Tempered fractional Sobolev spaces

The fractional Laplacian operator Delta(s) is the infinitesimal generator of the isotropic (2s)-stable Levy process on R-n, which is the scaling limit of the Levy flight with isotropic power law measure vertical bar x vertical bar(-n-2s). However, their second and higher moments are divergent, leading to the difficulty in the modelingof practical physical processes. Replacing vertical bar x vertical bar(-n-2s) by the measure of an isotropic tempered power law with the tempering exponent lambda (i.e., e(-lambda vertical bar x vertical bar)vertical bar x vertical bar(-n-2s)), the tempered fractional Laplacian operator (Delta + lambda)(s) was introduced in [17] as the infinitesimal generator of the tempered Levy process. In this paper, guided by the fractional Sobolev spaces W-s,W-p corresponding to the fractional Laplacian operator Delta(s), we deal with the tempered fractional Sobolev spaces W-s,W-lambda,W-p associated with the tempered fractional Laplacian (Delta + lambda)(s). First, the definition of the tempered fractional Sobolev spaces W-s,W-lambda,W-p is given via the Gagliardo approach, and some of their basic properties were studied. Subsequently, we focus on the Hilbert case H-s,H-lambda based on the Fourier transform. In particular, we deal with its relation with the tempered fractional Sobolev space W-s,W-lambda,W-2 and analyze their role in the trace theory, overcoming the challenges posed by the tempering exponent lambda. Then we investigate the asymptotic behavior of lambda -> 0(+), s -> 1(-) and s -> 0(+) that appear in the definition of the tempered fractional Laplacian operator (Delta + lambda)(s). Moreover, we show continuous and compact embeddings investigating the problem of the extension domains and the generalized Holder regularity results. As an application of the tempered fractional Sobolev spaces, we prove that the process defined by the tempered Levy process is a solution of some PDE with tempered fractional Lapalcian operator. (c) 2024 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.