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Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains.

Martin Dindos, Marius Mitrea (2002)

Publicacions Matemàtiques

Extending recent work for the linear Poisson problem for the Laplacian in the framework of Sobolev-Besov spaces on Lipschitz domains by Jerison and Kenig [16], Fabes, Mendez and Mitrea [9], and Mitrea and Taylor [30], here we take up the task of developing a similar sharp theory for semilinear problems of the type Δu - N(x,u) = F(x), equipped with Dirichlet and Neumann boundary conditions.

Separately superharmonic functions in product networks

Victor Anandam (2015)

Annales Polonici Mathematici

Let X×Y be the Cartesian product of two locally finite, connected networks that need not have reversible conductance. If X,Y represent random walks, it is known that if X×Y is recurrent, then X,Y are both recurrent. This fact is proved here by non-probabilistic methods, by using the properties of separately superharmonic functions. For this class of functions on the product network X×Y, the Dirichlet solution, balayage, minimum principle etc. are obtained. A unique integral representation is given...

Sets of determination for parabolic functions on a half-space

Jarmila Ranošová (1994)

Commentationes Mathematicae Universitatis Carolinae

We characterize all subsets M of n × + such that sup X n × + u ( X ) = sup X M u ( X ) for every bounded parabolic function u on n × + . The closely related problem of representing functions as sums of Weierstrass kernels corresponding to points of M is also considered. The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References. As a by-product the question of representability of probability continuous distributions as sums of multiples of normal distributions is investigated.

Sets of determination for solutions of the Helmholtz equation

Jarmila Ranošová (1997)

Commentationes Mathematicae Universitatis Carolinae

Let α > 0 , λ = ( 2 α ) - 1 / 2 , S n - 1 be the ( n - 1 ) -dimensional unit sphere, σ be the surface measure on S n - 1 and h ( x ) = S n - 1 e λ x , y d σ ( y ) . We characterize all subsets M of n such that inf x n u ( x ) h ( x ) = inf x M u ( x ) h ( x ) for every positive solution u of the Helmholtz equation on n . A closely related problem of representing functions of L 1 ( S n - 1 ) as sums of blocks of the form e λ x k , . / h ( x k ) corresponding to points of M is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References.

Sharp L log α L inequalities for conjugate functions

Matts Essén, Daniel F. Shea, Charles S. Stanton (2002)

Annales de l’institut Fourier

We give a method for constructing functions φ and ψ for which H ( x , y ) = φ ( x ) - ψ ( y ) has a specified subharmonic minorant h ( x , y ) . By a theorem of B. Cole, this procedure establishes integral mean inequalities for conjugate functions. We apply this method to deduce sharp inequalities for conjugates of functions in the class L log α L , for - 1 α < . In particular, the case α = 1 yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in L log L . We also apply the method to produce a new proof of the...

Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Adam Osękowski (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

Let ( h k ) k 0 be the Haar system on [0,1]. We show that for any vectors a k from a separable Hilbert space and any ε k [ - 1 , 1 ] , k = 0,1,2,..., we have the sharp inequality | | k = 0 n ε k a k h k | | W ( [ 0 , 1 ] ) 2 | | k = 0 n a k h k | | L ( [ 0 , 1 ] ) , n = 0,1,2,..., where W([0,1]) is the weak- L space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound | | Y | | W ( Ω ) 2 | | X | | L ( Ω ) , where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

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