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Introduction to Norm Inequalities and Extrapolation
The extrapolation theorem of Rubio de Francia is one of the deepest results in the study of weighted norm inequalities in harmonic analysis: it is...
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The Hardy-Littlewood Maximal Operator
In this chapter we begin the study of harmonic analysis on variable Lebesgue spaces. Our goal is to determine the behavior of some of the classical...
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Endpoint and A ∞ Extrapolation
In this chapter we consider further variations of the two-weight extrapolation theorem proved in Chapter 7.
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Self-improvement of Poincaré Type Inequalities Associated with Approximations of the Identity and Semigroups
The purpose of this paper is to present a general method that allows us to study self-improving properties of generalized Poincaré inequalities. When...
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Extrapolation on Function Spaces
In this chapter we extend our theory of extrapolation to get norm inequalities on Banach function spaces starting from inequalities in weighted L...
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Two-Weight Factorization
Our primary goal in this chapter is to define the appropriate analogs of A 1 weights and prove a “reverse...
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Mixed A p -A r Inequalities for Classical Singular Integrals and Littlewood–Paley Operators
We prove mixed A p - A r inequalities for several basic singular integrals, Littlewood–Paley operators, and the vector-valued maximal function. Our...
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Applications of Two-Weight Extrapolation
In this chapter and the next we apply the two-weight extrapolation theorems in Chapters 7 and 8 to the theory of two-weight norm inequalities.
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Volterra equations in Banach spaces with completely monotone kernels
We consider a class of infinite delay equations in Banach spaces that models arising in the theory of viscoelasticity, for instance. The equation...
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Further Applications of Two-Weight Extrapolation
In this chapter we continue to apply the extrapolation theorems in Chapters 7 and 8 to the theory of two-weight norm inequalities. We consider two...
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The maximal operator on weighted variable Lebesgue spaces
We study the boundedness of the maximal operator on the weighted variable exponent Lebesgue spaces L
ω p (·) (Ω). For a given log-Hölder continuous... -
Multilinear Calderón Zygmund operators on variable exponent Morrey spaces over domains
The boundedness of multilinear singular integrals of Calderón-Zygmund type on product of variable exponent Lebesgue spaces over both bounded and...
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Boundedness and applications of singular integrals and square functions: a survey
We present a survey of certain aspects of the theory of singular integrals and square functions, with emphasis on L 2 boundedness criteria and recent...
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Transfer Techniques
This chapter is a collection of various techniques with the common theme “transfer”. In other words we study methods which allow us to take results...
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Iterated maximal functions in variable exponent Lebesgue spaces
In this paper we study the iterated Hardy–Littlewood maximal operator in variable exponent Lebesgue spaces with exponent allowed to reach the value...
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Preliminary Results
In this chapter we gather together many of the basic facts we will use throughout Part II: Orlicz spaces, A...