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Multiple singular integrals and maximal functions along hypersurfaces

Javier Duoandikoetxea (1986)

Annales de l'institut Fourier

Maximal functions written as convolution with a multiparametric family of positive measures, and singular integrals whose kernel is decomposed as a multiple series of measures, are shown to be bounded in L p , 1 < p < . The proofs are based on the decomposition of the operators according to the size of the Fourier transform of the measures, assuming some regularity at zero and decay at infinity of these Fourier transforms. Applications are given to homogeneous singular integrals in product spaces with size...

Multiplicative square functions.

María José González, Artur Nicolau (2004)

Revista Matemática Iberoamericana

We study regularity properties of a positive measure in the euclidean space in terms of two square functions which are the multiplicative analogues of the usual martingale square function and of the Lusin area function of a harmonic function. The size of ...

Multipliers for Hermite expansions.

Sundaram Thangavelu (1987)

Revista Matemática Iberoamericana

The aim of this paper is to prove certain multiplier theorems for the Hermite series.

Multipliers of Hardy spaces, quadratic integrals and Foiaş-Williams-Peller operators

G. Blower (1998)

Studia Mathematica

We obtain a sufficient condition on a B(H)-valued function φ for the operator Γ φ ' ( S ) to be completely bounded on H B ( H ) ; the Foiaş-Williams-Peller operator | St Γφ | Rφ = | | | 0 S | is then similar to a contraction. We show that if ⨍ : D → B(H) is a bounded analytic function for which ( 1 - r ) | | ' ( r e i θ ) | | B ( H ) 2 r d r d θ and ( 1 - r ) | | " ( r e i θ ) | | B ( H ) r d r d θ are Carleson measures, then ⨍ multiplies ( H 1 c 1 ) ' to itself. Such ⨍ form an algebra A, and when φ’∈ BMO(B(H)), the map Γ φ ' ( S ) is bounded A B ( H 2 ( H ) , L 2 ( H ) H 2 ( H ) ) . Thus we construct a functional calculus for operators of Foiaş-Williams-Peller type.

Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

The Anh Bui, Jun Cao, Luong Dang Ky, Dachun Yang, Sibei Yang (2013)

Analysis and Geometry in Metric Spaces

Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∊ [1; 2) and qL ∊ (2;∞]. Let φ : X × [0;∞) → [0;∞) be a function such that φ (x;·) is an Orlicz function, φ(·;t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0;1] and φ(·; t) satisfies the uniformly reverse Hölder inequality of order...

Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

Viktor I. Burenkov, Huseyn V. Guliyev (2004)

Studia Mathematica

The problem of boundedness of the Hardy-Littewood maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions are also necessary.

New estimates for elliptic equations and Hodge type systems

Jean Bourgain, Haïm Brezis (2007)

Journal of the European Mathematical Society

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension n , with data in L 1 . We also present related results concerning differential forms with coefficients in the limiting Sobolev space W 1 , n .

Non-compact Littlewood-Paley theory for non-doubling measures

Michael Wilson (2007)

Studia Mathematica

We prove weighted Littlewood-Paley inequalities for linear sums of functions satisfying mild decay, smoothness, and cancelation conditions. We prove these for general “regular” measure spaces, in which the underlying measure is not assumed to satisfy any doubling condition. Our result generalizes an earlier result of the author, proved on d with Lebesgue measure. Our proof makes essential use of the technique of random dyadic grids, due to Nazarov, Treil, and Volberg.

Norm convergence of some power series of operators in L p with applications in ergodic theory

Christophe Cuny (2010)

Studia Mathematica

Let X be a closed subspace of L p ( μ ) , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that s u p n | | U | | < . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like n 1 ( U f ) / n 1 - α , 0 ≤ α < 1, in terms of | | f + + U n - 1 f | | p , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie....

Norm estimates for Bessel-Riesz operators on generalized Morrey spaces

Mochammad Idris, Hendra Gunawan, A. Eridani (2018)

Mathematica Bohemica

We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent 1 , and obtain a new estimate for their norm.

Norm inequalities for off-centered maximal operators.

Richard L. Wheeden (1993)

Publicacions Matemàtiques

Sufficient conditions are derived in order that there exist strong-type weighted norm inequalities for some off-centered maximal functions. The maximal functions are of Hardy-Littlewood and fractional types taken over starlike sets in Rn. The sufficient conditions are close to necessary and extend some previously known weak-type results.

Norm inequalities for the minimal and maximal operator, and differentiation of the integral.

David Cruz-Uribe, Christoph J. Neugebauer, Victor Olesen (1997)

Publicacions Matemàtiques

We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Hölder inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability of the...

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