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A second look on definition and equivalent norms of Sobolev spaces

Joachim Naumann, Christian G. Simader (1999)

Mathematica Bohemica

Sobolev’s original definition of his spaces L m , p ( Ω ) is revisited. It only assumed that Ω n is a domain. With elementary methods, essentially based on Poincare’s inequality for balls (or cubes), the existence of intermediate derivates of functions u L m , p ( Ω ) with respect to appropriate norms, and equivalence of these norms is proved.

A sharp form of an embedding into multiple exponential spaces

Robert Černý, Silvie Mašková (2010)

Czechoslovak Mathematical Journal

Let Ω be a bounded open set in n , n 2 . In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that sup Ω exp f ( x ) K n / ( n - 1 ) : f W 0 1 , n ( Ω ) , f L n 1 < . We extend this result to the situation in which the underlying space L n is replaced by the generalized Zygmund space L n log n - 1 L log α log L ( α < ...

A sharp iteration principle for higher-order Sobolev embeddings

Andrea Cianchi, Luboš Pick, Lenka Slavíková (2014)

Banach Center Publications

We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order involving...

A sharp rearrangement inequality for the fractional maximal operator

A. Cianchi, R. Kerman, B. Opic, L. Pick (2000)

Studia Mathematica

We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of ⨍, M γ , by an expression involving the nonincreasing rearrangement of ⨍. This estimate is used to obtain necessary and sufficient conditions for the boundedness of M γ between classical Lorentz spaces.

A short proof on lifting of projection properties in Riesz spaces

Marek Wójtowicz (1999)

Commentationes Mathematicae Universitatis Carolinae

Let L be an Archimedean Riesz space with a weak order unit u . A sufficient condition under which Dedekind [ σ -]completeness of the principal ideal A u can be lifted to L is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of C ( X ) -spaces. Similar results are obtained for the Riesz spaces B n ( T ) , n = 1 , 2 , , of all functions of the n th Baire class on a metric space T .

A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces

B. L. Chalmers, F. T. Metcalf (1992)

Annales Polonici Mathematici

It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant λ ( v ) = s u p X λ ( v ; X ) . The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in L ( ν ) and isometric to v and a projection P from C ⊕ V onto V such that P = P , where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if P = i = 1 2 U i v i , then P = i = 1 2 u i V i , where d V i = 2 v i d ν and d U i = - 2 u i d ν .

A strongly extreme point need not be a denting point in Orlicz spaces equipped with the Orlicz norm

Adam Bohonos, Ryszard Płuciennik (2011)

Banach Center Publications

There are necessary conditions for a point x from the unit sphere to be a denting point of the unit ball of Orlicz spaces equipped with the Orlicz norm generated by arbitrary Orlicz functions. In contrast to results in [12, 17, 16], we present also examples of Orlicz spaces in which strongly extreme points of the unit ball are not denting points.

A subelliptic Bourgain–Brezis inequality

Yi Wang, Po-Lam Yung (2014)

Journal of the European Mathematical Society

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space N L ˙ 1 , Q by L functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for on the Heisenberg group n .

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