Quantum mechanics

Bounded variation

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Philosophy

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Sobolev inequality

Omega

Bessel function

Nabla symbol

Bounded function

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Mathematical analysis

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Sobolev space

Fractional calculus

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Comptes Rendus Mathématique

We continue the study of the space BV α (R n ) of functions with bounded fractional variation in R n and of the distributional fractional Sobolev space S α,p (R n ), with p ∈ [1, +∞] and α ∈ (0, 1), considered in the previous works [27,28].We first define the space BV 0 (R n ) and establish the identificationswhere H 1 (R n ) and L α,p (R n ) are the (real) Hardy space and the Bessel potential space, respectively.We then prove that the fractional gradient ∇ α strongly converges to the Riesz transform as α → 0 + for H 1 ∩W α,1 and S α,p functions.We also study the convergence of the L 1 -norm of the α-rescaled fractional gradient of W α,1 functions.To achieve the strong limiting behavior of ∇ α as α → 0 + , we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II