Paley Theory Research Articles

On scales of Sobolev spaces associated to generalized Hardy operators

We consider the fractional Laplacian with Hardy potential and study the scale of homogeneous $L^p$ Sobolev spaces generated by this operator. Besides generalized and reversed Hardy inequalities, the analysis relies on a H\"ormander multiplier theorem which is crucial to construct a basic Littlewood--Paley theory. The results extend those obtained recently in $L^2$ but do not cover negative coupling constants in general due to the slow decay of the associated heat kernel.

Square function estimates for discrete Radon transforms

We show $\ell^p\big(\mathbb Z^d\big)$ boundedness, for $p\in(1, \infty)$, of discrete singular integrals of Radon type with the aid of appropriate square function estimates, which can be thought as a discrete counterpart of the Littlewood--Paley theory. It is a very powerful approach which allows us to proceed as in the continuous case.

Global mild solutions of the micropolar fluid system in critical spaces

We prove the global in time existence of a small solution for the 3D micropolar fluid system in critical Fourier–Herz spaces by using the Fourier localization method and Littlewood–Paley theory.

Hörmander type multiplier theorem on complex Iwasawa AN groups

We prove that, for a distinguished laplacian on an Iwasawa AN group corresponding to complex semisimple Lie group, a Hormander type multiplier theorem holds. Our argument is based on Littlewood- Paley theory.

Jacobi decomposition of weighted Triebel–Lizorkin and Besov spaces

The Littlewood–Paley theory is extended to weighted spaces of distributions on $[-1,1]$ with Jacobi weights $w(t)=(1-t)^\alpha(1+t)^\beta.$ Almost exponentially localized polynomial elements (needlets) $\{\varphi_\xi\}$, $\{\psi_\xi\}$ are construc

Beyond natural selection and divine intervention: The Lamarckian implication of Adam Smith's invisible hand

Adam Smith's invisible hand metaphor (IH) is examined in light of two different accounts of the origin of traits: Charles Darwin's theory of evolutionary optimization and William Paley's theory of divine intervention. Smith's stand supersedes both accounts. For Smith, intermediating drives, such as the sexual one, neither arise accidentally and favored according to their fitness a la Darwin nor planted by the Deity a la Paley. For Smith, such drives are adopted in light of their ultimate end. Smith did not provide an account of how the drives are connected to their far-reaching, invisible beneficial ends or why do agents become dimly aware of that causality.

A general theory of sampling synthesis

The theoretical foundations for synthesizing linear aperture distributions by means of pattern sampling are generalized to admit arbitrary location of the sampling points and arbitrary choice of the edge exponent \alpha . The generalization is based upon Paley and Wiener's theory of nonharmonic Fourier series. Explicit formulas are derived for the special case of semi-periodic sampling.