IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Fourier Analysis on Besov Spaces of Smoothness Zero

  • Jan. 31, 2017
  • 4:15 p.m.
  • LeConte 312

Abstract

There are many characterizations of Besov spaces $\mathbf{B}^s _ {p,q}$ with classical smoothness $s > 0$ given by differences in terms of other means. For instance, these spaces can be described by harmonic analysis methods through Fourier transform, wavelet bases, heat kernels, approximation by entire functions, etc. Such characterizations are useful in different areas of mathematics such as operator theory or analysis of PDEs.

In this talk we deal with Besov spaces $\mathbf{B}^{0,b} _ {p,q}$ with zero classical smoothness and logarithmic smoothness with exponent $b$, and show several equivalent characterizations. As we shall point out, the situation when $s=0$ is tricky because equivalent approaches in the classical setting differ in this limit case. More precisely, in some characterizations, a new ingredient appears: an additional truncated Littlewood-Paley construction (Carleson measures). This phenomenon arises in the Fourier-analytical decomposition and wavelet characterization of $\mathbf{B}^{0,b} _ {p,q}$. The latter allows us to derive some sharpness assertions on relationships between $\mathbf{B}^{0,b} _ {p,q}$ and their Fourier-analytically defined counterparts $B^{0,b} _ {p,q}$. On the other hand, from the abstract results on semi-groups we obtain descriptions in terms of the heat kernels and the Cauchy-Poisson semi-group. Moreover we discuss the structural differences of diverse norms of $\mathbf{B}^{0,b} _ {p,q}$ and their counterparts in $\mathbf{B}^s _ {p,q}$ and $B^{0,b} _ {p,q}$.

The talk is based on a joint work with F. Cobos (Madrid) and H. Triebel (Jena).

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