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Factorization and extension of positive homogeneous polynomials

100%

Defant A. , Mastyło M.

nr 1

87-99

We study the following problem: Given a homogeneous polynomial from a sublattice of a Banach lattice to a Banach lattice, under which additional hypotheses does this polynomial factorize through $L_{p}$-spaces involving multiplication operators? We prove that under some lattice convexity and concavity hypotheses, for polynomials certain vector-valued norm inequalities and weighted norm inequalities are equivalent. We combine these results and prove a factorization theorem for positive homogeneous polynomials which is a variant of a celebrated factorization theorem for linear operators due to Maurey and Rosenthal. Our main application is a Hahn-Banach extension theorem for positive homogeneous polynomials between Banach lattices.

Eigenvalues of Hille-Tamarkin operators and geometry of Banach function spaces

100%

Kühn T. , Mastyło M.

nr 3

275-296

We investigate how the asymptotic eigenvalue behaviour of Hille-Tamarkin operators in Banach function spaces depends on the geometry of the spaces involved. It turns out that the relevant properties are cotype p and p-concavity. We prove some eigenvalue estimates for Hille-Tamarkin operators in general Banach function spaces which extend the classical results in Lebesgue spaces. We specialize our results to Lorentz, Orlicz and Zygmund spaces and give applications to Fourier analysis. We are also able to show the optimality of our eigenvalue estimates in the Lorentz spaces $L_{2,q}$ with 1 ≤ q < 2 and in Zygmund spaces $L_{p}(log L)_a$ with 2 ≤ p < ∞ and a > 0.

Asymptotically isometric and isometric copies of $\ell_1$ in some Banach function lattices

100%

Kamińska A. , ~Mastyło M.

nr 2

We identify the class of Caldern-Lozanovskii spaces that do not contain an asymptotically isometric copy of $\ell_1$, and consequently we obtain the corresponding characterizations in the classes of Orlicz-Lorentz and Orlicz spaces equipped with the Luxemburg norm. We also give a~complete description of order continuous Orlicz-Lorentz spaces which contain (order) isometric copies of $\ell_1^{(n)}$ for each integer $n \geq 2$.~As an application we provide necessary and sufficient conditions for order continuous Orlicz-Lorentz spaces to contain an (order) isometric copy of $\ell_1$.~In particular we give criteria in Orlicz and Lorentz spaces for (order) isometric containment of $\ell_1^{(n)}$ and $\ell_1$.~The results are applied to obtain the~description of universal Orlicz-Lorentz spaces for all two-dimensional normed spaces.