# On the theory of general partial differential operators

@article{Hrmander1955OnTT,
title={On the theory of general partial differential operators},
author={Lars H{\"o}rmander},
journal={Acta Mathematica},
year={1955},
volume={94},
pages={161-248}
}
• L. Hörmander
• Published 1 December 1955
• Mathematics
• Acta Mathematica
I I . Min ima l d i f fe ren t i a l ope ra to r s w i t h c o n s t a n t coeff ic ients . I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 N o t a t i o n s a n d f o r m a l p r o p e r t i e s of d i f fe ren t ia l ope ra to r s w i t h c o n s t a n t coeff ic ients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 E s t i m a t e s b y Lap lace t r a n s f o r m s . . . . . . . . . . . . . . . . . . . . . . 177 T h e d i f fe…
296 Citations
A metric result about the zeros of a complex polynomial ideal
Let us begin by listing some notations. We shall denote by K the field of complex numbers, by K[x] = K [ x 1 . . . . . x ~] a polynemial ring over ~K in n variables, and by K n the n-dimensional
On the dominance of partial differential operators II
holding for all u e V, where the || Ili, i = 0, 1, ..., q, are suitable norms. In fact, one can safely say that it is a rare investigation in partial differential equations which does not require
A characterization of hypoelliptic differential operators with variable coefficients
Let P be a linear differential operator with coefficients in C?01c() where 0 c Rn. We characterize the hypoelliptic operators in terms of the *-hypoelliptic operators. P is defined to be
Equality of minimal and maximal extensions of partial differential operators in _{}(ⁿ)
It is known [1] that if Q is a bounded domain, and P=P(D) is a linear partial differential operator with constant coefficients, then every weak solution in L2(0) with compact support in Q, is also a
Cauchy’s problem for systems of PDE with constant coefficients and semigroups of operators
The paper deals with Cauchy’s problem ∂ ∂t u(t, x) = P (D)u(t, x), u(0, x) = u0(x), t ≥ 0, x ∈ R, for C-valued u and P (D) = ∑ |α|≤pAαi (∂/∂x1) α1 · · · (∂/∂xn) where Aα are m ×m matrices with
Solving Pseudo-Differential Equations
In 1957, Hans Lewy constructed a counterexample showing that very simple and natural differential equations can fail to have local solutions. A geo- metric interpretation and a generalization of this
On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power
• Mathematics
• 2018
A linear differential operator P(x, D) = P(x1,... xn, D1,..., Dn) = ∑αγα(x)Dα with coefficients γα(x) defined in En is called formally almost hypoelliptic in En if all the derivatives DνξP(x, ξ) can
Hypoellipticity of Fediĭ’s type operators under Morimoto’s logarithmic condition
• Mathematics
Journal of Pseudo-Differential Operators and Applications
• 2019
We prove hypoellipticity of second order linear operators on \$\$\mathbb {R}^{n+m}\$\$Rn+m of the form \$\$L(x,y,D_x,D_y) = L_1(x,D_x) + g(x) L_2(y,D_y)\$\$L(x,y,Dx,Dy)=L1(x,Dx)+g(x)L2(y,Dy), where
On Local Solvability of Linear Partial Differential Equations
• Mathematics
• 1973
The title indicates more or less what the talk is going to be about. I t is going to be about the problem which is probably the most primitive in partial differential equations theory, namely to know
On local solvability of linear partial differential equations
The title indicates more or less what the talk is going to be about. I t is going to be about the problem which is probably the most primitive in partial differential equations theory, namely to know

## References

SHOWING 1-10 OF 32 REFERENCES
Linear hyperbolic partial differential equations with constant coefficients
I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 C h a p t e r I. P r o o f of t h e o r e m I . . . . . . . . . . . . . . . . . . . . . 9 C h a p t e r 2. H y p e
A NEW DECISION METHOD FOR ELEMENTARY ALGEBRA
A. Tarski [4] has given a decision method for elementary algebra. In essence this comes to giving an algorithm for deciding whether a given finite set of polynomial inequalities has a solution. Below
Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution
© Annales de l’institut Fourier, 1956, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions
M~LGRA~GE, Equat ions aux d6riv6es partieUes k coefficients constants. 1. Solution 616mentaire
• C. R. Acad. Sci. Paris,
• 1953
On general boundary problems for elliptic differential equations
• Trudy Moskov. Mat. Obs~.,
• 1952
THE EIGENFUNCTION EXPANSION THEOREM FOR THE GENERAL SELF-ADJOINT SINGULAR ELLIPTIC PARTIAL DIFFERENTIAL OPERATOR. I. THE ANALYTICAL FOUNDATION.
• F. Browder
• Mathematics, Medicine
Proceedings of the National Academy of Sciences of the United States of America
• 1954
G3~t~DING, Linear hyperbolic partial differential equations with constant coefficients
• Acta Math.,
• 1951
EIGENFUNCTION EXPANSIONS FOR SINGULAR ELLIPTIC OPERATORS. II. THE HILBERT SPACE ARGUMENT; PARABOLIC EQUATIONS ON OPEN MANIFOLDS.
• F. Browder
• Mathematics, Medicine
Proceedings of the National Academy of Sciences of the United States of America
• 1954
GANELIUS, On the remainder in a Tauberian theorem
• Kungl. FysiograJiska Sdllskapets iLund FSrhandlingar,
• 1954
I)E RHAM, Solution ~16mentaire d '6quations aux d6riv6es partielles du second ordre coefficients constants
• Colloque Henri Poincard (Octobre
• 1954