Natan Kruglyak

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 ...

Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions

It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions f in L-p, then we have the sharp estimateparallel to(I - H)f parallel to(Lp) = 2.It is also shown, via a connection between the operator I - H and Laguerre functions, thatparallel to(1 - alpha)I + Phi(I - H)parallel to(L2 -> L2) = parallel to I - alpha H parallel to(L2 -> L2) = 1 for all a is an element of [ 0, 1].

Interpolation of Closed Subspaces and Invertibility of Operators

Let (Y0, Y1) be a Banach couple and let Xj be a closed complemented subspace of Yj ; (j = 0; 1). We present several results for the general problem of finding necessary and sufficient conditions on the parameters (θ, q) such that the real interpolation space (X0, X1)θ, q is a closed subspace of (Y0, Y1)θ, q : In particular, we establish conditions which are necessary and sufficient for the equality (X0, X1)θ, q =(Y0, Y1)θ, q, with the proof based on a previous result by Asekritova and Kruglyak on invertibility of operators. We also generalize the theorem by Ivanov and Kalton where this problem was solved under several rather restrictive conditions, such as that X1 = Y1 and X0 is a subspace of codimension one in Y0

Interpolation of Besov spaces in the nondiagonal case

In the nondiagonal case, interpolation spaces for a collection of Besov spaces are described. The results are consequences of the fact that, whenever the convex hull of points (¯ s0 ,η 0) ,..., (¯ sn ,η n) ∈ R m+1 includes a ball of R m+1 ,w e have (l ¯0

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 .