In this paper, we discuss relationships between the continuous embeddings of Dirichlet spaces (F, epsilon(1)) into Lebesgue spaces and the integrability of the associated resolvent kernel r(alpha)(x, y). For a positive measure mu, we consider the following two properties; the first one is that the Dirichlet space (F, epsilon(1)) is continuously embedded into L-2p(E; mu) (which we write as (Sob)(p)), and the second one is that the family of 1-order resolvent kernels {r(1)(x, y)}(x is an element of E) is uniformly p-th integrable in y with respect to the measure mu (which we write as (Dyn)(p)). Under some assumptions, for a measure mu, satisfying (Dyn)(1), we prove (Dyn)(p') implies (Sob)(p) for 1 <= p <= p' < infinity, and prove (Sob)(p'), implies (Dyn)(p) for 1 <= p <= p' < infinity. To prove these results we introduce L-p-Kato class, an L-p-version of the set of Kato class measures, and discuss its properties. We also give variants of such relations corresponding to the Gagliardo-Nirenberg type interpolation inequalities. As an application, we discuss the continuity of intersection measures in time. (C) 2021 Elsevier Inc. All rights reserved.
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