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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)
We show that essentially the speed of decay of the Fourier sine coefficients of a function in a Lebesgue space is comparable to that of the corresponding coefficients with respect to the basis formed by the generalized sine functions sin p,q .
Some three-dimensional analogues of the plane Darboux problems for hyperbolic equations with degeneracy are investigated. In 1954, Protter initiated the study of such three-dimensional problems, and it is now well known that for an infinite number of smooth right-hand sides these problems have solutions with a strong power-type singularity on the… (More)
Let I = [a, b] ⊂ R, let 1 < q, p < ∞, let u and v be positive functions with u ∈ L p ′ (I), v ∈ Lq(I) and let T : Lp(I) → Lq(I) be the Hardy-type operator given by (T f)(x) = v(x) Z x a f (t)u(t)dt, x ∈ I. We show that the asymptotic behavior of the eigenvalues λ of the non-linear integral system g(x) = (T f)(x) (f (x)) (p) = λ(T * (g (p)))(x) (where, for… (More)
Optimal sufficient conditions are found in weighted Lorentz spaces for weight functions which provide the boundedness of the Calderón– Zygmund singular integral operator defined on spaces of homogeneous and nonhomogeneous type.