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Regularity of the Hardy-Littlewood maximal operator on block decreasing functions

J. M. Aldaz, F. J. Pérez Lázaro (2009)

Studia Mathematica

We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the -norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily...

Relations between weighted Orlicz and B M O φ spaces through fractional integrals

Eleonor Ofelia Harboure, Oscar Salinas, Beatriz E. Viviani (1999)

Commentationes Mathematicae Universitatis Carolinae

We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator I α maps weak weighted Orlicz - φ spaces into appropriate weighted versions of the spaces B M O ψ , where ψ ( t ) = t α / n φ - 1 ( 1 / t ) . This generalizes known results about boundedness of I α from weak L p into Lipschitz spaces for p > n / α and from weak L n / α into B M O . It turns out that the class of weights corresponding to I α acting on weak - L φ for φ of lower type equal or greater than n / α , is the same as the one solving the problem for weak...

Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions

Fabio Berra (2022)

Czechoslovak Mathematical Journal

We give a quantitative characterization of the pairs of weights ( w , v ) for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak ( p , p ) type inequality for 1 p < . More precisely, given any measurable set E 0 , the estimate w ( { x n : M + , d ( 𝒳 E 0 ) ( x ) > t } ) C [ ( w , v ) ] A p + , d ( ) p t p v ( E 0 ) holds if and only if the pair ( w , v ) belongs to A p + , d ( ) , that is, | E | | Q | [ ( w , v ) ] A p + , d ( ) v ( E ) w ( Q ) 1 / p for every dyadic cube Q and every measurable set E Q + . The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the non-dyadic...

Reverse-Holder classes in the Orlicz spaces setting

E. Harboure, O. Salinas, B. Viviani (1998)

Studia Mathematica

In connection with the A ϕ classes of weights (see [K-T] and [B-K]), we study, in the context of Orlicz spaces, the corresponding reverse-Hölder classes R H ϕ . We prove that when ϕ is Δ 2 and has lower index greater than one, the class R H ϕ coincides with some reverse-Hölder class R H q , q > 1 . For more general ϕ we still get R H ϕ A = q > 1 R H q although the intersection of all these R H ϕ gives a proper subset of q > 1 R H q .

Rough Marcinkiewicz integral operators on product spaces.

Hussein M. Al-Qassem (2005)

Collectanea Mathematica

In this paper, we study the Marcinkiewicz integral operators MΩ,h on the product space Rn x Rm. We prove that MΩ,h is bounded on Lp(Rn x Rm) (1&lt; p &lt; ∞) provided that h is a bounded radial function and Ω is a function in certain block space Bq(0,0) (Sn−1 x Sm−1) for some q &gt; 1. We also establish the optimality of our condition in the sense that the space Bq(0,0) (Sn−1 x Sm−1) cannot be replaced by Bq(0,r) (Sn−1 x Sm−1) for any −1 &lt; r &lt; 0. Our results improve some...

Rough maximal functions and rough singular integral operators applied to integrable radial functions.

Peter Sjögren, Fernando Soria (1997)

Revista Matemática Iberoamericana

Let Ω be homogeneous of degree 0 in Rn and integrable on the unit sphere. A rough maximal operator is obtained by inserting a factor Ω in the definition of the ordinary maximal function. Rough singular integral operators are given by principal value kernels Ω(y) / |y|n, provided that the mean value of Ω vanishes. In an earlier paper, the authors showed that a two-dimensional rough maximal operator is of weak type (1,1) when restricted to radial functions. This result is now extended to arbitrary...

Rough Maximal Oscillatory Singular Integral Operators

Al-Salman, Ahmad (2005)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: Primary 42B20; Secondary 42B15, 42B25In this paper, we establish the L^p boundedness of certain maximal oscillatory singular integral operators with rough kernels belonging to certain block spaces. Our L^p boundedness result improves previously known results.

Second order elliptic operators with complex bounded measurable coefficients in  L p , Sobolev and Hardy spaces

Steve Hofmann, Svitlana Mayboroda, Alan McIntosh (2011)

Annales scientifiques de l'École Normale Supérieure

Let  L be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with L , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in  L p , Sobolev, and some new Hardy spaces naturally associated to  L . First, we show that the...

Sharp and weighted inequalities for multilinear integral operators.

Liu Lanzhe (2007)

RACSAM

In this paper, we prove some weighted inequalities for the multilinear operators related to certain integral operators on the generalized Morrey spaces by using the sharp estimates of the multilinear operators. The operators include Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Sharp L p -weighted Sobolev inequalities

Carlos Pérez (1995)

Annales de l'institut Fourier

We prove sharp weighted inequalities of the form R n | f ( x ) | p v ( x ) d x C R n | q ( D ) ( f ) ( x ) | p N ( v ) ( x ) d x where q ( D ) is a differential operator and N is a combination of maximal type operator related to q ( D ) and to p .

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