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We generalize a well-known separation condition of Everitt and Giertz to a class of weighted symmetric partial differential operators defined on domains in n . Also, for symmetric second-order ordinary differential operators we show that lim sup t→c (pq′)′/q2 = θ < 2 where c is a singular point guarantees separation of −(py′)′ + qy on its minimal domain and… (More)

- Alois Kufner, Lech Maligranda, Lars-Erik Persson
- The American Mathematical Monthly
- 2006

1. INTRODUCTION. The development of the famous Hardy inequality (in both its discrete and continuous forms) during the period 1906–1928 has its own history or, as we have called it, prehistory. Contributions of mathematicians other than G. are important here. In this article we describe some of those contributions. We also include and comment upon several… (More)

Including the previously untreated borderline cases, the trace spaces (in the distributional sense) of the Besov-Lizorkin-Triebel spaces are determined for the anisotropic (or quasi-homogeneous) version of these classes. The ranges of the traces are in all cases shown to be approximation spaces, and these are shown to be different from the usual spaces… (More)

- Hans P. Heinig, Alois Kufner
- 2002

- HANS P . HEINIGa, Alois Kufner, LARS - ERIK PERSSONc

Weighted inequalities for fractional derivatives (= fractional order Hardy-type inequalities) have recently been proved in [4] and 1]. In this paper, new inequalities of this type are proved and applied. In particular, the general mixed norm case and a general twodimensional weight are considered. Moreover, an Orlicz norm version and a multidimensional… (More)

- Vladimir Maz’ya, Tatyana Shaposhnikova, Alois Kufner
- 2002

Abstract. Pointwise interpolation inequalities, in particular, |∇ku(x)| c (Mu(x))1−k/m (M∇mu(x)) , k < m, and |Izf(x)| c(MIζf(x)) z/Re ζ(Mf(x))1−Re z/Re ζ , 0 < Re z < Re ζ < n, where ∇k is the gradient of order k, M is the Hardy-Littlewood maximal operator, and Iz is the Riesz potential of order z, are proved. Applications to the theory of multipliers in… (More)

- Vesa Mustonen, Matti Tienari, Alois Kufner
- 2002

We study the mappings of monotone type in Orlicz-Sobolev spaces. We introduce a new class (Sm) as a generalization of (S+) and extend the definition of quasimonotone map. We also prove existence results for equations involving monotone-like mappings.

- Hans Triebel, Alois Kufner
- 2002

- Peter Takáč, Alois Kufner
- 2005

We investigate the smoothing effect of the parabolic part of a quasilinear evolutionary equation on its solution as time evolves. More precisely, the following initial-boundary value problem with Dirichlet boundary conditions is considered: (P) 8 >< >: ∂u ∂t = ∆pu + f(x, t, u(x, t)) for (x, t) ∈ Ω × (0, T ); u(x, t) = 0 for (x, t) ∈ ∂Ω × (0, T ); u(x, 0) =… (More)

- Sorina Bârză, Alois Kufner
- 2002

Let + := {(x1, . . . , xN ) ; xi 0, i = 1, 2, . . . , N} and + := + . Assume that f : + → + is monotone which means that it is monotone with respect to each variable. We denote f ↓, when f is decreasing (= nonincreasing) and f ↑ when f is increasing (= nondecreasing). Throughout this paper ω, u, v are positive measurable functions defined on + , N 1. A… (More)