非局部扰动下的Laplace算子与分数Laplace算子的内在U超压缩性

Intrinsic ultracontractivity of Laplacian and fractional Laplacian perturbed by non-local operators

令d≥1,0<β<α≤2.考虑带非局部扰动的(分数)Laplace算子L^(b)=Δ^(α/2)+S^(b),其中低阶扰动项S^(b)f(x)∫_(Rd)(f(x+z)-f(x)-■f(x)·z■{|z|≤1}b(x,z)/|z|d+βdz,b(x,z)为有界Borel可测函数,且b(x,z)=b(x,z)对于任意x,z∈R^(d)都成立.本文使用概率方法,分别证明了在α=2和0<α<2两种情形下对b(x,z)加以一定的条件,则在任意有界开集D■R^(d)内,L^(b)具有内在U超压缩性.

Assume d≥1 and 0<β<α≤2.We consider the operator L=Δ+S,where S^(b)f(x)∫_(Rd)(f(x+z)-f(x)-■f(x)·z■{|z|≤1}b(x,z)/|z|d+βdz,and b(x,z)is a bounded Borel function with b(x,z)=b(x,-z)for x,z∈R^(d).The operator L^(b)can be seen as the Laplacian(α=2)or the fractional Laplacian(0<α<2)endowed with lower-order perturbation S.Under suitable conditions on b(x,z),L^(b)will determine a conservative Feller process.In this paper,we study the intrinsic ultracontractivity of L^(b)(and its corresponding process)in a bounded open set D■R^(d).We show that,under a moderate additional condition on b(x,z),L^(b)is intrinsically ultracontractive in an arbitrary bounded open set.The method we use is probabilistic.