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Some Estimates for Radial Fourier Multiplier Operators with Slowly Decaying Kernels
Author(s) -
Epperson Jay
Publication year - 1998
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19981910106
Subject(s) - mathematics , multiplier (economics) , bounded function , pointwise , pointwise convergence , operator (biology) , maximal operator , fourier transform , mathematical analysis , kernel (algebra) , type (biology) , pure mathematics , ecology , approx , biochemistry , chemistry , repressor , biology , computer science , transcription factor , gene , economics , macroeconomics , operating system
We consider the restriction to radial functions of a class of radial Fourier multiplier operators containing the Bochner‐Riesz multiplier operator. The convolution kernel K(x) of an operator in this class decays too slowly at infinity to be integrable, but has enough oscillation to achieve L p ‐boundedness for p inside a suitable interval (a, b). We prove boundedness results for the maximal operator Kf(x) = sup r>0 r n ∣K(r ) * f(x)∣ 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(x) leads to the pointwise convergence result lim γ→ γ n K (γ·) * f(x) = cf(x) a. e. for radial f ϵ L P (ℝ n ), a ≥ p > b .