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Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces

D. EdmundsYu. Netrusov — 1998

Studia Mathematica

Let id be the natural embedding of the Sobolev space W p l ( Ω ) in the Zygmund space L q ( l o g L ) a ( Ω ) , where Ω = ( 0 , 1 ) n , 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers e k ( i d ) of this embedding and show that e k ( i d ) k - η , where η = min(-a,l/n). Extensions to more general spaces are given. The results are applied to give information about the behaviour of the eigenvalues of certain operators of elliptic type.

Two-sided estimates of the approximation numbers of certain Volterra integral operators

D. EdmundsW. EvansD. Harris — 1997

Studia Mathematica

We consider the Volterra integral operator T : L p ( + ) L p ( + ) defined by ( T f ) ( x ) = v ( x ) ʃ 0 x u ( t ) f ( t ) d t . Under suitable conditions on u and v, upper and lower estimates for the approximation numbers a n ( T ) of T are established when 1 < p < ∞. When p = 2 these yield l i m n n a n ( T ) = π - 1 ʃ 0 | u ( t ) v ( t ) | d t . We also provide upper and lower estimates for the α and weak α norms of (an(T)) when 1 < α < ∞.

Fourier approximation and embeddings of Sobolev spaces

CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into W m , p ( Ω ) into L S ( Ω ) (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding W m , p ( Ω ) into L φ ( Ω ) ...............................................................

Spaces of Lipschitz type, embeddings and entropy numbers

AbstractWe establish the sharpness of the embedding of certain Besov and Triebel-Lizorkin spaces in spaces of Lipschitz type. In particular, this proves the sharpness of the Brézis-Wainger result concerning the “almost” Lipschitz continuity of elements of the Sobolev space H p 1 + n / p ( ) , where 1 < p < ∞. Upper and lower estimates are obtained for the entropy numbers of related embeddings of Besov spaces on bounded domains. CONTENTSIntroduction...........................................................51....

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