Spaces of Differentiable Functions of Several Variables and Imbedding Theorems

Estimates of difference norms for functions in anisotropic Sobolev spaces

“…Only at the end of the fifties Gagliardo and Nirenberg gave simple proofs of the inequality (1.3) for all 1 ≤ p < n. The inequality (1.3) has been generalized and developed in various directions (see [2,10,12,13,20,21] for details and references). It was proved that the left-hand side in (1.3) can be replaced by the stronger Lorentz norm; that is, there holds the inequality…”

“…The most extended theory of these classes is contained in the monography [2]. Furthermore, many authors have studied Sobolev and Nikol'skiiBesov spaces whose construction involves, instead of L p -norms, norms in more general spaces (see [12]). In this paper we suppose that derivatives belong to different Lorentz spaces L p k ,s k (R n ) (where 1 ≤ p k , s k < ∞ and s k = 1, if p k = 1).…”

Piecewise Polynomial Approximation Methods in the Theory of Nikol’Skiĭ–Besov Spaces

“…Recall the classical definition of Lizorkin-Triebel spaces (see, e.g., [69]). Expand a function f in a series with respect to f n , where functions f n are defined by means of the Fourier transformation.…”

“…3.6) and denoted by L λ p . Recall (see, e.g., [69]) that the Bessel-MacDonald kernel is defined by the relation G λ (ξ) = F −1 1 + |ξ| 2 − λ 2 , λ > 0.…”

Some notes on embedding for anisotropic Sobolev spaces

“…This embedding has been generalized and developed in different directions (see [7], [9], [12], [13], [14], [17], [18] for details and references). We denote by W r1,...,rn p,s…”