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.
For a precompact subset K of a Hilbert space we prove the following inequalities: , n ∈ ℕ, and , k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and and 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...
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.
We give very short and transparent proofs of extrapolation theorems of Yano type in the framework of Lorentz spaces. The decomposition technique developed in Edmunds-Krbec (2000) enables us to obtain known and new results in a unified manner.
Using the idea of the optimal decomposition developed in recent papers (Edmunds-Krbec, 2000) and in Cruz-Uribe-Krbec we study the boundedness of the operator Tg(x) = ∫ g(u)du / u, x ∈ (0,1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser's lemma, O'Neil's convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very...
Let and ; suppose that is a compact linear map with trivial kernel and range dense in . It is shown that if the Gelfand numbers of decay sufficiently quickly, then the action of is given by a series with calculable coefficients. The special properties of enable this to be established under weaker conditions on the Gelfand numbers than in earlier work set in the context of more general spaces.
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.
Sharp estimates are obtained for the rates of blow up of the norms of embeddings of Besov spaces in Lorentz spaces as the parameters approach critical values.
In this paper we consider a nonlinear elliptic equation with critical growth for the operator in a bounded domain . We state some existence results when . Moreover, we consider , expecially when is a ball in .
In this paper we consider a nonlinear elliptic equation with critical growth for the operator in a bounded domain . We state some existence results when . Moreover, we consider , expecially when is a ball in .
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