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For a prime number p, let Q_p denote the p-adic field and let
Q^d_p denote a vector space over Q_p which consists of all d-tuples of Q_p.
For a function f ∈L^1_loc(Q^d_p), we define the Hardy-Littlewood maximal
function of f on Q^d_p by
<수식>
where |E|_H denotes the Haar measure of a measurable subset E of Q^d_p
and B_γ(x) denotes the p-adic ball with center x∈Q^d_p
and radius p^γ. If 1 < q ≤ ∞, then we prove that M_p is a bounded operator of L_q(Q^d_p) into
Lq(Q^d_p); moreover, M_p is of weak type (1, 1) on L^1(Q^d_p), that is to say,
<수식>
for any f ∈ L^1(Q^d_p).
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참고문헌 (8)
참고문헌 신청
E. M. Stein / 1993 / Harmonic Analysis: Real variable methods, orthogonality, and oscillatory integrals / Princeton Univ. Press
J. Marcinkiewicz / 1939 / Sur l’interpolation d’operateurs / C. R. Acad. Sci. Paris 208 : 1272 ~ 1273
M. A. Naimark / 1968 / Normed Rings / Nauka
S. Haran / 1990 / Riesz potentials and explicit sums in arithmetic / Invent. Math. 101 : 697 ~ 703
S. Haran / 1993 / Analytic potential theory over the p-adics / Ann. Inst. Fourier(Grenoble) 43 (4) : 905 ~ 944
J. Marcinkiewicz / 1939 / Sur l’interpolation d’operateurs / C. R. Acad. Sci. Paris 208 : 1272 ~ 1273
M. A. Naimark / 1968 / Normed Rings / Nauka
S. Haran / 1990 / Riesz potentials and explicit sums in arithmetic / Invent. Math. 101 : 697 ~ 703
S. Haran / 1993 / Analytic potential theory over the p-adics / Ann. Inst. Fourier(Grenoble) 43 (4) : 905 ~ 944
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