Kakeya极大算子及其分数次情形的正则性

研究中心Kakeya(Nikodym)极大算子K_N(N2)及其分数次情形K_(α,N)(0αd)的正则性.特别地,建立了中心分数次Kakeya极大算子K_(α,N)是从W~(1,p)(R~d)到W~(1,q)(R~d)上的有界连续算子,其中1p∞,q=dp/(d-αp)和0≤αd/p.还证明了中心Kakeya极大算子K_N是分数次Sobolev空间W~(s,p)(R~d),非齐次Triebel-Lizorkin空间F_s~(p,q)(R~d)以及非齐次Besov空间B_s~(p,q)(R~d)上的有界连续算子,其中0s1,1p,q∞.此外,也考虑分数次Kakeya极大函数的弱导数的两种点态估计以及其离散情形的正则性.

Regularity of the Kakeya Maximal Operator and Its Fractional Variant

In this article, the authors investigate the regularity properties of the centered Kakeya (Nikodym) maximal operator K_N(with N>2) and its fractional variant Kα,N (with 0 < α < d). More precisely, the authors prove that, the operator Kα,N is bounded and continuous from W^1,p(R^d to W^1,p(R^d for 1 < p < ∞ and q=dp/(d -αp) with 0 ≤ α < d/p, and the operator K_N is bounded and continuous on the fractional Sobolev spaces W_s,p(R^d, inhomogeneous Triebel-Lizorkin spaces F_s^p,q(R^d and inhomogeneous Besov spaces B_s^p,q(R^d for all 0 < s < 1 and 1 < p, q < ∞. In addition, two pointwise estimates for the derivatives of the fractional Kakeya maximal functions and the regularity properties for the discrete versions of these operators are also presented.