38 research outputs found
Gradient estimates of q-harmonic functions of fractional Schrodinger operator
We study gradient estimates of -harmonic functions of the fractional
Schr{\"o}dinger operator , in bounded
domains . For nonnegative we show that if is H{\"o}lder
continuous of order then exists for any and |\nabla u(x)| \le c u(x)/ (\dist(x,\partial D) \wedge 1). The
exponent is critical i.e. when is only H{\"o}lder
continuous may not exist. The above gradient estimates are well
known for under the assumption that belongs to the Kato
class \calJ^{\alpha - 1}. The case is different. To obtain
results for we use probabilistic methods. As a corollary, we
obtain for that a weak solution of is in fact a strong solution