Differential expressions with mixed homogeneity and spaces of smooth functions they generate in arbitrary dimension

Abstract

Let {T 1 , . . . , T l } be a collection of differential operators with constant coefficients on the torus T n . Consider the Banach space X of functions f on the torus for which all functions T j f , j = 1, . . . , l, are continuous. Extending the previous work of the first two authors, we analyse the embeddability of X into some space C(K) as a complemented subspace. We prove the following. Fix some pattern of mixed homogeneity and extract the senior homogeneous parts (relative to the pattern chosen) {τ 1 , . . . , τ l } from the initial operators {T 1 , . . . , T l }. Let N be the dimension of the linear span of {τ 1 , . . . , τ l }. If N 2, then X is not isomorphic to a complemented subspace of C(K) for any compact space K.The main ingredient of the proof of this fact is a new Sobolev-type embedding theorem. In general, we use the mapping t → e 2πit to parametrise the unit circle; so, the trigonometric system is {e 2πikt } k∈Z , t ∈ [0, 1), and the (multiple) Fourier series are handled accordingly.

Anisotropic Ornstein noninequalities

“…where the coefficients c α ′ ,β come from the Leibniz formula. The error is O(ε ′ ) in absolute value by (8) and equals to zero on the set…”

“…due to Gagliardo and Nirenberg had been generalized to the anisotropic case only in [16] and finally in [9]; if one deals with similar embeddings for vector fields, the isotropic case was successfully considered in [14] (see also the survey [15]), and there is almost no progress for anisotropic case (however, see [7,8]).…”

Restrictions of higher derivatives of the Fourier transform

“…Proposition 1.11. If HDR(Σ, k, s, , p) holds true and p > 1, then (17) where the numbers σ p and κ p are defined by (4) and (5), respectively. In the case p = 1, equality in (15) may also occur.…”

Gennadi Markovich Henkin (obituary)

“…Г. М. Хенкин более не возвращался к этой теме, однако его статья породила целую серию исследований, в основном в России и в Польше, -cм. последнюю по времени в этой серии работу [35], где изложена также история вопроса и приведены ссылки на предшествующие публикации. Отметим только, что отправной точкой в этой деятельности было наблюдение (С. В. Кисляков, 1974), что результат Хенкина можно вывести и даже существенно усилить, просто посмотрев пристально на соболевские вложения с предельным показателем (сам Хенкин пользовался совсем другим методом).…”