A class of parabolic maximal functions. (English) Zbl 1333.42039

Shakkah, Ghada; Al-Salman, Ahmad

Commun. Math. Anal. 19, No. 2, 1-31 (2016).

Summary: In this paper, we prove \(L^{p}\) estimates of a class of parabolic maximal functions provided that their kernels are in \(L^{q}\). Using the obtained estimates, we prove the boundedness of the maximal functions under very weak conditions on the kernel. In particular, we prove the\(\;L^{p}\)-boundedness of our maximal functions when their kernels are in \(L\log L^{\frac{1}{2}}(\mathbb{S}^{n-1})\) or in the block space \(B_{q}^{0,-1/2}(\mathbb{S}^{n-1}),\) \(q>1\).

MSC:

42B25	Maximal functions, Littlewood-Paley theory
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35	Function spaces arising in harmonic analysis

Keywords:

parabolic maximal functions; oscillatory integrals; singular integrals; block space