Weak type inequalities of maximal Hankel convolution operators

In this paper we characterize weak type (1,1) inequalities for Hankel convolution operators by means of discrete methods. Partially supported by DGICYT Grant PB 94-0591 (Spain).     

Norm inequalities for convolution operators

We study norm convolution inequalities in Lebesgue and Lorentz spaces. First, we improve the well-known O'Neil's inequality for the convolution operators and prove corresponding estimate from below. Second, we obtain Young–O'Neil-type estimate in the Lorentz spaces for the limit value parameters, i.e., $6K1f{6}_{L(p,h_1)\to L(p,h_2)}$. Finally, similar estimates in the weighted Lorentz spaces are presented. To cite this article: E. Nursultanov et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Weak type estimates for maximal operators over convex hypersurfaces

We prove sharp weak type (p,p) estimates on H ^p spaces for the maximal operators with a rough distance function over convex hypersurfaces.     

Weak type (1,1) estimates for some integral operators related to rough maximal functions

We study the problem of determining for which integrable functionsG:R → (0, ∞) the operatorf → 1/yG(y.) *f(x), which maps functions on the real line into functions defined on the upper half-planeR ₊ ² , is of weak type (1,1). Here,R ₊ ² is endowed with the measurey dx dy. The conditions we will impose are related to the distribution of the mass ofG. One of the motivations for this study comes from the problem of deciding whether there is a weak type (1,1) inequality for the “rough” modification of the standard maximal function, obtained by inserting in the mean values a factor Ω which depends only on the angle. Here, Ω≥0 is any integrable function on the sphere. Our estimates for the first-mentioned problem allow us to answer in the affirmative, the second one in dimension two, when we restrict the operator to radial functions. Some extensions to higher dimensions in the context of both problems are also discussed. Both authors were partially supported by DGICYT PB90/187.     

Hardy type inequalities with exponential weights for a class of convolution operators

In this paper we consider operators of the form H=λ(-i∇), with λ analytic in a strip and with some specific growth conditions at infinity, and prove Hardy type estimates in L ²(ℝ ⁿ ) with exponential weights. In fact we extend our previous results [19] from bounded analytic functions on a strip to analytic functions with polynomial growth in that strip.     

Weak type estimates for some maximal operators on Hardy spaces

We show that the lacunary maximal operator associated to a compact smooth hypersurface on which the Gaussian curvature nowhere vanishes to infinite order maps the standard Hardy space H ¹ to L ^1,∞ . (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak type inequalities for fractional maximal operator on weighted orlicz spaces

The fractional maximal operator on homogeneous space (X, d, μ) is de fined as . In this paper, the sufficient and necessary conditions for the to be of weak type and extra weak type will be given. In Memory of Professor M. T. Cheng     

Weak type inequalities for fractional maximal operator on weighted orlicz spaces

The fractional maximal operator on homogeneous space (X, d, μ) is de fined as      

Weak type estimates for the maximal operators associated to plane curves

For the plane curves Γ,the maximal operator associated to it is defined byMf(x)=sup|∫f(x-Γ(t))(r~(-1)t)r~(-1)dt|where is a Schwartz function.For a certain class of curves in R~2,M is shown to boundedon (H(R~2),Weak L~1(R~2).This extends the theorem of Stein & Wainger and the theo-rem of Weinberg.     

Weak type estimates for maximal operators with a cylindric distance function

We establish sharp weak-type estimates for the maximal operators T^λ_* associated with cylindric Riesz means for functions on H^p(ℝ³) when 4/5 <p<1 and λ=3/p−5/2, and when p=4/5 and λ>3/p−5/2. The first author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) No. R04-2002-000-20028-0. The third author was supported by a Korea University Grant.     

Weighted norm inequalities for operators of potential type and fractional maximal functions

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.Research of the first author was supported in part by NSERC grant A5149.Research of the second author was supported in part by NSF grant DMS93-02991.     

Two-weight inequalities for convolution operators in Lebesgue space

In this paper, we prove a theorem on the boundedness of a convolution operator in a weighted Lebesgue space with kernel satisfying a certain version of Hörmander’s condition.     

Weak type estimates for some maximal operators on the weighted Hardy spaces

Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spacesH _ω ^p (0 <p < 1, ω ∈A ₁) (0<p<1, ω∞A₁); in particular, weighted weak type endpoint estimates are obtained for the maximal operators arising from the Bochner-Riesz means and the spherical means onH _ω ^p .     

Two-weight norm inequalities for fractional maximal operators on spaces of generalized homogeneous type

In this paper a characterization is given for a pairs of weights (w,v) for which the fractional maximal operator is bounded from when is a space of generalized homogeneous type introduced by A. Carbery et al. [4].     

Weighted norm inequalities for the maximal singular integral operators on spaces of homogeneous type

Weighted norm inequalities with general weights are established for the maximal singular integral operators on spaces of homogeneous type, when the kernel satisfies a Hörmander regularity condition on one variable and a Hölder regularity condition on the other variable.     

Weighted norm inequalities for geometric fractional maximal operators

For 0 < let Tf denote one of the operators

粗糙平方函数及极大算子的弱型估计

§ 1　 PreliminariesWe considerψ( x)∈ L1 ( Rn) satisfying the mean valuezero,i.e.∫Rnψdx=0 ,and definethe square function g( f) on Rnbyg( f) ( x) =( k|ψk* f|2 ) 1 2 ( x)for f∈ S( Rn) ,the Schwartz space,whereψk( x) =ψ2 k( x) .　　 Whenψ has some smooth property,one can obtain the weak type estimate by viewingthe square function g( f) as the vector-valued singularintegrals,which the readercan referto [1 ,2 ] .As for the results aboutthe Lp-estimates,see [3,4 ] .In this paper,we sha…     

The Bellman functions of dyadic-like maximal operators and related inequalities

For each p>1 we precisely evaluate the main Bellman functions associated with the dyadic maximal operator on and the dyadic Carleson imbedding theorem. Actually, we do that in the more general setting of tree-like maximal operators. These provide refinements of the sharp L^p inequalities for those operators. For this we introduce an effective linearization for such maximal operators on an adequate set of functions.