L. Hormander, Notions of convexity, Birkhauser, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of M there exists a potential function v satisfying = dd c v then for psh the function v is a true plurisubharmonic function. Thus such properties of plurisubharmonic functions as Hartogs lemma or the theorem saying that the weak convergence implies convergence in L p loc (see e.g. [H]) hold also for psh functions. The same goes for the convergence theorems, proved by Bedford and Taylor [BT2] which say that the Monge Amp ere measures associated to a convergent monotone sequence of plurisubharmonic functions converge. For a Borel set E ae M one can define a capacity cap (E) ....

....Hence, applying Holder s inequality we infer s n 1 N Gamman a j Z M j Gamma j jf j n ( Z M j Gamma j j q n ) 1=q ( Z M f p j n ) 1=p C( Z M j Gamma j j q n ) 1=q : Thus we have obtained the estimate from the statement. It is known (see e.g. [H]) that a sequence of plurisubharmonic functions which is weakly convergent converges also in L q loc for any 1 q 1. The same is true for psh functions if we apply the above statement for v j v where v is a local potential for . Therefore lim j 1 a j lim j 1 CN n s ....

L. Hormander, Notions of convexity, Birkhauser, 1996.

....0. Introduction. It is a well known property of plurisubharmonic functions that if a sequence u j 2 PSH( Omega Gamma tends to u 2 PSH( Omega Gamma in the sense of distribution theory then it is convergent in a much stronger sense; for instance, u j u in L p loc for any p 2 [p; 1) see e.g. [H]) However, the Monge Amp ere measures (dd c u j ) n may still have no limit (see [L] C] CK] To be sure that (dd c u j ) n converge as measures we need to know that u j u with respect to capacity [X] In the present paper we study the convergence of plurisubharmonic functions with ....

....(u Gamma u j ) Z Omega (f j ) dV are uniformly bounded. Using the assumptions on we can make M (M) arbitrarily small by taking M big enough. We choose M so that the first term on the right hand side (2.3) is less than ffl=2 for any j. Since u j u in L 1( Omega Gamma (see e.g. [H]) the other term is less than ffl=2 for j j 0 . Therefore a j (ffi) fflffi Gamman Gamma1 for j j 0 : Suppose for a while that E j (3ffi) were nonempty. Then, applying Lemma 1 we would get ffi (a j (ffi) fflffi Gamman Gamma1 ) j j 0 : Since, by the assumption on h we have lim ....

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L. Hormander, Notions of convexity, Birkhauser, 1996.

....bilinear forms r[s i ] by Q [s 1 ; s n ] 1 n X oe; 2Sn ( Gamma1) sgn( sgn(oe) r [s 1 ] oe(1) 1) r [s n ] oe(n) n) 6) where the sum is over all pairs of permutations on n elements. The operator Q is called the mixed discriminant of s 1 ; s n (see [Al2] and [Ho], Prop. 2.1.31) Proposition 3. 1) Q is symmetric: Q [f 1 ; fn ] Q [f oe 1 ; f oe n ] for any permutation oe; 2) Q[f 1 ; fn ] 0 for any f 1 ; fn with r[f i ] positive definite; 3) If r[f i ] is positive definite for i = 2; n, then Q[f ] Q ....

L. Hormander, "Notions of convexity", Birkhauser Boston 1994.

....hyperplane for Omega if y 2 Omega and Omega P y = We have the following technical lemmas. Lemma 1. Let y 2 Omega : Then there exists 2 R n with 6= 0 such that P y is a supporting hyperplane for Omega : Proof. This is well known, see for example Theorem 2.1. 10, page 44 of [16]. Lemma 2. Let y 2 Omega : Assume that P y is a supporting hyperplane for Omega : Then there exists 2 R; 6= 0 such that = dr(y) Proof. Let x 2 P y U: Since we have Omega P y = it follows that r(x) 0: Since r(y) 0; it follows that r has a minimum at y along the line with ....

L. Hormander, Notions of convexity, Birkhauser, 1994.

....the fundamental solution E(p; x) is the Fourier transform of the distribution p Gamma ( Gamma1 . 3 Hyperbolicity Cone The following fundamental result is first proved by Garding [7] There are at present many proofs of this [8] 16] Lemma 8.7.3) 1] Lemma 3.22 and Corollary 3. 23) and [17] (Prop. 2.1.31) It also follows immediately from Bochner s Tube Theorem [2] Chap. 5) see [8] We include, for readers convenience, and because the same kinds of arguments will be needed later, the following elementary proof which combines arguments from [1] and [17] Theorem 3.1 Let p 2 Hyp(d; ....

....and Corollary 3.23) and [17] Prop. 2.1.31) It also follows immediately from Bochner s Tube Theorem [2] Chap. 5) see [8] We include, for readers convenience, and because the same kinds of arguments will be needed later, the following elementary proof which combines arguments from [1] and [17]. Theorem 3.1 Let p 2 Hyp(d; m) Then, the cone K(p; d) is convex and K(p; d) K d where K d : fx 2 R n : t 7 p(x td) has negative rootsg = fx 2 R n : p(x td) 6= 0; t 0g: The polynomial p is positive on K, and equal to zero on the boundary of K. Furthermore, p is hyperbolic with ....

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Hormander, L. (1994). Notions of Convexity, Birkhauser, Boston.

....class of initial functions. We want to refer the readers to [4] 8] 14] 24] for the study of viscosity solutions, 1] 9] 17] 18] for multifuntions, 3] for quasiconvex conjugate, 2] 12] 17] 23] for minimax solutions, globally Lipschitz solutions, quasiclassical solutions. and [10] [11], 13] 19] for the facts concerning convex functions, difference of convex functions, semicontinuous functions and normal cones. 2. PRELIMINARIES In this section, we recall the definition of quasiconvex conjugates according to the view of the paper [3] and refer the readers to the book [9] ....

L. Hormander, Notions of Convexity, Birkhauser, (1994).

....the fundamental solution E(p; x) is the Fourier transform of the distribution p Gamma ( Gamma1 . 3 Hyperbolicity Cone The following fundamental result is first proved by Garding [6] There are at present many proofs of this [7] 14] Lemma 8.7.3) 1] Lemma 3.22 and Corollary 3. 23) and [15] (Prop. 2.1.31) It also follows immediately from Bochner s Tube Theorem [2] Chap. 5) see [7] We include, for readers convenience, and because the same kinds of arguments will be needed later, the following elementary proof which combines arguments from [1] and [15] Theorem 3.1 Let p 2 Hyp(d; ....

....and Corollary 3.23) and [15] Prop. 2.1.31) It also follows immediately from Bochner s Tube Theorem [2] Chap. 5) see [7] We include, for readers convenience, and because the same kinds of arguments will be needed later, the following elementary proof which combines arguments from [1] and [15]. Theorem 3.1 Let p 2 Hyp(d; m) Then, the cone K(p; d) is convex and K(p; d) K d where K d = fx 2 R n : t 7 p(x td) has negative rootsg = fx 2 R n : p(x td) 6= 0; t 0g: The polynomial p is positive on K, and equal to zero on the boundary of K. Furthermore, p is hyperbolic with ....

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Hormander, L. (1994). Notions of Convexity, Birkhauser, Boston, Basel, Berlin.

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Hormander, Lars, Notions of Convexity, Birkhauser, Boston, 1994.

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L. Hormander. Notions of Convexity. Birkhauser, Basel, 1994.

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L. Hormander, Notions of convexity, Birkhauser, Basel, 1994.

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