- and $^{∞}$-hypoellipticity of partial differential operators with constant Colombeau coefficients

Claudia Garetto

Displaying similar documents to “- and $^{\infty}$ -hypoellipticity of partial differential operators with constant Colombeau coefficients”

On the fundamental solution of the equation $∆^{p} u (X) + k (r) u (X) = 0$

J. Musiałek (1969)

Annales Polonici Mathematici

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The norm of the polynomial truncation operator on the unit disk and on [-1,1]

Tamás Erdélyi (2001)

Colloquium Mathematicae

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Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. $ₙ^{c}$ ) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $P ₙ \in ₙ^{c}$ of the form $P ₙ (z) : = \sum_{j = 0}^{n} a_{j} z_{j}$ , $a_{j} \in C$ , by $S ₙ (P ₙ) (z) : = \sum_{j = 0}^{n} a ̃_{j} z_{j}$ , $a ̃_{j} : = a_{j} | a_{j} | m i n | a_{j} |, 1$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $∥ S ₙ ∥_{\infty, \partial D}^{r e a l} : = s u p_{P ₙ \in ₙ} (m a x_{z \in \partial D} | S ₙ (P ₙ) (z) |) / (m a x_{z \in \partial D} | P ₙ (z) |)$ , $∥ S ₙ ∥_{\infty, \partial D}^{c o m p} : = s u p_{P ₙ \in ₙ^{c}} (m a x_{z \in \partial D} | S ₙ (P ₙ) (z) |) / (m a x_{z \in \partial D} | P ₙ (z) |$ . Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

Surjectivity of partial differential operators on ultradistributions of Beurling type in two dimensions

Thomas Kalmes (2010)

Studia Mathematica

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We show that if Ω is an open subset of ℝ², then the surjectivity of a partial differential operator P(D) on the space of ultradistributions $_{(ω)}^{'} (Ω)$ of Beurling type is equivalent to the surjectivity of P(D) on $C^{\infty} (Ω)$ .

Multiplication operators on $L (L_{p})$ and $ℓ_{p}$ -strictly singular operators

William Johnson, Gideon Schechtman (2008)

Journal of the European Mathematical Society

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A classification of weakly compact multiplication operators on $L (L_{p}),$ 1<p< $, i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f$ p $- s t r i c t l y s i n g u l a r o p e r a t o r s, a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l$ p $- s t r i c t l y s i n g u l a r o p e r a t o r s o n$ Lp $. T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r$ T $o n$ Lp $,$ 1<p<2 $, i s$ p $- s t r i c t l y s i n g u l a r a n d$ T|X $i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e$ X $o f$ Lp $, t h e n$ X $e m b e d s i n t o$ Lr $f o r a l l$ r<2 $, b u t$ X $n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e .$ It is also shown that if $T$ is convolution by a biased coin on $L_{p}$ of the Cantor group, $1 \leq p < 2$ , and $T_{| X}$ is an isomorphism for some reflexive subspace $X$ of $L_{p}$ , then $X$ is isomorphic to a Hilbert space. The case $p = 1$ answers a question asked by Rosenthal in 1976.

2-summing multiplication operators

Dumitru Popa (2013)

Studia Mathematica

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Let 1 ≤ p < ∞, $= {(X ₙ)}_{n \in ℕ}$ be a sequence of Banach spaces and $l_{p} ()$ the coresponding vector valued sequence space. Let $= {(X ₙ)}_{n \in ℕ}$ , $= {(Y ₙ)}_{n \in ℕ}$ be two sequences of Banach spaces, $= {(V ₙ)}_{n \in ℕ}$ , Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{} : l_{p} () \to l_{q} ()$ by $M_{} ({(x ₙ)}_{n \in ℕ}) : = {(V ₙ (x ₙ))}_{n \in ℕ}$ . We give necessary and sufficient conditions for $M_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...

