On the weak (1,1) boundedness of a class of oscillatory singular integrals

Yibiao Pan

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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.

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.

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...

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.

$B M O_{ψ}$ -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.

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.

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.

On weighted Bergman kernels of bounded domains

Sorin Dragomir (1994)

Studia Mathematica

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We build on work by Z. Pasternak-Winiarski [PW2], and study a-Bergman kernels of bounded domains $Ω \subset ℂ^{N}$ for admissible weights $a \in L ¹ (Ω)$ .

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.

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).

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.

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.

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

A. Pełczyński, M. Wojciechowski (1993)

Studia Mathematica

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Let E be a Banach space. Let $L ¹_{(1)} (ℝ^{d}, E)$ be the Sobolev space of E-valued functions on $ℝ^{d}$ with the norm $ʃ_{ℝ^{d}} ∥ f ∥_{E} d x + ʃ_{ℝ^{d}} ∥ \nabla f ∥_{E} d x = ∥ f ∥ ₁ + ∥ \nabla f ∥ ₁$ . It is proved that if $f \in L ¹_{(1)} (ℝ^{d}, E)$ then there exists a sequence $(g_{m}) \subset L_{(1)} ¹ (ℝ^{d}, E)$ such that $f = \sum_{m} g_{m}$ ; $\sum_{m} (∥ g_{m} ∥ ₁ + ∥ \nabla g_{m} ∥ ₁) < \infty$ ; and $∥ g_{m} ∥_{\infty}^{1 / d} ∥ g_{m} ∥ ₁^{(d - 1) / d} \leq b ∥ \nabla g_{m} ∥ ₁$ for m = 1, 2,..., where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding $L_{(1)} ¹ (ℝ^{d}, E) ↪ L ² (ℝ^{d}, E)$ . In particular, the embedding into Besov spaces $L ¹_{(1)} (ℝ^{d}, E) ↪ B_{p, 1}^{θ (p, d)} (ℝ^{d}, E)$ is proved, where $θ (p, d) = d (p^{- 1} + d^{- 1} - 1)$ for 1 < p ≤ d/(d-1), d=1,2,... The latter embedding in the scalar case is due to Bourgain...