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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.

Necessary conditions for the L p -convergence ( 0 < p < 1 ) of single and double trigonometric series

Xhevat Z. Krasniqi, Péter Kórus, Ferenc Móricz (2014)

Mathematica Bohemica

We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the L p -metric, where 0 < p < 1 . The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in H p and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the L p -metric, where 0 < p < 1 .

New Calderón-Zygmund decomposition for Sobolev functions

N. Badr, F. Bernicot (2010)

Colloquium Mathematicae

We give a new Calderón-Zygmund decomposition for Sobolev spaces on a doubling Riemannian manifold. Our hypotheses are weaker than those of the already known decomposition which used classical Poincaré inequalities.

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.

Nonconvolution transforms with oscillating kernels that map 1 0 , 1 into itself

G. Sampson (1993)

Studia Mathematica

We consider operators of the form ( Ω f ) ( y ) = ʃ - Ω ( y , u ) f ( u ) d u with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and h L (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space 1 0 , 1 (= B) into itself. In particular, all operators with h ( y ) = e i | y | a , a > 0, a ≠ 1, map B into itself.

Non-homogeneous strongly singular integrals

Bassam Shayya (2008)

Studia Mathematica

We study the L p mapping properties of a family of strongly singular oscillatory integral operators on ℝⁿ which are non-homogeneous in the sense that their kernels have isotropic oscillations but non-isotropic singularities.

Non-isotropic distance measures for lattice-generated sets.

Alexander Iosevich, Misha Rudnev (2005)

Publicacions Matemàtiques

We study distance measures for lattice-generated sets in Rd, d&gt;=3, with respect to non-isotropic distances l-l.K, induced by smooth symmetric convex bodies K. An effective Fourier-analytic approach is developed to get sharp upper bounds for the second moment of the weighted distance measure.

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