Advanced Search

We consider a generalized Hardy operator $Tf\left(x\right)=\varphi \left(x\right){ʃ}_0^x\psi fv$. For T to be bounded from a weighted Banach function space (X,v) into another, (Y,w), it is always necessary that the Muckenhoupt-type condition $ℬ=su{p}_{R>0}\parallel \varphi {\chi }_{(R,\infty )}{\parallel }_Y\parallel \psi {\chi }_{(0,R)}{\parallel }_{X^{\text{'}}}<\infty$ be satisfied. We say that (X,Y) belongs to the category M(T) if this Muckenhoupt condition is also sufficient. We prove a general criterion for compactness of T from X to Y when (X,Y) ∈ M(T) and give an estimate for the distance of T from the finite rank operators. We apply the results to Lorentz spaces and characterize...

This paper is a continuation of [5] and provides necessary and sufficient conditions for double exponential integrability of the Bessel potential of functions from suitable (generalized) Lorentz-Zygmund spaces. These results are used to establish embedding theorems for Bessel potential spaces which extend Trudinger's result.

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: $n^{1/2}cₙ\left(cov\left(K\right)\right)\le c_K(1+{\sum }_{k=1}^{ⁿ}{k}^{-1/2}{e}_k\left(K\right))$, n ∈ ℕ, and $k^{1/2}{c}_{k+n}\left(cov\left(K\right)\right)\le c[lo{g}^{1/2}(n+1)\epsilon ₙ\left(K\right)+{\sum }_{j=n+1}^{\infty }{\epsilon }_j\left(K\right)/\left(jlo{g}^{1/2}(j+1)\right)]$, k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and ${\epsilon }_k\left(K\right)$ and $e_k\left(K\right)$ 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 $p\in (1,\infty )$ and $I=(0,1)$; suppose that $T:L_p\left(I\right)\to L_p\left(I\right)$ is a compact linear map with trivial kernel and range dense in $L_p\left(I\right)$. 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. The special properties of $L_p\left(I\right)$ 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 present new formulae providing equivalent quasi-norms on Lorentz-Karamata spaces. Our results are based on properties of certain averaging operators on the cone of non-negative and non-increasing functions in convenient weighted Lebesgue spaces. We also illustrate connections between our results and mapping properties of such classical operators as the fractional maximal operator and the Riesz potential (and their variants) on the Lorentz-Karamata spaces.

Upper estimates are obtained for approximation and entropy numbers of the embeddings of weighted Sobolev spaces into appropriate weighted Orlicz spaces. Results are given when the underlying space domain is bounded and for certain unbounded domains.

The Hardy inequality ${\int }_{\Omega }{\left|u\left(x\right)\right|}^{p}d{\left(x\right)}^{-p}\ddot{x}\le c{\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^p\ddot{x}$ with $d\left(x\right)=dist(x,\partial \Omega )$ holds for $u\in C_0^{\infty }\left(\Omega \right)$ if $\Omega \subset {ℝ}^n$ is an open set with a sufficiently smooth boundary and if $1<p<\infty$. P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for $p=1$.

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 ${\mathrm{\Delta }}^2$ in a bounded domain $\mathrm{\Omega }\subset {ℝ}^n$. We state some existence results when $n\ge 8$. Moreover, we consider $5\le n\le 7$, expecially when $\mathrm{\Omega }$ is a ball in ${ℝ}^n$.