On pointwise interpolation inequalities for derivatives

Vladimir G. Maz'ya; Tatjana Olegovna Shaposhnikova

Displaying similar documents to “On pointwise interpolation inequalities for derivatives”

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 $p = 1$ .

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 $F_{p q}^{s} (Ω)$ .

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...

On systems of linear algebraic equations in the Colombeau algebra

Jan Ligęza, Milan Tvrdý (1999)

Mathematica Bohemica

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From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra $\bar{ℝ}$ of generalized real numbers. It is worth mentioning that the algebra $\bar{ℝ}$ is not a field.