Atlas: Some properties of $B_{k,n}$-maximal functions and $B_{k,n}$-Riesz potentials by N.N. Garahanova

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3rd International ISAAC Congress
August 20-25, 2001
Freie Universitaet Berlin
Berlin, Germany

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Some properties of B_{k, n}-maximal functions and B_{k, n}-Riesz potentials
by
N.N. Garahanova
Baku State University, Baku, 370069, Azerbaijan, Rasim Mukthar str. 10

Let Rⁿ-n-dimentional Euclidian spaces, |x|² = \sum_i=1ⁿ x_i², 1 <= k <= n-1, x'=x_{1, k}=(x₁, ... , x_k) in R^k, x''=x_{k, n}=(x_k+1, ... , x_n) in R^n-k, x=(x', x'')=(x_{1, k}, x_{k, n}) in Rⁿ , R_{k, +}ⁿ={x=(x_{1, k}, x_{k, n}) in Rⁿ; x_k+1 > 0, ... , x_n > 0}, E_{k, +}(x, r)={y-x in R_{k, +}ⁿ ; |x-y| < r}, | E_{k, +}(0, r)|_{\gamma_{k, n}} = \int_{E_{k, +}(0, r)} y_{k, n}^{\gamma_{k, n}}dy, \gamma_{k, n}=(\gamma_k+1, ... , \gamma_n), |\gamma_{k, n}| = \sum_i=k+1ⁿ \gamma_i, \gamma_k+1 > 0, ... , \gamma_n > 0, B_{k, n}=(B_k+1, ... , B_n), B_i=\frac\partial²\partialx_i²+ \frac\gamma_ix_i\frac \partial\partialx_i, i=k+1, ... , n, x_{k, n}^{\gamma_{k, n}}=x_k+1^\gamma_k+1· ... ·x_n^\gamma_n. In the case k=0 x=x''=x_{0, n} in Rⁿ, R₊ⁿ \equiv R_{0, +}ⁿ = {x in Rⁿ; x₁ > 0, ... , x_n > 0}, \gamma = \gamma_{0, n}=(\gamma₁, ... , \gamma_n), B=B_{0, n}=(B₁, ... , B_n). For 1 <= p <= \infty, L_{p, \gamma_{k, n}} \equiv L_p(R_{k, +}ⁿ, B_{k, n}) || f|| _{p, \gamma_{k, n}}^p = \int_{R_{k, +}ⁿ}| f(x)|^p x_{k, n}^{\gamma_{k, n}} dx, 1 <= p < \infty.

Denote the T^y the B_{k, n}-shift operator acting according to the law
T^yf(x) = C_{\gamma_{k, n}} \int₀^\pi ... \int₀^\pi f( x'-y', \surd{x_k+1²-2x_k+1y_k+1cos\alpha_k+1+y_k+1²},
... , \surd{x_n²-2x_ny_ncos\alpha_n+y_n²}) ·\prod_i=k+1ⁿ sin^\gamma_i-1\alpha_i d\alpha_k+1 ... d\alpha_n.

In the note, we investigate the boundedness of B_{k, n}-maximal operators
M_{B_{k, n}}f(x)=sup_{r > 0}| E_{k, +}(0, r)|_{\gamma_{k, n}}^-1\int_{E_{k, +}(0, r)}T^y|f(x)| y_{k, n}^{\gamma_{k, n}}dy
and the B_{k, n}-Riesz potentials
I_{B_{k, n}}^\alpha f(x)=\int_{R_{k, +}ⁿ}T^y |x|^{\alpha-n-|\gamma_{k, n}|} f(y)y_{k, n}^{\gamma_{k, n}} dy, 0 < \alpha < n+|\gamma_{k, n}|
on the L_p(R_{k, +}ⁿ, B_{k, n}) and B_{k, n}- BMO spaces BMO(R_{k, +}ⁿ, B_{k, n}). Note that, the B_{k, n}-maximal operator M_{B_{k, n}} and the B_{k, n}-Morrey spaces L_{p, \lambda}(R_{k, +}ⁿ, B_{k, n}), B_{k, n}- BMO spaces BMO(R_{k, +}ⁿ, B_{k, n}) are introduced and investigated by V.S.Guliev [1] (see also [2]) in the case k=0.

[1]. Guliev V.S. Sobolev theorems for B-Riesz potentials.
Dokl. RAN, 1998, v.358, 4, c.450-451. (Russian)

[2]. Guliev V.S. Some aspects of B-harmonic analysis.
Proceeding of the Second ISAAC Congress. Edited by H. Begehr, R.Gilbert, J.Kajiwara. Kluwer Acad. Publ., Dordrecht / Boston / London, 2000, Vol. 2, p.1223-1240.

Date received: June 13, 2001