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Sobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the Hardy–Littlewood–Sobolev fractional integration theorem. An equivalent statement is known as the Sobolev lemma in (Aubin 1982, Chapter 2). A proof is in (Stein 1970, Chapter V, §1.3). Let 0 < α < n and 1 < p < q < ∞.
In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the -norms of different weak derivatives of a function through an interpolation inequality.
In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations , including the theory of harmonic maps .
Whether a space supports a Poincaré inequality has turned out to have deep connections to the geometry and analysis of the space. For example, Cheeger has shown that a doubling space satisfying a Poincaré inequality admits a notion of differentiation. [3] Such spaces include sub-Riemannian manifolds and Laakso spaces.
In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a
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In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient . These inequalities were discovered and named by Leonard Gross , who established them in dimension-independent form, [ 1 ] [ 2 ] in the context of constructive quantum field theory .
U.S. President-elect Donald Trump said on Tuesday he will create a new government agency called the External Revenue Service "to collect tariffs, duties, and all revenue" from foreign sources as ...