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)     

Endpoint estimates for the circular maximal function

We consider the problem of endpoint estimates for the circular maximal function defined by

where is the normalized surface area measure on . Let be the closed triangle with vertices . We prove that for , there is a constant such that Furthermore, .

     

Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators

In this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. As a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional maximal operator or function. In particular, we extend some results given in Carro et al. (2005) [7] to the multilinear context. On the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator Mα,B associated to a Young function B and the multilinear maximal operators Mψ=M0,ψ, ψ(t)=B(t1−α/(nm))^nm/(nm−α). As an application of these estimate we obtain a direct proof of the Lp−Lq boundedness results of Mα,B for the case B(t)=t and Bk(t)=tk(1+log⁺t) when 1/q=1/p−α/n. We also give sufficient conditions on the weights involved in the boundedness results of Mα,B that generalizes those given in Moen (2009) [22] for B(t)=t. Finally, we prove some boundedness results in Banach function spaces for a generalized version of the multilinear fractional maximal operator.     

The boundedness of maximal operators and singular integrals via Fourier transform estimates

In this paper, the author studies the mapping properties for some general maximal operators and singular integrals on certain function spaces via Fourier transform estimates. Also, some concrete maximal operators and singular integrals are studied as applications.     

Maximal functions associated to surfaces of revolution on product domains

The author establishes the Lp boundedness for a class of maximal functions related to singular integrals associated to surfaces of revolution on product domains with rough kernels in L(logL)(Sn−1).     

Boundedness and unboundedness results for some maximal operators on functions of bounded variation

We characterize the space BV(I) of functions of bounded variation on an arbitrary interval I⊂R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV(I) into the Sobolev space W1,1(I). By restriction, the corresponding characterization holds for W1,1(I). We also show that if U is open in Rd, d>1, then boundedness from BV(U) into W1,1(U) fails for the local directional maximal operator , the local strong maximal operator , and the iterated local directional maximal operator . Nevertheless, if U satisfies a cone condition, then boundedly, and the same happens with , , and MR.     

Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.     

Estimates for multiparameter maximal operators of Schrödinger type

Multiparameter maximal estimates are considered for operators of Schrödinger type. Sharp and almost sharp results, that extend work by Rogers and Villarroya, are obtained. We provide new estimates via the integrability of the kernel which naturally appears with a TT^?$T{T}^{?}$ argument and discuss the behavior at the endpoints. We treat in particular the case of global integrability of the maximal operator on finite time for solutions to the linear Schrödinger equation and make some comments on an open problem.     

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 ).     

Sharp general local estimates for dyadic-like maximal operators and related Bellman functions

For each 1?q<p we precisely evaluate the main Bellman functions associated with the local Lp→Lq estimates of the dyadic maximal operator on Rn. Actually we do that in the more general setting of tree-like maximal operators and with respect to general convex and increasing growth functions. We prove that these Bellman functions equal to analogous extremal problems for the Hardy operator which can be viewed as a symmetrization principle for such operators. Under certain mild conditions on the growth functions we show that for the latter extremals exist (although for the original Bellman functions do not) and analyzing them we give a determination of the corresponding Bellman function.     

Maximal operators with rough kernels on product domains

In this paper, we study the Lp boundedness of certain maximal operators on product domains with rough kernels in L(logL). We prove that our operators are bounded on Lp for all 2?p<∞. Moreover, we show that our condition on the kernel is optimal in the sense that the space L(logL) cannot be replaced by Lr(logL) for any r<1. Our results resolve a problem left open in [Y. Ding, A note on a class of rough maximal operators on product domains, J. Math. Anal. Appl. 232 (1999) 222-228].     

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.     

An Endpoint Estimate for the Commutator of Singular Integrals

It is well known that the commutator Tb of the singular integral operator T with a BMO function b is bounded on L^P（R^n）, 1 〈 p 〈 ∞. In this paper, we consider the endpoint estimates for a kind of commutator of singular integrals. A BMO-type estimate for Tb is obtained under the assumption b ∈ LMO.     

Extremal problems related to maximal dyadic-like operators

We obtain sharp estimates for the localized distribution function of the dyadic maximal function Md?, when ? belongs to Lp,∞. Using this we obtain sharp estimates for the quasi-norm of Md? in Lp,∞ given the localized L¹-norm and certain weak Lp-conditions.     

Error estimates in the numerical evaluation of some BEM singular integrals

In some applications of Galerkin boundary element methods one has to compute integrals which, after proper normalization, are of the form where , or , or , and is a smooth function.

In this paper we derive error estimates for a numerical approach recently proposed to evaluate the above integral when a , or , formulation of a Galerkin method is used. This approach suggests approximating the inner integral by a quadrature formula of interpolatory type that exactly integrates the Cauchy kernel, and the outer integral by a rule which takes into account the endpoint singularities of its integrand. Some numerical examples are also given.

     

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.     

Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms

For all d?2 and p∈(1,max(2,(d+1)/2)], we prove sharp Lp to Lp(Lq) estimates (modulo an endpoint) for a directional maximal operator associated to curves generated by the dilation matrices exp((logt)P), where P has real entries and eigenvalues with positive real part. For the corresponding Hilbert transform we prove an analogous result for all d?2 and p∈(1,2]. As corollaries, we prove Lp bounds for variable kernel singular integral operators and Nikodym-type maximal operators taking averages over certain families of curved sets in Rd.     

Weighted estimates for singular integral operators and commutators associated with the sections

In this paper we prove weighted estimates for singular integral operators and commutators associated with the sections.     

Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.