A. Saginashvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON NORMS OF SOME INTEGRAL OPERATORS

Theorem 1. Let f ∈ L_{1}(ℝ), f(x) ≥ 0 a.e. Then the norm of the convolution operator

(W_{f}φ)(t) = ∫_{−∞}^{+∞}f(t − τ)φ(τ) dτ

in Lebesgue space L_{p}(ℝ), p ≥ 1, is equal to f̂(0), where

f̂(t) = ∫_{−∞}^{+∞}e^{itτ} φ(τ) dτ.

Let p ≥ 1, α, r ∈ ℝ. Denote by L_{p,α} the Banach space with the norm

∥φ∥_{p,α} = ( ∫_{0}^{1} |x^{α}φ(x)|^{p})^{1/p}

and by H_{r} denote the Hardy operator

(H_{r}φ)(x) = ∫_{0}^{x} x^{r−1}y^{−r}φ(y) dy.

Theorem 2. The operator H_{r} is bounded in the space L_{p,α} if and only if r < 1 − α − 1/p. In this case

∥H_{r}∥_{p,α} = \frac{1}{1 − α − 1/p − r} .

Let r > 0. Denote by R_{r} the Riemann-Liouville operator

(R_{r}φ)(t) = ∫_{0}^{x}(x − y)^{r−1}x^{−r}φ(y) dy.

Theorem 3. The operator R_{r} is bounded in the space L_{p,α} if and only if α + 1/p < 1. In this case

∥R_{r}∥_{p,α} = B(1 − α − 1/p, r),

where B(a, b) is the Beta-function.