Estimates of -Harmonic Conjugate Operator
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Research Article Estimates of M-Harmonic Conjugate Operator Jaesung Lee and Kyung Soo Rim Department of Mathematics, Sogang University, 1 Sinsu-dong, Mapo-gu, Seoul 121-742, South Korea Correspondence should be addressed to Jaesung Lee, [email protected] Received 30 November 2009; Revised 23 February 2010; Accepted 17 March 2010 Academic Editor: Shusen Ding Copyright q 2010 J. Lee and K. S. Rim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We define the M-harmonic conjugate operator K and prove that for 1 < p < ∞, there is a constant weight Cp such that S |Kf|p ωdσ ≤ Cp S |f|p ωdσ for all f ∈ Lp ω if and only if the nonnegative p -condition. Also, we prove that if there is a constant C such that |Kf| vdσ ≤ ω satisfies the A p p S Cp S |f|p wdσ for all f ∈ Lp w, then the pair of weights v, w satisfies the Ap -condition.
1. Introduction Let B be the unit ball of Cn with norm |z| z, z1/2 where , is the Hermitian inner product, let S be the unit sphere, and, σ be the rotation-invariant probability measure on S. In 1, for z ∈ B, ξ ∈ S, we defined the kernel Kz, ξ by iKz, ξ 2Cz, ξ − P z, ξ − 1,
1.1
where Cz, ξ 1 − z, ξ−n is the Cauchy kernel and P z, ξ 1 − |z|2 n /|1 − z, ξ|2 n is the invariant Poisson kernel. Thus for each ξ ∈ S, the kernel K, ξ is M-harmonic. And for all f ∈ AB, the ball algebra, such that f0 is real, the reproducing property of 2Cz, ξ − 1 3.2.5 of 2 gives
Kz, ξ Re fξdσξ −i fz − Re fz Im fz. S
For that reason, Kz, ξ is called the M-harmonic conjugate kernel.
1.2
2
Journal of Inequalities and Applications For f ∈ L1 S, Kf, the M-harmonic conjugate function of f, on S is defined by
Kf ζ lim
r →1
1.3
Krζ, ξfξdσξ, S
since the limit exists almost everywhere. For n 1, the definition of Kf is the same as the classical harmonic conjugate function 3, 4. Many properties of M-harmonic conjugate function come from those of Cauchy integral and invariant Poisson integral. Indeed the following properties of Kf follow directly from Chapters 5 and 6 of 2. 1 As an operator, K is of weak type 1.5 and bounded on Lp S for 1 < p < ∞. 2 If f ∈ L1 S, then Kf ∈ Lp S for all 0 < p < 1 and if f ∈ L log L, then Kf ∈ L1 S. 3 If f is in the Euclidean Lipschitz space of order α for 0 < α < 1, then so is Kf. Also, in 1, it is shown that K is bounded on the Euclidean Lipschitz space of order α for 0 < α < 1/2, and bounded on BMO. In this paper, we focus on the weighted norm inequality for M-harmonic conjugate functions. In the past, there have been many results on weighted norm inequalities and related subjects, for which the two books 3, 4 provide good references. Some classical results include those of M. Riesz in 1924 about the Lp boundedness of harmonic conjugate functions on the unit circle for 1 < p < ∞ 3, Theore
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