We investigate the boundedness, compactness, invertibility and Fredholmness of weighted composition operators between Lorentz spaces. It is also shown that the classes of Fredholm and invertible weighted composition maps between Lorentz spaces coincide when the underlying measure space is nonatomic.

In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality:

$$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$

${\cal D}_0^{2,2} $

We study several questions about the weak-type boundedness of the Fourier transform ℱ on rearrangement invariant spaces. In particular, we characterize the action of ℱ as a bounded operator from the minimal Lorentz space Λ(X) into the Marcinkiewicz maximal space M(X), both associated with a rearrangement invariant space X. Finally, we also prove some results establishing that the weak-type boundedness of ℱ, in certain weighted Lorentz spaces, is equivalent to the corresponding strong-type estimates.

We find new properties for the space R(X), introduced by Soria in the study of the best constant for the Hardy operator minus the identity. In particular, we characterize when R(X) coincides with the minimal Lorentz space Λ(X). The condition that R(X) ≠ {0} is also described in terms of the embedding (L1, ∞ ∩ L∞) ⊂ X. Finally, we also show the existence of a minimal rearrangement-invariant Banach function space (RIBFS) X among those for which R(X) ≠ {0} (which is the RIBFS envelope of the quasi-Banach space L1, ∞ ∩ L∞).

Let ℳ be a semi-finite von Neumann algebra equipped with a faithful normal trace τ. We prove a Kadec-Pelczyński type dichotomy principle for subspaces of symmetric space of measurable operators of Rademacher type 2. We study subspace structures of non-commutative Lorentz spaces Lp, q, (ℳ, τ), extending some results of Carothers and Dilworth to the non-commutative settings. In particular, we show that, under natural conditions on indices, ℓp cannot be embedded into Lp, q (ℳ, τ). As applications, we prove that for 0 < p < ∞ with p ≠ 2, ℓp cannot be strongly embedded into Lp(ℳ, τ). This provides a non-commutative extension of a result of Kalton for 0 < p < 1 and a result of Rosenthal for 1 ≦ p < 2 on Lp [0, 1].

We study when the spaces of Lorentz multipliers from ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ are distinct. Our main interest is the case when $s\,<\,t$, the Lorentz-improving multipliers. We prove, for example, that the space of multipliers which map ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ is different from those mapping ${{L}^{r,v}}\,\to \,{{L}^{r,u}}$ if either $r\,=\,p$ or ${p}'$ and $1/s\,-\,1/t\,\ne \,1/u\,-\,1/v$, or $r\,\ne \,p$ or ${p}'$. These results are obtained by making careful estimates of the Lorentz multiplier norms of certain linear combinations of Fejer or Dirichlet kernels. For the case when the first indices are different the linear combination we analyze is in the spirit of a Rudin-Shapiro polynomial.

We consider the notion of a uniformly concave function, using it to characterize those Lorentz spaces Lw,1 that have the weak-star uniform Kadec-Klee property as precisely those for which the antiderivative ϕ of w is uniformly concave; building on recent work of Dilworth and Hsu. We also derive a quite general sufficient condition for a twice-differentiable ϕ to be uniformly concave; and explore the extent to which this condition is necessary.

In this paper we characterize weighted Lorentz norm inequalities for the one sided Hardy-Littlewood maximal function

Similar questions are discussed for the maximal operator associated to an invertible measure preserving transformation of a measure space.