Andrei K. Lerner

AN ELEMENTARY APPROACH TO SEVERAL RESULTS ON THE HARDY-LITTLEWOOD MAXIMAL OPERATOR

We give new elementary proofs of theorems due to B. Mucken-houpt, B. Jawerth, and S. Buckley. By means of our approach we answer a question raised by J. Orobitg and C. Perez.

Commutators of singular integrals on generalized

A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized

L^p

L^p

spaces with variable exponent

spaces with variable exponent.

MULTIDIMENSIONAL HAUSDORFF OPERATORS ON THE REAL HARDY SPACE

For a wide family of multivariate Hausdorff operators, the boundedness of an operator from this family is proved on the real Hardy space. By this we extend and strengthen previous results due to Andersen and Moricz.

On some questions related to the maximal operator on variable

Let P(ℝ n ) be the class of all exponents p for which the Hardy-Littlewood maximal operator M is bounded on Lp (·) (ℝ n ). A recent result by T. Kopaliani provides a characterization of P in terms of the Muckenhoupt-type condition A under some restrictions on the behavior of p at infinity. We give a different proof of a slightly extended version of this result. Then we characterize a weak type (p(·), p(·)) property of M in terms of A for radially decreasing p. Finally, we construct an example showing that p ∈ P(ℝ n ) does not imply p(·) - α ∈ P(ℝ n ) for all α 1/p - .

L^p

We consider maximal operators MB with respect to a basis B. In the case when MB satisfies a reversed weak type inequality, we obtain a boundedness criterion for MB on an arbitrary quasiBanach function space X. Being applied to specific B and X this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function M + on L p.

spaces

Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.

A boundedness criterion for general maximal operators

A multidimensional version of the Riesz rising sun lemma is proved by means of a generalized dyadic process.

Weighted rearrangement inequalities for local sharp maximal functions

A general, albeit simple, approach to obtaining rearrangement inequalities of maximal operators is given. Being applied to specific operators it leads not only to new proofs of well-known relations in a simpler and shorter way, but also to new, profitable results.