Sergey Kislyakov

Extremal Problems in Interpolation Theory, Whitney-Besicovitch Coverings, and Singular Integrals

In this book we suggest a unified method of constructing near-minimizers for certain important functionals arising in approximation, harmonic analysis and ill-posed problems and most widely used in ...

Classical Calderón–Zygmund decomposition and real interpolation

We start with the Riesz rising sun lemma, and then pass to its many-dimensional substitute, i.e., the Calderon–Zygmund lemma. We discuss some applications of the latter, the John–Nirenberg and Campanato inequalities being among them. To relate the matter to real interpolation, we observe that the so-called “good” part of the Calderon–Zygmund decomposition is a near-minimizer for the couple (L 1, L ∞). Next, in a rather elementary setting, we prove that certain nearminimizers are stable under the action of certain unbounded operators that are nevertheless “bounded far away” from the support of the function to which they are applied. They will be called long-range regular operators.

Stability of near-minimizers

Near-minimizers constructed in the preceding chapter have the form

Classical covering theorems

f_t = \sum_{i\in I} P_i \psi_i + f \chi_\mathbb{R}/\cup K_i ,

Construction of near-minimizers

where {K i} i∈I is a WB-covering, \(\{\psi_i\}i \in I\) is a smooth partition of unity adjusted to it, and the {P i } i∈I are polynomials of degree strictly smaller than k that provide a nearly optimal approximation of f on the cubes K i .

The omitted case of a limit exponent

In this chapter we summarize some information from interpolation theory. In distinction to the preceding chapters, here we mainly avoid giving proofs of general results. Good references for that stuff are [BL], [BK], [BSh1], and [KPS]. To the contrary, we do give detailed proofs of many statements directly related to nearminimizers. It should be noted that, combined with explicit constructions of nearminimizers and with stability theorems presented in this book, these results give additional interesting information.

Regularization for Banach spaces

In this chapter we prove classical covering theorems. The arguments will be not quite standard, and these new thoughts will be used in Part 2 in the proofs of controlled covering theorems.

Stability for analytic Hardy spaces

In this chapter we present some very basic facts about spaces of smooth functions (specifically, about Sobolev and Morrey–Campanato spaces). Next, we give some additional information about singular integrals (note that in the stability theorem in the last chapter we employed some facts not covered before). We prove that the adjoints to singular integral operators take L ∞ to BMO and discuss the weak L 1-boundedness condition (3.45). More generally, we shall see that, under some additional assumptions, the adjoints to singular integral operators take some Morrey-Campanato spaces to themselves. Also, we analyze partial sum operators for wavelet expansions from a singular integral theory viewpoint. All this will be of much importance in Part 2 of the book.