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We study Dirichlet forms defined by nonintegrable Levy kernels whose singularity at the origin can be weaker than that of any fractional Laplacian. We show some properties of the associated Sobolev type spaces in a bounded domain, such as symmetrization estimates, Hardy inequalities, compact inclusion in L-2 or the inclusion in some Lorentz space. We then apply those properties to study the associated nonlocal operator L and the Dirichlet and Neumann problems related to the equations L-u = f(x) and L-u = f(u) in Omega. (C) 2017 Elsevier Inc. All rights reserved.
dirichlet forms; integral operators; nonlocal equations; compact embeddings