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The inequality expressing this fact has constants that do not involve the dimension of the space and, thus, the inequality holds in the setting of a Gaussian measure on an infinite-dimensional space. It is now known that logarithmic Sobolev inequalities hold for many different types of measures, not just Gaussian measures.
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 .
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, the Poincaré inequality [1] is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition.
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
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L p-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete , i.e. a Banach space .
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting.
The trace operator can be defined for functions in the Sobolev spaces , with <, see the section below for possible extensions of the trace to other spaces. Let Ω ⊂ R n {\textstyle \Omega \subset \mathbb {R} ^{n}} for n ∈ N {\textstyle n\in \mathbb {N} } be a bounded domain with Lipschitz boundary.