaces”

On a higher-order Hardy inequality

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

Linear integral equations in the space of regulated functions

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

On the set of solutions of the system x 1 + x 2 + x 3 = 1 , x 1 x 2 x 3 = 1

On pointwise interpolation inequalities for derivatives

On the set of solutions of the system $x_{1} + x_{2} + x_{3} = 1, x_{1} x_{2} x_{3} = 1$