Displaying similar documents to “Separation of ( n + 1 ) -families of sets in general position in 𝐑 n

On Boman's theorem on partial regularity of mappings

Tejinder S. Neelon (2011)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let Λ n × m and k be a positive integer. Let f : n m be a locally bounded map such that for each ( ξ , η ) Λ , the derivatives D ξ j f ( x ) : = d j d t j f ( x + t ξ ) | t = 0 , j = 1 , 2 , k , exist and are continuous. In order to conclude that any such map f is necessarily of class C k it is necessary and sufficient that Λ be not contained in the zero-set of a nonzero homogenous polynomial Φ ( ξ , η ) which is linear in η = ( η 1 , η 2 , , η m ) and homogeneous of degree k in ξ = ( ξ 1 , ξ 2 , , ξ n ) . This generalizes a result of J. Boman for the case k = 1 . The statement and the proof of a theorem of Boman for the case k = is...

Vector integral equations with discontinuous right-hand side

Filippo Cammaroto, Paolo Cubiotti (1999)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We deal with the integral equation u ( t ) = f ( I g ( t , z ) u ( z ) d z ) , with t I = [ 0 , 1 ] , f : 𝐑 n 𝐑 n and g : I × I [ 0 , + [ . We prove an existence theorem for solutions u L ( I , 𝐑 n ) where the function f is not assumed to be continuous, extending a result previously obtained for the case n = 1 .

On the H p - L q boundedness of some fractional integral operators

Pablo Rocha, Marta Urciuolo (2012)

Czechoslovak Mathematical Journal

Similarity:

Let A 1 , , A m be n × n real matrices such that for each 1 i m , A i is invertible and A i - A j is invertible for i j . In this paper we study integral operators of the form T f ( x ) = k 1 ( x - A 1 y ) k 2 ( x - A 2 y ) k m ( x - A m y ) f ( y ) d y , k i ( y ) = j 2 j n / q i ϕ i , j ( 2 j y ) , 1 q i < , 1 / q 1 + 1 / q 2 + + 1 / q m = 1 - r , 0 r < 1 , and ϕ i , j satisfying suitable regularity conditions. We obtain the boundedness of T : H p ( n ) L q ( n ) for 0 < p < 1 / r and 1 / q = 1 / p - r . We also show that we can not expect the H p - H q boundedness of this kind of operators.