On a higher-order Hardy inequality

David Eric Edmunds; Jiří Rákosník

Displaying similar documents to “On a higher-order Hardy inequality”

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.

Hardy inequalities in function spaces

Hans Triebel (1999)

Mathematica Bohemica

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Let $Ω$ be a bounded $C^{\infty}$ domain in $ℝ^{n}$ . The paper deals with inequalities of Hardy type related to the function spaces $B_{p q}^{s} (Ω)$ and $F_{p q}^{s} (Ω)$ .

On weighted estimates of solutions of nonlinear elliptic problems

Igor V. Skrypnik, Dmitry V. Larin (1999)

Mathematica Bohemica

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The paper is devoted to the estimate u(x,k)Kk{capp,w(F)pw(B(x,))} 1p-1, $2 p < n$ for a solution of a degenerate nonlinear elliptic equation in a domain $B (x_{0}, 1) ∖ F$ , $F \subset B (x_{0}, d) = {x \in ℝ^{n} | x_{0} - x | < d}$ , $d < \frac{1}{2}$ , under the boundary-value conditions $u (x, k) = k$ for $x \in \partial F$ , $u (x, k) = 0$ for $x \in \partial B (x_{0}, 1)$ and where $0 < ρ \leq d i s t (x, F)$ , $w (x)$ is a weighted function from some Muckenhoupt class, and $c a p_{p, w} (F)$ , $w (B (x, ρ))$ are weighted capacity and measure of the corresponding sets.

A second look on definition and equivalent norms of Sobolev spaces

Joachim Naumann, Christian G. Simader (1999)

Mathematica Bohemica

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Sobolev’s original definition of his spaces $L^{m, p} (Ω)$ is revisited. It only assumed that $Ω \subseteq ℝ^{n}$ is a domain. With elementary methods, essentially based on Poincare’s inequality for balls (or cubes), the existence of intermediate derivates of functions $u \in L^{m, p} (Ω)$ with respect to appropriate norms, and equivalence of these norms is proved.