David Cruz-Uribe

Weights, extrapolation and the theory of Rubio de Francia

Preface.- Preliminaries.- Part I. One-Weight Extrapolation.- Chapter 1. Introduction to Norm Inequalities and Extrapolation.- Chapter 2. The Essential Theorem.- Chapter 3. Extrapolation for Muckenhoupt Bases.- Chapter 4. Extrapolation on Function Spaces.- Part II. Two-Weight Factorization and Extrapolation.- Chapter 5. Preliminary Results.- Chapter 6. Two-Weight Factorization.- Chapter 7. Two-Weight Extrapolation.- Chapter 8. Endpoint and A1 Extrapolation.- Chapter 9. Applications of Two-Weight Extrapolation.- Chapter 10. Further Applications of Two-Weight Extrapolation.- Appendix A. The Calderon-Zygmund Decomposition.- Bibliography.- Index.

THE STRUCTURE OF THE REVERSE HOLDER CLASSES

In this paper we study the structure of the class of functions (RHS) which satisfy the reverse Holder inequality with exponent s > I. To do so we introduce a new operator, the minimal operator, which is analogous to the Hardy-Littlewood maximal operator, and a new class of functions, {RHcc), which plays the same role for (RHS) that the class (Ax) does for the (Ap) classes.

Sharp weighted estimates for classical operators

Abstract We give a general method based on dyadic Calderon–Zygmund theory to prove sharp one- and two-weight norm inequalities for some of the classical operators of harmonic analysis: the Hilbert and Riesz transforms, the Beurling–Ahlfors operator, the maximal singular integrals associated to these operators, the dyadic square function and the vector-valued maximal operator. In the one-weight case we prove the sharp dependence on the A p constant by finding the best value for the exponent α ( p ) such that ‖ T f ‖ L p ( w ) ⩽ C n , T [ w ] A p α ( p ) ‖ f ‖ L p ( w ) . For the Hilbert transform, the Riesz transforms and the Beurling–Ahlfors operator the sharp value of α ( p ) was found by Petermichl and Volberg (2007, 2008, 2002) [47] , [48] , [49] ; their proofs used approximations by the dyadic Haar shift operators, Bellman function techniques, and two-weight norm inequalities. Our proofs again depend on dyadic approximation, but avoid Bellman functions and two-weight norm inequalities. We instead use a recent result due to A. Lerner (2010) [34] to estimate the oscillation of dyadic operators. By applying this we get a straightforward proof of the sharp dependence on the A p constant for any operator that can be approximated by Haar shift operators. In particular, we provide a unified approach for the Hilbert and Riesz transforms, the Beurling–Ahlfors operator (and their corresponding maximal singular integrals), dyadic paraproducts and Haar multipliers. Furthermore, we completely solve the open problem of sharp dependence for the dyadic square functions and vector-valued Hardy–Littlewood maximal function. In the two-weight case we use the very same techniques to prove sharp results in the scale of A p bump conditions. For the singular integrals considered above, we show they map L p ( v ) into L p ( u ) , 1 p ∞ , if the pair ( u , v ) satisfies sup Q ‖ u 1 / p ‖ A , Q ‖ v − 1 / p ‖ B , Q ∞ , where A ¯ ∈ B p ′ and B ¯ ∈ B p are Orlicz functions. This condition is sharp. Furthermore, this condition characterizes (in the scale of these A p bump conditions) the corresponding two-weight norm inequality for the Hardy–Littlewood maximal operator M and its dual: i.e., M : L p ( v ) → L p ( u ) and M : L p ′ ( u 1 − p ′ ) → L p ( v 1 − p ′ ) . Muckenhoupt and Wheeden conjectured that these two inequalities for M are sufficient for the Hilbert transform to be bounded from L p ( v ) into L p ( u ) . Thus, in the scale of A p bump conditions, we prove their conjecture. We prove similar, sharp two-weight results for the dyadic square function and the vector-valued maximal operator.

The fractional maximal operator and fractional integrals on variable

We prove that if the exponent function p(·) satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maximal operator Mα, 0 <α< n, maps Lp(·) to Lq(·), where 1 p(x)− 1 q(x) = αn . We also prove a weak-type inequality corresponding to the weak (1, n/(n − α)) inequality for Mα. We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [3]. As a consequence of these results for Mα, we show that the fractional integral operator Iα satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable Lp spaces.

L^p

We prove that the commutator [b, Iα], b ∈ BMO, Iα the fractional integral operator, satisfies the sharp, modular weak-type inequality f(x) tdx, where B(t) = tlog(e + t) and Ψ(t)=[tlog(e + tα/n)]n/(n−α). These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality, M#([b, Iα]f)(x) ≤ CbBMO [Iαf(x) + Mα,Bf(x)], where M# is the sharp maximal operator, and Mα,B is a generalization of the fractional maximal operator in the scale of Orlicz spaces. Using this inequality we also prove one-weight inequalities for the commutator; to do so we prove one and two-weight norm inequalities for Mα,B which are of interest in their own right.[b, Iα]f(x)

spaces.

We study the boundedness of the maximal operator on the weighted variable exponent Lebesgue spaces Lωp(·) (Ω). For a given log-Hölder continuous exponent p with 1 < inf p ⩽ supp < ∞ we present a necessary and sufficient condition on the weight ω for the boundedness of M. This condition is a generalization of the classical Muckenhoupt condition.

Endpoint estimates and weighted norm inequalities for commutators of fractional integrals

Abstract We give a new and simpler proof of Sawyers theorem characterizing the weights governing the two-weight, strong-type norm inequality for the Hardy-Littlewood maximal operator and the fractional maximal operator. As a further application of our techniques, we give new proofs of two sufficient conditions for such weights due to Wheeden and Sawyer.

The maximal operator on weighted variable Lebesgue spaces

We consider two closely related but distinct operators,

Weighted norm inequalities for the geometric maximal operator

\align M_0f(x)&= \sup_{I\ni x}\exp\left(\frac{1}{|I|}\int_I\log|f|\,dy\right) \quad\text{and}\\ M_0^*f(x) &= \lim_{r\rightarrow0} \sup_{I\ni x}\left(\frac{1}{|I|}\int_I|f|^r\,dy\right)^{1/r}. \endalign