Uniqueness of complete norms for quotients of Banach function algebras

W. Bade; H. Dales

Displaying similar documents to “Uniqueness of complete norms for quotients of Banach function algebras”

Mixed-norm spaces and interpolation

Joaquín Ortega, Joan Fàbrega (1994)

Studia Mathematica

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Let D be a bounded strictly pseudoconvex domain of $ℂ^{n}$ with smooth boundary. We consider the weighted mixed-norm spaces $A_{δ, k}^{p, q} (D)$ of holomorphic functions with norm $∥ f ∥_{p, q, δ, k} = (\sum_{| α | \leq k} ʃ_{0}^{r_{0}} (ʃ_{\partial D_{r}} | D^{α} {f |}^{p} d σ_{r} {)^{q / p} r^{δ q / p - 1} d r)}^{1 / q}$ . We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces $A_{δ, k}^{p} (D)$ and we give results about real and complex interpolation between them. We apply these results to prove that $A_{δ, k}^{p, q} (D)$ is the intersection of a Besov space $B_{s}^{p, q} (D)$ with the space of holomorphic functions on D. Further, we obtain several properties...

Subanalytic version of Whitney's extension theorem

Krzysztof Kurdyka, Wiesław Pawłucki (1997)

Studia Mathematica

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For any subanalytic $C^{k}$ -Whitney field (k finite), we construct its subanalytic $C^{k}$ -extension to $ℝ^{n}$ . Our method also applies to other o-minimal structures; e.g., to semialgebraic Whitney fields.

$B M O_{ψ}$ -spaces and applications to extrapolation theory

Stefan Geiss (1997)

Studia Mathematica

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We investigate a scale of $B M O_{ψ}$ -spaces defined with the help of certain Lorentz norms. The results are applied to extrapolation techniques concerning operators defined on adapted sequences. Our extrapolation works simultaneously with two operators, starts with $B M O_{ψ}$ - $L_{\infty}$ -estimates, and arrives at $L_{p}$ - $L_{p}$ -estimates, or more generally, at estimates between K-functionals from interpolation theory.

Representations of the spaces $C^{\infty} (ℝ^{N}) \cap H^{k, p} (ℝ^{N})$

A. Albanese, V. Moscatelli (2000)

Studia Mathematica

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We give a representation of the spaces $C^{\infty} (ℝ^{N}) \cap H^{k, p} (ℝ^{N})$ as spaces of vector-valued sequences and use it to investigate their topological properties and isomorphic classification. In particular, it is proved that $C^{\infty} (ℝ^{N}) \cap H^{k, 2} (ℝ^{N})$ is isomorphic to the sequence space $s^{ℕ} \cap l^{2} (l^{2})$ , thereby showing that the isomorphy class does not depend on the dimension N if p=2.

Comparing gaussian and Rademacher cotype for operators on the space of continuous functions

Marius Junge (1996)

Studia Mathematica

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We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if ${(\sum_{k} {((∥ T x_{k} ∥_{F}) / (\sqrt l o g (k + 1)))}^{q})}^{1 / q} \leq c ∥ \sum_{k} ɛ_{k} x_{k} ∥_{L_{2} (C (K))}$ , for all sequences ${(x_{k})}_{k \in ℕ} \subset C (K)$ with ${(∥ T x_{k} ∥)}_{k = 1}^{n}$ decreasing. (2) T is of Rademacher cotype q if and only if $(\sum_{k} {(∥ T x_{k} ∥_{F} \sqrt ({(l o g (k + 1))}^{q}))}^{1 / q} \leq c ∥ \sum_{k} g_{k} x_{k} ∥_{L_{2} (C (K))}$ , for all sequences ${(x_{k})}_{k \in ℕ} \subset C (K)$ with ${(∥ T x_{k} ∥)}_{k = 1}^{n}$ decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of...

High order representation formulas and embedding theorems on stratified groups and generalizations

Guozhen Lu, Richard Wheeden (2000)

Studia Mathematica

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We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable $L^{1}$ to $L^{1}$ Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and $L^{1}$ ...

