Boundedness conditions and spectrum estimates for the operators 𝑏(𝑋)𝑎(𝐷) and their analogs

Cwikel Type Estimates in Non‐Power Ideals

“…As already mentioned in Section 1, the generalized Cwikel inequality ( [7], which has been proven in [5]. Notice that (4.1) and (4.4) implies ( This estimate can also be obtained by bilinear complex interpolation methods ([6], [2], [5]).…”

“…Previously the correlated estimate on the number of bound states of Schrodinger operators had been obtained by ROZENBLYUM [14] and LIEB [lo]. For generalizations see [2], [5].…”

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators I

“…q,.ql(el(x)) = BE,,,(e2(x)) 3 then the embedding is compact if (4) holds. This 'follows immediately from part (ii) of the above theorem, the embeddings for the unweighted spaces and FSp,q, and Theorem 2.2 (ii).…”

“…This point of view is supported by the applications we have in mind and by what has been done in this direction in literature, see [16, Ch. 41, [3], [4]. However we shift this task to a later occasion and simplify our situation by assuming el(x) = (x)' for some CI > 0.…”

Sharp Constants in the Hardy–Rellich Inequalities

“…If l<dÂ2 is not an integer, then (1.1) with some estimating constant and its different generalizations (in particular, to weighted L p -norms for p{2) were also discussed extensively in the literature. Actually, the functions |x| &2l and |!| 2l can be replaced in (1.1) by more general functions (see [4] and references therein). Inequality (1.1) can also be considered as a particular case of the Stein and Weiss inequality (see, e.g., [8] and references therein) which is formulated as the boundedness of an integral operator with kernel |x& y| &s in different L p -spaces with power weights.…”