The natural operators $T_{| ℳ f ₙ} ⇝ T * T^{r }$ and $T_{| ℳ f ₙ} ⇝ Λ ² T T^{r *}$

W. M. Mikulski (2002)

Colloquium Mathematicae

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Let r and n be natural numbers. For n ≥ 2 all natural operators $T_{| ℳ f ₙ} ⇝ T * T^{r *}$ transforming vector fields on n-manifolds M to 1-forms on $T^{r *} M = J^{r} (M, ℝ) ₀$ are classified. For n ≥ 3 all natural operators $T_{| ℳ f ₙ} ⇝ Λ ² T * T^{r *}$ transforming vector fields on n-manifolds M to 2-forms on $T^{r *} M$ are completely described.

On the range-kernel orthogonality of elementary operators

Said Bouali, Youssef Bouhafsi (2015)

Mathematica Bohemica

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Let $L (H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$ . For $A, B \in L (H)$ , the generalized derivation $δ_{A, B}$ and the elementary operator $Δ_{A, B}$ are defined by $δ_{A, B} (X) = A X - X B$ and $Δ_{A, B} (X) = A X B - X$ for all $X \in L (H)$ . In this paper, we exhibit pairs $(A, B)$ of operators such that the range-kernel orthogonality of $δ_{A, B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $Δ_{A, B}$ with respect to the wider class of unitarily invariant...

On certain general integral operators of analytic functions

B. A. Frasin (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain new sufficient conditions for the operators $F_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ and $G_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ to be univalent in the open unit disc $𝒰$ , where the functions $f_{1}, f_{2}, . . ., f_{n}$ belong to the classes $S^{*} (a, b)$ and $𝒦 (a, b)$ . The order of convexity for the operators $F_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ and $G_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ is also determined. Furthermore, and for $β = 1$ , we obtain sufficient conditions for the operators $F_{n} (z)$ and $G_{n} (z)$ to be in the class $𝒦 (a, b)$ . Several corollaries and consequences of the main results are also considered.

On spirallike functions of order $α$ and type $β$ with fixed second coefficient

M. K. Aouf (1988)

Matematički Vesnik

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Some types of points in $N^{*}$

Gryzlov, A.

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The multiplicity of the zero at 1 of polynomials with constrained coefficients

Peter Borwein, Tamás Erdélyi, Géza Kós (2013)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let $κ_{p} (n, L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P (x) = \sum_{j = 0}^{n} a_{j} x^{j}$ , $| a_{0} | \geq L (\sum_{j = 1}^{n} | a_{j} |^{p}$ 1/p $,$ aj ∈ ℂ $,$ such that ${(x - 1)}^{k}$ divides P(x). For n ∈ ℕ and L > 0 let $κ_{\infty} (n, L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P (x) = \sum_{j = 0}^{n} a_{j} x^{j}$ , $| a_{0} | \geq L m a x_{1 \leq j \leq n} | a_{j} |$ , $a_{j} \in ℂ$ , such that ${(x - 1)}^{k}$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_{1} \sqrt (n / L) - 1 \leq κ_{\infty} (n, L) \leq c_{2} \sqrt (n / L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

Linear maps preserving $A$ -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

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Let $ℋ$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ (ℋ)$ and $ℬ_{A} (ℋ)$ the sub-algebra of $ℬ (ℋ)$ of all $A$ -self-adjoint operators. Assume $φ : ℬ_{A} (ℋ)$ onto itself is a linear continuous map. This paper shows that if $φ$ preserves $A$ -unitary operators such that $φ (I) = P$ then $ψ$ defined by $ψ (T) = P φ (P T)$ is a homomorphism or an anti-homomorphism and $ψ (T^{♯}) = ψ {(T)}^{♯}$ for all $T \in ℬ_{A} (ℋ)$ , where $P = A^{+} A$ and $A^{+}$ is the Moore-Penrose inverse of $A$ . A similar result is also true if $φ$ preserves $A$ -quasi-unitary operators in both directions such that there...