Wyniki wyszukiwania

1

Boundedness of fractional maximal operators between classical and weak-type Lorentz spaces

100%

Edmunds D., Opic B.

Instytut Matematyczny Polskiej Akademii Nauk

EN

We derive conditions, which are both necessary and sufficient, for the boundedness of (power-logarithmic) fractional maximal operators between classical and weak-type Lorentz spaces.

2

Explicit representation of compact linear operators in Banach spaces via polar sets

94%

Edmunds D., Lang J.

Studia Mathematica

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2013

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tom 214

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nr 3

265-278

EN

We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. This provides a Banach space version of the well-known Hilbert space result of E. Schmidt.

3

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

94%

Carl B., Edmunds D.

Studia Mathematica

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2003

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tom 159

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nr 3

391-402

EN

For a precompact subset K of a Hilbert space we prove the following inequalities: $n^{1/2} cₙ(cov(K)) ≤ c_{K}(1 + ∑^{ⁿ}_{k=1} k^{-1/2} e_k(K))$, n ∈ ℕ, and $k^{1/2} c_{k+n}(cov(K)) ≤ c[log^{1/2}(n+1)εₙ(K) + ∑_{j=n+1}^{∞} ε_j(K)/(j log^{1/2}(j+1))]$, k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and $ε_k(K)$ and $e_k(K)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K) are slowly decreasing. For example, we get optimal estimates in the non-critical case where $εₙ(K) ⪯ log^{-α}(n + 1)$, α ≠ 1/2, 0 < α < ∞, as well as in the critical case where α = 1/2. For α = 1/2 we show the asymptotically optimal estimate $cₙ(cov(K)) ⪯ n^{-1/2} log(n + 1)$, which refines the corresponding result of Gao [Ga] obtained for entropy numbers. Furthermore, we establish inequalities similar to that of Creutzig and Steinwart [CrSt] in the critical as well as non-critical cases. Finally, we give an alternative proof of a result by Li and Linde [LL] for Gelfand and entropy numbers of the absolutely convex hull of K when K has the shape K = {t₁,t₂,...}, where ||tₙ|| ≤ σₙ, σₙ↓ 0. In particular, for $σₙ ≤ log^{-1/2}(n + 1)$, which corresponds to the critical case, we get a better asymptotic behaviour of Gelfand numbers, $cₙ(cov(K)) ⪯ n^{-1/2}$.

4

Compact and continuous embeddings of logarithmic Bessel potential spaces

76%

Edmunds D., Gurka P., Opic B.

Studia Mathematica

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2005

|

tom 168

|

nr 3

229-250

EN

We establish compact and continuous embeddings for Bessel potential spaces modelled upon generalized Lorentz-Zygmund spaces. The target spaces are either of Lorentz-Zygmund or Hölder type.

Ograniczanie wyników

3 Studia Mathematica

4 Edmunds D.

2 Opic B.

1 Carl B.

1 Gurka P.

1 Lang J.

1 2013

1 2005

1 2003

1 2002

Boundedness of fractional maximal operators between classical and weak-type Lorentz spaces

Explicit representation of compact linear operators in Banach spaces via polar sets

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Compact and continuous embeddings of logarithmic Bessel potential spaces