Abstract

In the first chapter of this dissertation, we introduce the parabolic Harnack inequality and the Caccioppoli inequality for stable-like processes.

In the second chapter, we let [special characters omitted] be the operator defined by [special characters omitted]and consider the space-time process Y_t = (X_t, V_t), where X_t is the process that corresponds to the operator [special characters omitted], and V_t = V₀ + t. Under the assumption that 0 < k ₁ ≤ a(x, h) ≤ k ₂ and a(x, h) = a( x, –h), we prove a parabolic Harnack inequality for non-negative functions that are parabolic in a domain. We also prove some estimates on equicontinuity of resolvents.

In the third chapter, we let f : [special characters omitted] and consider the following operators defined by [special characters omitted]and[special characters omitted]

Under the assumption that 0 < k₁ ≤ A(x, y) ≤ k₂ and A(x, y) = A(y, x), we establish a Caccioppoli inequality for powers of non-negative functions that are harmonic with respect to [special characters omitted].