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

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
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