Integral operators and weighted amalgams

C. Carton-Lebrun, H. Heinig, S. Hofmann (1994)

Studia Mathematica

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For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from $ℓ^{q ̅} (L_{v}^{p ̅})$ into $ℓ^{q} (L_{u}^{p})$ . For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted $L^{p}$ -spaces. Amalgams of the form $ℓ^{q} (L_{w}^{p})$ , 1 < p,q < ∞ , q ≠ p, $w \in A_{p}$ , are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.

Moment inequalities for sums of certain independent symmetric random variables

P. Hitczenko, S. Montgomery-Smith, K. Oleszkiewicz (1997)

Studia Mathematica

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This paper gives upper and lower bounds for moments of sums of independent random variables $(X_{k})$ which satisfy the condition ${P (| X |}_{k} \geq t) = e x p (- N_{k} (t))$ , where $N_{k}$ are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which $N (t) = {| t |}^{r}$ for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.

A characterization of some weighted norm inequalities for the fractional maximal function

Richard Wheeden (1993)

Studia Mathematica

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A new characterization is given for the pairs of weight functions v, w for which the fractional maximal function is a bounded operator from $L_{v}^{p} (X)$ to $L_{w}^{q} (X)$ when 1 < p < q < ∞ and X is a homogeneous space with a group structure. The case when X is n-dimensional Euclidean space is included.

A new Taylor type formula and $C^{\infty}$ extensions for asymptotically developable functions

M. Zurro (1997)

Studia Mathematica

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The paper studies the relation between asymptotically developable functions in several complex variables and their extensions as functions of real variables. A new Taylor type formula with integral remainder in several variables is an essential tool. We prove that strongly asymptotically developable functions defined on polysectors have $C^{\infty}$ extensions from any subpolysector; the Gevrey case is included.

Divergence of the Bochner-Riesz means in the weighted Hardy spaces

Shuichi Sato (1996)

Studia Mathematica

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We costruct functions in $H_{w}^{1}$ ( $w \in A_{1}$ ) whose Fourier integral expansions are almost everywhere non-summable with respect to the Bochner-Riesz means of the critical order.

Uniform convergence of double trigonometric series

Chang-Pao Chen, Gwo-Bin Chen (1996)

Studia Mathematica

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It is shown that under certain conditions on $c_{j k}$ , the rectangular partial sums $s_{m n} (x, y)$ converge uniformly on $T^{2}$ . These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is $\sum_{| k | = n}^{\infty} | Δ c_{k} | = o (1 / n)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition:...

$L^{p}$ -improving properties of measures supported on curves on the Heisenberg group

Silvia Secco (1999)

Studia Mathematica

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$L^{p}$ - $L^{q}$ boundedness properties are obtained for operators defined by convolution with measures supported on certain curves on the Heisenberg group. We find the curvature condition for which the type set of these operators can be the full optimal trapezoid with vertices A=(0,0), B=(1,1), C=(2/3,1/2), D=(1/2,1/3). We also give notions of right curvature and left curvature which are not mutually equivalent.

A condition implying boundedness and VMO for a function f

Michelangelo Franciosi (1997)

Studia Mathematica

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Some boundedness and VMO results are proved for a function f integrable on a cube $Q_{0}$ , starting from an integral bound.

Discrete Hardy spaces

Santiago Boza, María Carro (1998)

Studia Mathematica

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We study various characterizations of the Hardy spaces $H^{p} (ℤ)$ via the discrete Hilbert transform and via maximal and square functions. Finally, we present the equivalence with the classical atomic characterization of $H^{p} (ℤ)$ given by Coifman and Weiss in [CW]. Our proofs are based on some results concerning functions of exponential type.

On the size of approximately convex sets in normed spaces

S. Dilworth, Ralph Howard, James Roberts (2000)

Studia Mathematica

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Let X be a normed space. A set A ⊆ X is approximately convexif d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with $ℋ (A, C o (A)) \geq l o g_{2} n - 1$ and $d i a m (A) \leq C \sqrt n {(l n n)}^{2}$ , where ℋ denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0, we construct worst possible approximately convex sets in C[0,1] such that ℋ(A,Co(A))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.

Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces

J. García-Cuerva, K. Kazarian (1994)

Studia Mathematica

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We study sufficient conditions on the weight w, in terms of membership in the $A_{p}$ classes, for the spline wavelet systems to be unconditional bases of the weighted space $H^{p} (w)$ . The main tool to obtain these results is a very simple theory of regular Calderón-Zygmund operators.