Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm

A. Pełczyński; M. Wojciechowski

Displaying similar documents to “Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm”

Oscillatory kernels in certain Hardy-type spaces

Lung-Kee Chen, Dashan Fan (1994)

Studia Mathematica

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We consider a convolution operator Tf = p.v. Ω ⁎ f with $Ω (x) = K (x) e^{i h (x)}$ , where K(x) is an (n,β) kernel near the origin and an (α,β), α ≥ n, kernel away from the origin; h(x) is a real-valued $C^{\infty}$ function on $ℝ^{n} ∖ 0$ . We give a criterion for such an operator to be bounded from the space $H_{0}^{p} (ℝ^{n})$ into itself.

Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis

Paolo Terenzi (1994)

Studia Mathematica

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Every separable, infinite-dimensional Banach space X has a biorthogonal sequence $z_{n}, z *_{n}$ , with $s p a n z *_{n}$ norming on X and $∥ z_{n} ∥ + ∥ z *_{n} ∥$ bounded, so that, for every x in X and x* in X*, there exists a permutation π(n) of n so that $x \in \bar{c o n v} f i n i t e s u b s e r i e s o f \sum_{n = 1}^{\infty} z *_{n} (x) z_{n} a n d x *_{n} (x) = \sum_{n = 1}^{\infty} z *_{π (n)} (x) x * (z_{π (n)})$ .

Montel and reflexive preduals of spaces of holomorphic functions on Fréchet spaces

Christopher Boyd (1993)

Studia Mathematica

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For U open in a locally convex space E it is shown in [31] that there is a complete locally convex space G(U) such that $G {(U)}_{i}^{'} = (ℋ (U), τ_{δ})$ . Here, we assume U is balanced open in a Fréchet space and give necessary and sufficient conditions for G(U) to be Montel and reflexive. These results give an insight into the relationship between the $τ_{0}$ and $τ_{ω}$ topologies on ℋ (U).

Hull-minimal ideals in the Schwartz algebra of the Heisenberg group

J. Ludwig (1998)

Studia Mathematica

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For every closed subset C in the dual space $Ĥ_{n}$ of the Heisenberg group $H_{n}$ we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra $S (H_{n})$ and we show that in general for two closed subsets $C_{1}, C_{2}$ of $Ĥ_{n}$ the product of $j (C_{1})$ and $j (C_{2})$ is different from $j (C_{1} \cap C_{2})$ .

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.

Holomorphic motions commuting with semigroups

Zbigniew Słodkowski (1996)

Studia Mathematica

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A holomorphic family $f_{z}$ , |z|<1, of injections of a compact set E into the Riemann sphere can be extended to a holomorphic family of homeomorphisms $F_{z}$ , |z|<1, of the Riemann sphere. (An earlier result of the author.) It is shown below that there exist extensions $F_{z}$ which, in addition, commute with some holomorphic families of holomorphic endomorphisms of $ℂ ̅ ̅ ̅ ̅ ̅ ̅ f_{z} (E)$ , |z|<1 (under suitable assumptions). The classes of covering maps and maps with the path lifting property are discussed. ...

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.

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.

A compact set without Markov’s property but with an extension operator for $C^{\infty}$ -functions

Alexander Goncharov (1996)

Studia Mathematica

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We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator $L : ℇ (K) \to C^{\infty} [0, 1]$ . At the same time, Markov’s inequality is not satisfied for certain polynomials on K.

Uniqueness of complete norms for quotients of Banach function algebras

W. Bade, H. Dales (1993)

Studia Mathematica

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We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra $L^{1} (G)$ of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients $A (Γ) / \bar{J (E)}$ arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed....

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...

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:...

On the type constants with respect to systems of characters of a compact abelian group

Aicke Hinrichs (1996)

Studia Mathematica

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We prove that there exists an absolute constant c such that for any positive integer n and any system Φ of $2^{n}$ characters of a compact abelian group, $2^{- n / 2} t_{Φ} (T) \leq c n^{- 1 / 2} t_{n} (T)$ , where T is an arbitrary operator between Banach spaces, $t_{Φ} (T)$ is the type norm of T with respect to Φ and $t_{n} (T)$ is the usual Rademacher type-2 norm computed with n vectors. For the system of the first $2^{n}$ Walsh functions this is even true with c=1. This result combined with known properties of such type norms provides easy access to quantitative...

Commutants of certain multiplication operators on Hilbert spaces of analytic functions

K. Seddighi, S. Vaezpour (1999)

Studia Mathematica

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This paper characterizes the commutant of certain multiplication operators on Hilbert spaces of analytic functions. Let $A = M_{z}$ be the operator of multiplication by z on the underlying Hilbert space. We give sufficient conditions for an operator essentially commuting with A and commuting with $A^{n}$ for some n>1 to be the operator of multiplication by an analytic symbol. This extends a result of Shields and Wallen.

Structure of maximal sum-free sets in $C_{p}$

H. Yap (1970)

Acta Arithmetica

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$L^{p}$ -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.

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...