Displaying similar documents to “ L p -inequalities for the laplacian and unique continuation”

On some singular integral operatorsclose to the Hilbert transform

T. Godoy, L. Saal, M. Urciuolo (1997)

Colloquium Mathematicae

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Let m: ℝ → ℝ be a function of bounded variation. We prove the L p ( ) -boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by T m f ( x ) = p . v . k ( x - y ) m ( x + y ) f ( y ) d y where k ( x ) = j 2 j φ j ( 2 j x ) for a family of functions φ j j satisfying conditions (1.1)-(1.3) given below.

Estimates of Fourier transforms in Sobolev spaces

V. Kolyada (1997)

Studia Mathematica

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We investigate the Fourier transforms of functions in the Sobolev spaces W 1 r 1 , . . . , r n . It is proved that for any function f W 1 r 1 , . . . , r n the Fourier transform f̂ belongs to the Lorentz space L n / r , 1 , where r = n ( j = 1 n 1 / r j ) - 1 n . Furthermore, we derive from this result that for any mixed derivative D s f ( f C 0 , s = ( s 1 , . . . , s n ) ) the weighted norm ( D s f ) L 1 ( w ) ( w ( ξ ) = | ξ | - n ) can be estimated by the sum of L 1 -norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

Partial differential operators depending analytically on a parameter

Frank Mantlik (1991)

Annales de l'institut Fourier

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Let P ( λ , D ) = | α | m a α ( λ ) D α be a differential operator with constant coefficients a α depending analytically on a parameter λ . Assume that the family { P( λ ,D) } is of constant strength. We investigate the equation P ( λ , D ) 𝔣 λ g λ where 𝔤 λ is a given analytic function of λ with values in some space of distributions and the solution 𝔣 λ is required to depend analytically on λ , too. As a special case we obtain a regular fundamental solution of P( λ ,D) which depends analytically on λ . This result answers a question of L. Hörmander. ...

On the representation of functions by orthogonal series in weighted L p spaces

M. Grigorian (1999)

Studia Mathematica

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It is proved that if φ n is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form k = 1 c k φ k ( x ) , where c k l q for all q>2, with the following properties: 1. For any p ∈ [1,2) and f L μ p [ 0 , 1 ] = f : ʃ 0 1 | f ( x ) | p μ ( x ) d x < there are numbers ɛ k , k=1,2,…, ɛ k = 1 or 0, such that l i m n ʃ 0 1 | k = 1 n ɛ k c k φ k ( x ) - f ( x ) | p μ ( x ) d x = 0 . 2. For every p ∈ [1,2) and f L μ p [ 0 , 1 ] there are a function g L 1 [ 0 , 1 ] with g(x) = f(x) on E and numbers δ k , k=1,2,…, δ k = 1 or 0,...

The null space of the ¯ -Neumann operator

Lars Hörmander (2004)

Annales de l’institut Fourier

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Let Ω be a complex analytic manifold of dimension n with a hermitian metric and C boundary, and let = ¯ ¯ * + ¯ * ¯ be the self-adjoint ¯ -Neumann operator on the space L 0 , q 2 ( Ω ) of forms of type ( 0 , q ) . If the Levi form of Ω has everywhere at least n - q positive or at least q + 1 negative eigenvalues, it is well known that Ker has finite dimension and that the range of is the orthogonal complement. In this paper it is proved that dim Ker = if the range of is closed and the Levi form of...

On invariant measures for power bounded positive operators

Ryotaro Sato (1996)

Studia Mathematica

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We give a counterexample showing that ( I - T * ) L ¯ L + = 0 does not imply the existence of a strictly positive function u in L 1 with Tu = u, where T is a power bounded positive linear operator on L 1 of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.

Weighted integrability and L¹-convergence of multiple trigonometric series

Chang-Pao Chen (1994)

Studia Mathematica

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We prove that if c j k 0 as max(|j|,|k|) → ∞, and | j | = 0 ± | k | = 0 ± θ ( | j | ) ϑ ( | k | ) | Δ 12 c j k | < , then f(x,y)ϕ(x)ψ(y) ∈ L¹(T²) and T ² | s m n ( x , y ) - f ( x , y ) | · | ϕ ( x ) ψ ( y ) | d x d y 0 as min(m,n) → ∞, where f(x,y) is the limiting function of the rectangular partial sums s m n ( x , y ) , (ϕ,θ) and (ψ,ϑ) are pairs of type I. A generalization of this result concerning L¹-convergence is also established. Extensions of these results to double series of orthogonal functions are also considered. These results can be extended to n-dimensional case. The aforementioned results generalize work of Balashov [1],...

On the absence of the one-sided Poincaré lemma in Cauchy-Riemann manifolds

Fabio Nicola (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Given an embeddable C R manifold M and a non-characteristic hypersurface S M we present a necessary condition for the tangential Cauchy-Riemann operator ¯ M on M to be locally solvable near a point x 0 S determined by S .