A supercritical estimate for Bessel potentials on Lorentz spaces

In this paper we give a simple proof of an estimate for Bessel potentials acting on Lorentz spaces in the supercritical exponent: let 1 < p = d/alpha-1 and 1 <= q <= +infinity. If f is an element of L-p,L-q (R-d), then there exists a constant C = C(alpha, d, p, q) such that vertical bar g(alpha) * f(x) - g(alpha) * f(z)vertical bar <= C vertical bar x - z| (vertical bar ln(vertical bar x - z vertical bar)vertical bar + 1)1/q' parallel to f parallel to (Lp,q).