A Note on a Maximal Function Over Arbitrary Sets of Directions (original) (raw)
A unifying approach for certain class of maximal functions
Journal of Inequalities and Applications, 2006
We establish L p estimates for certain class of maximal functions with kernels in L q (S n−1). As a consequence of such L p estimates, we obtain the L p boundedness of our maximal functions when their kernels are in L(logL) 1/2 (S n−1) or in the block space B 0,−1/2 q (S n−1), q > 1. Several applications of our results are also presented.
Inequalities for Some Maximal Functions. I
Transactions of The American Mathematical Society, 1985
This paper presents a new approach to maximal functions on R". Our method is based on Fourier analysis, but is slightly sharper than the techniques based on square functions. In this paper, we reprove a theorem of E. M. Stein [16] on spherical maximal functions and improve marginally work of N. E. Aguilera [1] on the spherical maximal function in L2(R2). We prove results on the maximal function relative to rectangles of arbitrary direction and fixed eccentricity; as far as we know, these have not appeared in print for the case where n > 3, though they were certainly known to the experts. Finally, we obtain a best possible theorem on the pointwise convergence of singular integrals, answering a question of A. P. Calderón and A. Zygmund [3, 4] to which N. E. Aguilera and E. O. Harboure [2] had provided a partial response.
A REMARK ON MAXIMAL OPERATORS ALONG DIRECTIONS IN R2
1960
In this paper we give a simple proof of a long-standing conjecture, recently proved by N. Katz, on the weak-type norm of a maximal operator associated with an arbitrary collection of directions in the plane. The proof relies upon a geometric argument and on induction with respect to the number of directions. Applications are given to estimate the behavior of several types of maximal operators.
2018
We prove nontangential and radial maximal function characterizations for Hardy spaces associated to a non-negative self-adjoint operator satisfying Gaussian estimates on a space of homogeneous type with finite measure. This not only addresses an open point in the literature, but also gives a complete answer to the question posed by Coifman and Weiss in the case of finite measure. We then apply our results to give maximal function characterizations for Hardy spaces associated to second order elliptic operators with Neumann and Dirichlet boundary conditions, Schrödinger operators with Dirichlet boundary conditions, and Fourier– Bessel operators.
On the Lp boundedness of the non-centered GaussianHardy-Littlewood Maximal
1999
The purpose of this paper is to prove the L p (R n ; dd) boundedness, for p > 1, of the non-centered Hardy-Littlewood maximal operator associated with the Gaussian measure dd = e ?jxj 2 dx. Let dd = e ?jxj 2 dx be a Gaussian measure in Euclidean space R n. We consider the non-centered maximal function deened by Mf(x) = sup x2B 1 (B) Z B jfj dd; where the supremum is taken over all balls B in R n containing x. P. Sjj ogren 2] proved that M is not of weak type (1,1) with respect to dd. A more general result was obtained by A. Vargas 3], who characterized those radial and strictly positive measures for which the corresponding maximal operator is of weak type (1,1). However, these papers leave open the question of the L p (dd) boundedness of M for p > 1 and n > 1: The main result in this paper is Theorem 1 M is a bounded operator on L p (dd) for p > 1, that is, there exists a constant C = C(n; p) such that for f 2 L p (dd); kMfk L p (dd) Ckfk L p (dd) : We denote S n?1
A Class of Parabolic Maximal Functions
Communications in Mathematical Analysis, 2016
In this paper, we prove L p estimates of a class of parabolic maximal functions provided that their kernels are in L q. Using the obtained estimates, we prove the boundedness of the maximal functions under very weak conditions on the kernel. In particular, we prove the L p-boundedness of our maximal functions when their kernels are in L log L 1 2 (S n−1