Abstract
In this article, first, we prove that weighted-norm inequalities for the M-harmonic conjugate operator on the unit sphere whenever the pair (u, v) of weights satisfies the Ap-condition, and uds ,v ds are doubling measures, where ds is the rotationinvariant positive Borel measure on the unit sphere with total measure 1. Then, we drive cross-weighted norm inequalities between the Hardy-Littlewood maximal function and the sharp maximal function whenever (u, v) satisfies the Ap-condition, and vds does a certain regular condition. 2000 MSC: primary 32A70; secondary 47G10.
Highlights
In 1973, Hunt et al [4] proved that, for 1
Let B be the unit ball of Cn with norm |z| = 〈z, z〉1/2 where 〈, 〉 is the Hermitian inner product, S be the unit sphere and s be the rotation-invariant probability measure on S.For z Î B, ξ Î S, we define the M-harmonic conjugate kernel K(z, ξ) by iK(z, ξ ) = 2C(z, ξ ) − P(z, ξ ) − 1, where C(z, ξ) = (1 - 〈z, ξ〉)-n is the Cauchy kernel and P(z, ξ) = (1 - |z|2)n/|1 - 〈z, ξ〉 | 2n is the invariant Poisson kernel [1].For the kernels, C and P, refer to [2]
If (u, v) satisfies two-weighted Ap’ (S)-condition for some p’ < p and uds, vds are doubling measures, there exists a constant C which depends on u, v and p, such that for all function f, K f pu dσ ≤ C f pv dσ for all f ∈ Lp(v)
Summary
In 1973, Hunt et al [4] proved that, for 1
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