David E. Edmunds

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Let m and n be positive integers with n 2 and 1 m n ? 1. We study rearrangement-invariant quasinorms % R and % D on functions f : (0; 1) ! R such that to each bounded domain in R n , with Lebesgue measure jj, there corresponds C = C(jj) > 0 for which one has the Sobolev imbedding inequality % R ? u (jjt) C% D ? jr m uj (jjt) ; u 2 C m 0 ((); involving the(More)
Various properties of the generalized trigonometric functions sinp,q are established. In particular, it is shown that those functions can approximate functions from every space Lr (0, 1) (1 < r < ∞) provided that p and q ′ (p, q ∈ (1, ∞)) are not too far apart (in fact we prove that these functions form a basis in every space Lr (0, 1)). An addition formula(More)
The Hardy inequality ∫ Ω |u(x)|pd(x)−p dx c ∫ Ω |∇u(x)|p dx with d(x) = dist(x, ∂Ω) holds for u ∈ C∞ 0 (Ω) if Ω ⊂ n is an open set with a sufficiently smooth boundary and if 1 < p < ∞. P.Haj lasz 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(More)