Suppose that 1 ≤ p < ∞, and M is the maximal operator on Lp(G) defined by a sequence {ψn}∞n = 1 of strong type Fourier multipliers which are continuous functions on γ.
where M u is the maximal operator with respect to udσ, the second inequality follows from the doubling condition of udσ.
The operator (L_{0}) defined by D(L_{0})=W_{2}^{n}(0,a), quadquad L_{0}y=ell y, quad yin W_{2}^{n}(0,a), (2.2) is called the maximal operator associated with the differential expression ℓ on ([0,a]).
Theorem 1 Let u ∈ L loc t ( Ω, ∧ l ), l = 1, 2, …, n, be a smooth differential form satisfying A-harmonic equation (1), let G be Green's operator, and let M s ♯ be the sharp maximal operator defined in (4) with 1 < s < t < ∞.
Lemma 2 Let u be a smooth differential form satisfying A-harmonic equation (1) in a bounded domain Ω, let G be Green's operator, and let M s ♯ be the sharp maximal operator defined in (4) with 1 < s ≤ p, q < ∞.
Theorem 2 Let φ be a Young function in the class G ( p, q, C 0 ), 1 ≤ p < q < ∞, C 0 ≥ 1, let u ∈ L loc t ( Ω, ∧ l ), l = 1, 2, …, n, be a smooth differential form satisfying A-harmonic equation (1) in Ω, let G be Green's operator, and let M s ♯ be the sharp maximal operator defined in (4) with 1 < s ≤ t < ∞.
Remark 3.7 Note that from Theorems 3.1 and 3.2 that it is found that the maximal operator M and the singular integral operator K are the weak Φ-admissible singular operators for any Young function Φ and Φ ∈ Δ 2, respectively.
Given (p cdot)in{mathcal {P}}(mathbb {R}^{n})) and a weight w, we say ((p cdot),w)) is an M-pair if the maximal operator M is bounded on (L^{p cdot)}(w^{p cdot)})) and on (L^{p'(cdot)}(w^{-p'(cdot)})).
Denote D : = y ∈ l w 2 : w - ℒ y ∈ l w 2 , which is the domain of the maximal operator corresponding to operator ℒ.
In [13] and [14] it was also proved that the maximal operator widetilde{sigma}_{p}^{alpha,ast}:=sup_{ninmathbb{N}} biglvert sigma_{n}^{alpha}f bigrvert / bigl( (n+1 )^{1/p-alpha-1} log^{ (1+alpha ) [p+alpha (1+alpha ) ] } (n+1 ) bigr) is bounded from the Hardy space (H_{p}) to the Lebesgue space (L_{p}), where (0< pleq1/ ( 1+alpha ) ).
