Two weighted inequalities for fractional maximal functions and fractional integrals, generated by Bessel differential operators
by
A.I. Gadziyev
Baku State University, Baku, 370069, Azerbaijan, Rasim Mukthar str. 10

In this note we prove two-weight inequalities for the fractional maximal functions and fractional integrals, generated by Bessel differential operators B=[(d²)/(d x²)]+ [(\gamma)/x][ d/d x] (B-fractional maximal functions and B-fractional integrals). In some special case we have found the necessary and sufficent conditions for pairs of weights ensuring the validity of strong inequalities for B-fractional integrals.

Let R₊=]0, \infty[, \gamma > 0, E₊(x, r) = {y in R₊  : |x-y| < r}, A_p^\gamma=A_p(R₊, x^\gamma dx) is a B.Muckenhoupt classes. Denote the T^y the B-shift operator acting according to the law
T^yf(x) = C_\gamma \int₀^\pi f(\surd{x²+y²-2xycos\alpha}) sin^\gamma-1\alphad\alpha.

We remark that T^y is closely connected with the B=[(d²)/(d x²)]+ [(\gamma)/x][ d/dx]. Let w be a positive meusurable function on R₊. Denote by L_{p, w}^\gamma(R₊) the set of meusurable functions f(x), x in R₊, with finite norm ||f||_{Lp, w^\gamma(R+)}^p=\int_R+|f(x)|^pw(x)x^\gammadx < \infty,     1 <= p < \infty.

Let's determine B-fractional integral by following way:
I_B^\alphaf(x) = \int₀^\infty T^yx^{\alpha-1-\gamma} f(y)y^\gammady,        0 < \alpha < 1+\gamma.

Theorem 1 Let 0 < \alpha < 1+\gamma, 1 < p < [(1+\gamma)/(\alpha)], [ 1/p]-[ 1/q]=[(\alpha)/(1+\gamma)], the function v in A_1+[ q/(p')]^\gamma(R₊), w₁=\sigmav and w=uv, the weight pairs (w₁, w) satisfies condition a) or b):
a)  v(t) and \sigma(t), be positive incresing function on (0, \infty), and
sup_{x > 0}(\int_x^\inftyw₁(y)y^{-1-[((1+\gamma)q)/(p')]}dy)^[ p/q](\int₀^x/2(v^{-[(\alphap)/(1+\gamma)]}(y)w(y))^1-p'y^\gammady)^p-1 < \infty,
b)  v(t) and \sigma(t), be positive decresing function on (0, \infty), and
sup_{x > 0}(\int₀^x/2w₁(y)y^\gammady)^[ p/q](\int_x^\infty(v^{-[(\alphap)/(1+\gamma)]}(y)w(y))^1-p'y^{-1-[((1+\gamma)p')/q]}dy)^p-1 < \infty.
Then there exists c > 0 such that an arbitrary f in L_{p, w}^\gamma(R₊) the inequality
(\int_R+|I_B^\alpha (f·v^{[(\alpha)/(1+\gamma)]})(x)|^q w₁(x)x^\gammadx)^[ 1/q] <= c(\int_R+ |f(x)|^p w(x)x^\gammadx)^[ 1/p] is valid.

Date received: June 18, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-27.