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The trilinear embedding theorem

Hitoshi Tanaka (2015)

Studia Mathematica

Let σ i , i = 1,2,3, denote positive Borel measures on ℝⁿ, let denote the usual collection of dyadic cubes in ℝⁿ and let K: → [0,∞) be a map. We give a characterization of a trilinear embedding theorem, that is, of the inequality Q K ( Q ) i = 1 3 | Q f i d σ i | C i = 1 3 | | f i | | L p i ( d σ i ) in terms of a discrete Wolff potential and Sawyer’s checking condition, when 1 < p₁,p₂,p₃ < ∞ and 1/p₁ + 1/p₂ + 1/p₃ ≥ 1.

The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of p -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Theory of Bessel potentials. III : potentials on regular manifolds

Robert Adams, Nachman Aronszajn, M. S. Hanna (1969)

Annales de l'institut Fourier

In this paper Bessel potentials on C -Riemannian manifolds (open or bordered) are studied. Let M be an n -dimensional manifold, and N a submanifold of M of dimension k . Sufficient conditions are given for: 1) the restriction to N of any potential of order α on M to be a potential of order α - n - k 2 on N  ; 2) any potential of order α - n - k 2 on N to be extendable to a potential of order α on M . It is also proved that for a bordered manifold M the restriction to its interior M i is an isometric isomorphism between the...

Thinness and non-tangential limit associated to coupled PDE

Allami Benyaiche, Salma Ghiate (2013)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we study the reduit, the thinness and the non-tangential limit associated to a harmonic structure given by coupled partial differential equations. In particular, we obtain such results for biharmonic equation (i.e. 2 ϕ = 0 ) and equations of 2 ϕ = ϕ type.

Topologies fines et compactifications associées à certains espaces de Dirichlet

Denis Feyel, A. de La Pradelle (1977)

Annales de l'institut Fourier

Nous commençons par définir la notion d’espaces L 1 ( γ ) γ est une capacité, ce qui permet d’introduire la notion de mesure d’énergie finie par rapport à γ , et de parler d’espaces de Dirichlet basés sur γ .Soit d’autre part un espace de Dirichlet en ce sens avec potentiels s.c.i. : on étudie les espaces de Dirichlet sur les ouverts fins correspondants à l’aide d’une compactification. On retrouve plus facilement et on généralise les résultats de D. Feyel et A. de La Pradelle, (Lecture Notes).

Topologies semi-vectorielles. Application à l'analyse complexe

Pierre Lelong (1975)

Annales de l'institut Fourier

On définit sur un espace vectoriel E une classe de topologies qui rendent la multiplication continue, mais ne sont pas vectorielles en général. Sur un espace complexe E elles permettent d’obtenir encore les principales propriétés des fonctions plurisousharmoniques. De telles topologies séparées sont localement pseudo-convexes (mais non localement convexes en général) : cette notion intervient dans les extensions données récemment par l’auteur du théorème de Banach-Steinhaus aux familles de polynômes...

Transitions on a noncompact Cantor set and random walks on its defining tree

Jun Kigami (2013)

Annales de l'I.H.P. Probabilités et statistiques

First, noncompact Cantor sets along with their defining trees are introduced as a natural generalization of p -adic numbers. Secondly we construct a class of jump processes on a noncompact Cantor set from given pairs of eigenvalues and measures. At the same time, we have concrete expressions of the associated jump kernels and transition densities. Then we construct intrinsic metrics on noncompact Cantor set to obtain estimates of transition densities and jump kernels under some regularity conditions...

Trudinger's inequality for double phase functionals with variable exponents

Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno, Tetsu Shimomura (2021)

Czechoslovak Mathematical Journal

Our aim in this paper is to establish Trudinger’s inequality on Musielak-Orlicz-Morrey spaces L Φ , κ ( G ) under conditions on Φ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger’s inequality for double phase functionals Φ ( x , t ) = t p ( x ) + a ( x ) t q ( x ) , where p ( · ) and q ( · ) satisfy log-Hölder conditions and a ( · ) is nonnegative, bounded and Hölder continuous.

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