Jiří Rákosník

Sobolev Embeddings with Variable Exponent, II

Let Ω be a bounded open subset of Rn with Lipschitz boundary, let n 0 such that for all f ∈ W1,p(x)(Ω), where ∥·∥M,Ω is the norm on a certain space of Orlicz-Musielak type and ∥·∥1,p,Ω is the norm on W1,p(x)(Ω). This inequality reduces to the usual Sobolev inequality when supΩp < n. The paper extends earlier work of the authors ([ER]) in which it was assumed that p was Lipschitz-continuous.

Poincaré and Friedrichs inequalities in abstract Sobolev spaces II

This paper is a continuation of [ 4 ]; its aim is to extend the results of that paper to include abstract Sobolev spaces of higher order and even anisotropic spaces. Let Ω be a domain in ℝ N , let X = X(Ω) and Y = Y(Ω) be Banach function spaces in the sense of Luxemburg (see [ 4 ] for details), and let W(X, Y) be the abstract Sobolev space consisting of all f ∈ X such that for each i ∈ {l, …, N} the distributional derivative belongs to Y; equipped with the norm W(X, Y) is a Banach space. Given any weight function w on Ω, the triple [ w, X, Y ] is said to support the Poincare inequality on Ω. if there is a positive constant K such that for all u ∈ W(X, Y) the pair [ X, Y ] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W 0 (X, Y) (the closure of in W(X, Y))

Weighted estimates for the averaging integral operator

AbstractLet 1 <p≤q<+∞ and letv, w be weights on (0, + ∞) satisfying: (*)v(x)xρ is equivalent to a non-decreasing function on (0, +∞) for someρ ≥ 0 and

DML-CZ: The Objectives and the First Steps

[w(x)x]^{1/q} \approx [v(x)x]^{1/p} for all x \in (0, + \infty ).

From Pixels and Minds to the Mathematical Knowledge in a Digital Library

We prove that if the averaging operator

DML-CZ: the experience of a medium-sized digital mathematics library

(Af)(x): = \frac{1}{x}\int_0^x f (t) dt