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Let 4#0(x) E 5S(R') and let fR4'o(y) dy4# 0. Forf E S'(R'), x E Rn and M > 0, let f + (x) = sup I * 40t(x) t>O and letf*M(x) = sup{If * 4'1(x)I: t > 0, 4#(y) E S(Rn), supp 4 c {y E Rn: IYI 0. Then there exists M(p, n), depending only on p and n, such that if M > M(p, n), then CIU' JII < lifW |1LP < Cl|f ' ILP for any f E S, '(R n), where c and C are positive constants depending only on 4'0, p, M and n. We investigate this on the space of homogeneous type with certain assumptions.

A maximal function characterization of 𝐻^{𝑝} on the space of homogeneous type