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 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 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.     

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 .     

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.     

Sharp estimates for maximal operators associated to the wave equation

The wave equation, ∂ _tt u=Δu, in ℝ ⁿ⁺¹, considered with initial data u(x,0)=f∈H ^s (ℝ ⁿ ) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H ^s (ℝ ⁿ ) to L ^q (ℝ ⁿ ).     

Endpoint estimates for some maximal operators associated to the circular conical surface

For and m>0 we consider the maximal operator given by

     

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).     

Two estimates for curves in the plane

We obtain a Fourier transform estimate and an convolution estimate for certain measures on a class of convex curves in the plane.

     

Pointwise estimates for the maximal multilinear singular integral operators

Two pointwise estimates relating the maximal multilinear singular integral operators and some classical maximal operators are established. These pointwise estimates imply the rearrangement estimate and the BLO(Rn) estimate for the maximal multilinear singular integral 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…     

Weak estimates for commutators of fractional integral operators

By introducing a kind of maximal operator of the fractional order associated with the mean Luxemburg norm and using the technique of the sharp function, the weak type LlogL estimates for the commutators of the fractional integral operator and the related maximal operator are established.     

Weak type (1, 1) inequalities of maximal convolution operators

In this paper we study the behavior of the constants which appear in the weak type (1, 1) inequalities for maximal convolution operators by means of discrete methods. One of the first applications of these techniques will give us a very simple proof of the ergodic theorem. We also present partial results in order to investigate the best constant in the weak type (1, 1) inequality for the Hardy-Littlewood centered maximal operator in dimension one. In dimension bigger than one we also obtain some lower bounds for that constant.     

Weak boundedness properties for maximal operators defined on weighted spaces

The purpose of this paper is to obtain characterizations of weak type (1,q) inequalities,q ≥ 1, for maximal operators defined on weighted spaces by means of the corresponding operator acting over Dirac deltas. We present a technical theorem which allows us to obtain characterizations for a pair of weights belonging to the classA ₁ of weights by means of the fractional maximal operator. Analogous results are obtained for the one-sided fractional maximal operator.     

Two-weighted estimates for generalized fractional maximal operators on non-homogeneous spaces

Let μ be a nonnegative Borel measure on R ^d satisfying that μ(Q) ? l(Q)ⁿ for every cube Q ? R ⁿ , where l(Q) is the side length of the cube Q and 0 < n ? d.We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function B in the context of non-homogeneous spaces related to the measure μ. Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result for certain fractional maximal operator proved in W.Wang, C. Tan, Z. Lou (2012).     

Pointwise gradient estimates for evolution operators associated with Kolmogorov operators

We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated with nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover, we exhibit a class of operators which satisfy our conditions.     

Weighted L p estimates for the area integral associated to self-adjoint operators

This article is concerned with some weighted norm inequalities for the so-called horizontal (i.e., involving time derivatives) area integrals associated to a non-negative self-adjoint operator satisfying a pointwise Gaussian estimate for its heat kernel, as well as the corresponding vertical (i.e., involving space derivatives) area integrals associated to a non-negative self-adjoint operator satisfying in addition a pointwise upper bounds for the gradient of the heat kernel. As applications, we obtain sharp estimates for the operator norm of the area integrals on ${L^p(\mathbb{R}^N)}$ as p becomes large, and the growth of the A _p constant on estimates of the area integrals on the weighted L ^p spaces.     

Weighted endpoint estimates for maximal commutators associated with the sections

We prove endpoint estimates for maximal commutators for a class of singular integral operators related to the real analysis of the Monge Ampere equation.