We prove sharp Lp(w) norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the Ap characteristic of w for all 1 < p <∞. This implies the same sharp inequalities for the classical Lusin area integral S(f ), the Littlewood–Paley g-function, and their continuous analogs Sψ and gψ . Also, as a corollary, we obtain sharp weighted inequalities for any conv...

to demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the banach-zareckitheorem is presented on the basis of the radon-nikodym theoremwhich emphasizes on measure-type properties of the lebesgueintegral. the banach-zarecki theorem says that a real-valuedfunction $f$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

For distributions, we build a theory of higher order pointwise differentiability comprising, for zero, {\L}ojasiewicz's notion point value. Results include Borel regularity differentials, rectifiability the associated jets, Rademacher-Stepanov type theorem, and Lusin approximation. A substantial part this development is new also zeroth order. Moreover, establish Poincar\'e inequality involving ...

We extend work of the author to show that the “intrinsic” square function, which pointwise dominates (and, in almost all ways, is easier to work with than) the familiar Lusin area function and its real-variable analogues, also defines a bounded operator on L(R, w dx) for all weights w in the Muckenhoupt Ap class (1 < p < ∞). We apply this result to get a strong weighted Littlewood-Paley inequal...