Search results for "Lp space"
showing 10 items of 58 documents
Two-dimensional Banach spaces with polynomial numerical index zero
2009
We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.
Beyond frames: Semi-frames and reproducing pairs
2017
Frames are nowadays a standard tool in many areas of mathematics, physics, and engineering. However, there are situations where it is difficult, even impossible, to design an appropriate frame. Thus there is room for generalizations, obtained by relaxing the constraints. A first case is that of semi-frames, in which one frame bound only is satisfied. Accordingly, one has to distinguish between upper and lower semi-frames. We will summarize this construction. Even more, one may get rid of both bounds, but then one needs two basic functions and one is led to the notion of reproducing pair. It turns out that every reproducing pair generates two Hilbert spaces, conjugate dual of each other. We …
Topological Dual Systems for Spaces of Vector Measure p-Integrable Functions
2016
[EN] We show a Dvoretzky-Rogers type theorem for the adapted version of the q-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector value…
A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions
2010
Abstract In this paper, we establish the equivalence between the Hajlasz–Sobolev spaces or classical Triebel–Lizorkin spaces and a class of grand Triebel–Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when p ∈ ( n / ( n + 1 ) , ∞ ) , we give a new characterization of the Hajlasz–Sobolev spaces M ˙ 1 , p ( R n ) via a grand Littlewood–Paley function.
The Bohr Radius of a Banach Space
2009
Following the scalar-valued case considered by Djakow and Ramanujan (A remark on Bohr’s theorem and its generalizations 14:175–178, 2000) we introduce, for each complex Banach space X and each \(1\le p0\). We study the p-Bohr radius of the Lebesgue spaces \(L^q(\mu )\) for different values of p and q. In particular we show that \(r_p(L^q(\mu ))=0\) whenever \(p<2\) and \(dim(L^q(\mu ))\ge 2\) and \(r_p(L^q(\mu ))=1\) whenever \(p\ge 2\) and \(p'\le q\le p\). We also provide some lower estimates for \(r_2(L^q(\mu ))\) for the values \(1\le q<2\).
Stochastic Processes on Ends of Tree and Dirichlet Forms
2016
We present main ideas and compare two constructions of stochastic processes on the ends (leaves) of the trees with varying numbers of edges at the nods. In one of them the trees are represented by spaces of numerical sequences and the processes are obtained by solving a class of Chapman-Kolmogorov Equations. In the other the trees are described by the set of nodes and edges. To each node there is naturally associated a finite dimensional function space and the Dirichlet form on it. Having a class of Dirichlet forms at the nodes one can under certain conditions build a Dirichlet form on L2 space of funcions on the ends of the trees. We show that the state spaces of two approaches are homeomo…
Remarks on the semivariation of vector measures with respect to Banach spaces.
2007
Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.
Dissipative operators and differential equations on Banach spaces
1991
If we consider the initial value problem Inline Equation $$x'(t) = f(t,x(t)),{\text{ }}x(0) = {x_0}$$ on the real line, it is well known that one—sided bounds like Inline Equation $$\left[ {f(t,x) - f\left( {t,y} \right)} \right]\left( {x - {\text{y}}} \right) \leqslant \omega {\left( {x - y} \right)^2}$$ give much better information about the behaviour of solutions than the Lipschitz- type estimates Inline Equation $$ \left| {f\left( {t,x} \right) - f\left( {t,y} \right)} \right| \leqslant L\left| {x - y} \right|,$$ because ω, unlike L, may be negative.
On the Euler-Lagrange inequality of a convex variational integral in Orlicz spaces
1987
A note on the Banach space of preregular maps
2011
The aim of this paper is to give simple proofs for Jeurnink's characterizations of preregular maps in terms of Θ-maps acting between Banach lattices. For Banach lattices E and F, we achieve our goal by considering the space Lβ(E, F) of all those linear maps T: E → F for which there exists a constant K such that {double pipe}Vn i=1 {pipe}Txi{pipe} ≤ K {double pipe}Vn i=1{pipe}xi for all finite sequences x1, ..., xn e{open}E. We show that, if Lβ(E; F), and the spaces L Θ (E; F) of Θ -map and Lpr(E; F) of preregular maps are respectively endowed with their canonical norms, then they are identical Banach spaces