J. M. Aldaz

Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities

We prove that if f: I ⊂ R→ R is of bounded variation, then the uncentered maximal function Mf is absolutely continuous, and its derivative satisfies the sharp inequality ∥DMf∥ L 1(I) < |Df|(I). This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.

Bernstein Operators for Exponential Polynomials

AbstractLet L be a linear differential operator with constant coefficients of order n and complex eigenvalues λ0,…,λn. Assume that the set Un of all solutions of the equation Lf=0 is closed under complex conjugation. If the length of the interval [a,b] is smaller than π/Mn, where Mn:=max {|Im λj|:j=0,…,n}, then there exists a basis pn,k , k=0,…,n, of the space Un with the property that each pn,k has a zero of order k at a and a zero of order n−k at b, and each pn,k is positive on the open interval (a,b). Under the additional assumption that λ0 and λ1 are real and distinct, our first main result states that there exist points a=t0<t1<⋅⋅⋅<tn=b and positive numbers α0,…,αn, such that the operator

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

B_{n}f:=\sum_{k=0}^{n}\alpha _{k}f(t_{k})p_{n,k}(x)

A measure-theoretic version of the Dragomir-Jensen inequality

satisfies

Singular Measures and Convolution Operators

B_{n}e^{\lambda _{j}x}=e^{\lambda _{j}x}

On the pointwise domination of a function by its maximal function

, for j=0,1. The second main result gives a sufficient condition guaranteeing the uniform convergence of Bnf to f for each f∈C[a,b].

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

We show that a certain optimality property of the classical Bernstein operator also holds, when suitably reinterpreted, for generalized Bernstein operators on extended Chebyshev systems.

THE WEAK TYPE (1,1) BOUNDS FOR THE MAXIMAL FUNCTION ASSOCIATED TO CUBES GROW TO INFINITY WITH THE DIMENSION

Abstract 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 M R from BV ( I ) into the Sobolev space W 1 , 1 ( I ) . By restriction, the corresponding characterization holds for W 1 , 1 ( I ) . We also show that if U is open in R d , d > 1 , then boundedness from BV ( U ) into W 1 , 1 ( U ) fails for the local directional maximal operator M T v , the local strong maximal operator M T S , and the iterated local directional maximal operator M T d ○ ⋯ ○ M T 1 . Nevertheless, if U satisfies a cone condition, then M T S : BV ( U ) → L 1 ( U ) boundedly, and the same happens with M T v , M T d ○ ⋯ ○ M T 1 , and M R .