Some Estimates for Radial Fourier Multiplier Operators with Slowly Decaying Kernels
✍ Scribed by Jay Epperson
- Publisher
- John Wiley and Sons
- Year
- 1998
- Tongue
- English
- Weight
- 503 KB
- Volume
- 191
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
We consider the restriction to radial functions of a class of radial Fourier multiplier operators containing the Bochner-Ftiesz multiplier operator. The convolution kernel K ( z ) of an operator in this class decays too slowly at infinity to be integrable, but has enough oscillation to achieve LP-boundedness for p inside a suitable interval (a, b). We prove boundedness results for the maximal operator rCf(z) = s~p,.,~y~/K(7-) * f(z)l associated with such a kernel. The maximal operator is shown to be weak type bounded at the lower critical index a, restricted weak type bounded at the upper critical index b, and strong type bounded between. This together with our assumptions on K ( z ) leads to the pointwise convergence result l i q + , y n K ( 7 -) * f(z) = cf(z) a.e. for radial f E LP(R"), a 5 p < b.
1991 Mathematics Subject Cloreijicotion. Primary: 42B 15. Keywords ond phrases. Radial Fourier multipliers, Bochner -Riesa multipliers.