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

Richard Wheeden

Displaying similar documents to “A characterization of some weighted norm inequalities for the fractional maximal function”

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

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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.

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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.

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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.

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

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Studia Mathematica

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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.

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Studia Mathematica

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We prove two-weight norm estimates for fractional integrals and fractional maximal functions associated with starlike sets in Euclidean space. This is seen to include general positive homogeneous fractional integrals and fractional integrals on product spaces. We consider both weak type and strong type results, and we show that the conditions imposed on the weight functions are fairly sharp.

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

M. Zurro (1997)

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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.

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Subanalytic version of Whitney's extension theorem

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Andrew Kepert (1994)

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The existence of a projection onto an ideal I of a commutative group algebra $L^{1} (G)$ depends on its hull Z(I) ⊆ Ĝ. Existing methods for constructing a projection onto I rely on a decomposition of Z(I) into simpler hulls, which are then reassembled one at a time, resulting in a chain of projections which can be composed to give a projection onto I. These methods are refined and examples are constructed to show that this approach does not work in general. Some answers are also given to previously...

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.

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.