V. Kokilashvili and A. Meskhi

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

ON A TRACE INEQUALITY FOR ONE-SIDED POTENTIALS AND APPLICATIONS TO THE SOLVABILITY

OF NONLINEAR INTEGRAL EQUATIONS

In the present note criteria governing the trace inequality

∫_{0}^{∞} |R_{α}f(x)|^{p}v(x) dx ≤ c_{0} ∫_{0}^{∞} |f(x)|^{p} dx, f ∈ L^{p}(R_{+}),  (1)

for the Riemann–Liouville integral operator

R_{α}f(x) = ∫_{0}^{x} f(y)(x − y)^{α−1} dy, x > 0,

are establish, where 0 < α < 1/p. Some applications to the solvability of a certain nonlinear integral equation are presented. The following statement holds:

Theorem 1. Let 1 < p < ∞ and let 0 < α < 1/p. Then inequality (1) holds if and only if W_{α}v ∈ L_{loc}^{p′}(R_{+}) and W_{α}[W_{α}v]^{p′}(x) ≤ cW_{α}v(x) a.e. where p′ = p/(p − 1) and

W_{α}f(x) = ∫_{x}^{∞} f(y)(y − x)^{α−1} dy.

An analogous problem for the Volterra integral operator

Kf(x) = ∫_{0}^{x} f(y)k(x, y) dμ(y)

is discussed.

Theorem 2. Let 1 < p < ∞, 0 < α < 1, and A_{p} = (p′ − 1)(p′)^{−p}.

(i) If W_{α}v ∈ L_{loc}^{p}(R_{+}) and the inequality W_{α}[W_{α}v]^{p}(x) ≤ A_{p}W_{α}v(x) a.e. holds, then the integral equation

φ(x) = ∫_{x}^{∞} \frac{φ^{p}(t)}{(t − x)^{1−α}} dt + ∫_{x}^{∞} \frac{v(t)}{(t − x)^{1−α}} dt, 0 < α < 1  (2)

has a non-negative solution φ ∈ L_{loc}^{p}(R_{+}). Moreover, (W_{α}v)(x) ≤ φ(x) ≤ p′(W_{α}v)(x).

(ii) If 0 < α < 1/p′ and (2) has a non-negative solution in L_{loc}^{p}(R_{+}), then W_{α}v ∈ L_{loc}^{p}(R_{+}) and W_{α}[(W_{α}v)^{p}](x) ≤ cW_{α}v(x) a.e. for some constant c > 0.