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GT
science forum beginner

Joined: 06 Oct 2005
Posts: 4

Posted: Thu May 25, 2006 3:00 pm    Post subject: characterization of W^{1,p} norms

Hi,

I have a question concerning W^{1,p} norms:

is it true that the W^{1,p} norm of a function $v$ can be
characterized as

\int u v + \nabla u \nabla v

with $u$ ranging in W^{1,q}?

(q is the dual exponent of p)

A possible proof is the followinig: a standard representation theorem
for the dual of W^{1,p} in N dimensions states that any dual F can be
represented through a N+1 ple of L^q functions f_i, so that

<F,v>=\int f_0 u + (f_1,...f_N) \nabla u

So, if I want to compute the norm of u, I must test its gradient
against all vector fields. On the other hand the vector field
(f_1,...,f_N) can be decomposed into a sum of a solenoidal field and a
gradient. When testing with such decomposition, the solenoidal field is
calceled out, and we remain with the scalar product of two gradients.

Any opinions?

joe

Thanks.

Joe
Stephen Montgomery-Smith1
science forum Guru

Joined: 01 May 2005
Posts: 487

Posted: Sat May 27, 2006 4:00 am    Post subject: Re: characterization of W^{1,p} norms

GT wrote:
 Quote: Hi, I have a question concerning W^{1,p} norms: is it true that the W^{1,p} norm of a function $v$ can be characterized as \int u v + \nabla u \nabla v with $u$ ranging in W^{1,q}? (q is the dual exponent of p) A possible proof is the followinig: a standard representation theorem for the dual of W^{1,p} in N dimensions states that any dual F can be represented through a N+1 ple of L^q functions f_i, so that F,v>=\int f_0 u + (f_1,...f_N) \nabla u So, if I want to compute the norm of u, I must test its gradient against all vector fields. On the other hand the vector field (f_1,...,f_N) can be decomposed into a sum of a solenoidal field and a gradient. When testing with such decomposition, the solenoidal field is calceled out, and we remain with the scalar product of two gradients. Any opinions?

I haven't worked out the solution, but I have a feeling that the

If a vector f with components in L_p(R^N) is decomposed into its
gradient part nabla u and its solenoidal part v (i.e. the Hodge
decomposition), then for 1<p<infinity, ||f||_p is equivalent (i.e. the
ratio is bounded above and below by constants depending only upon p and
possibly N) to ||nabla u||_p + ||v||_p.

This follows basically because the decomposition is a singular integral
- indeed it can be made up from compositions of so called Riesz
tranforms. The result may in effect be found in
http://www.ams.org/mathscinet-getitem?mr=1442167
but this is somewhat using a sledgehammer to crack a nut.

My guess is that the result you are looking for is false if p=1 or
p=infinity.

I hope this helps.

Stephen
don11154
science forum beginner

Joined: 05 Jul 2005
Posts: 39

Posted: Fri Jul 07, 2006 9:13 pm    Post subject: Re: characterization of W^{1,p} norms

 Quote: Hi, I have a question concerning W^{1,p} norms: is it true that the W^{1,p} norm of a function $v$ can be characterized as \int u v + \nabla u \nabla v with $u$ ranging in W^{1,q}? (q is the dual exponent of p) A possible proof is the followinig: a standard representation theorem for the dual of W^{1,p} in N dimensions states that any dual F can be represented through a N+1 ple of L^q functions f_i, so that F,v>=\int f_0 u + (f_1,...f_N) \nabla u So, if I want to compute the norm of u, I must test its gradient against all vector fields. On the other hand the vector field (f_1,...,f_N) can be decomposed into a sum of a solenoidal field and a gradient. When testing with such decomposition, the solenoidal field is calceled out, and we remain with the scalar product of two gradients. Any opinions? Thanks in advance. joe Thanks. Joe

I was flipping through a book and saw something along the lines of

sup( \int_\Omega grad u grad v ) >= c || u ||

where the norm is the W_0^{1,p}(\Omega) one and we are supping over v in W_0^{1,q} with norm of v <=1.

Here q is conjugate of p. They said result followed

from some Zygmund - Calderon theorem but I never figure out which one.

If you have this figure out I would really like to know some of the details or where to look.

I am in some open , bounded subset Omega of R^n .

thanks

craig

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From: Aaron Bergman <abergman@physics.utexas.edu>
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"Agusti Roig" <soquiso@hotmail.com> wrote:

 Quote: Aaron Bergman ha escrit: I've been working on doing some computations in Hochschild cohomology, and all the definitions I've seen of the Gerstenhaber bracket define it either hopelessly abstractly or in terms of the bar resolution. I have a different resolution I've been using to compute HH^*, and I was hoping to be able to compute the Gerstenhaber bracket (or HH^2 to a specific formal deformation of the algebra) without having to compute an explicit quasi-isomorphism to the bar resolution. Any ideas or references would be appreciated. I don't know if I understand the problem: do you mean you have a resolution that allows you the computation of the Gerstenhaber bracket and you want an explicit quasi-isomorphism with the bar resolution because your are not sure that your resolution actually computes Hochschild cohomology?

No. I have an explicit projective resolution that isn't the bar
resolution. I was hoping to be able to compute the Gerstenhaber bracket
from it, but judging from the other response, I guess that's not so
likely.

The parenthetical comment is somewhat unclear, so, to elaborate, I was
also hoping to describe the explicit deformation for a given element of
HH^2 using my resolution, but perhaps that is also hard.

Thanx,
Aaron
don11154
science forum beginner

Joined: 05 Jul 2005
Posts: 39

Posted: Fri Jul 07, 2006 9:13 pm    Post subject: Re: characterization of W^{1,p} norms

 Quote: Hi, I have a question concerning W^{1,p} norms: is it true that the W^{1,p} norm of a function $v$ can be characterized as \int u v + \nabla u \nabla v with $u$ ranging in W^{1,q}? (q is the dual exponent of p) A possible proof is the followinig: a standard representation theorem for the dual of W^{1,p} in N dimensions states that any dual F can be represented through a N+1 ple of L^q functions f_i, so that F,v>=\int f_0 u + (f_1,...f_N) \nabla u So, if I want to compute the norm of u, I must test its gradient against all vector fields. On the other hand the vector field (f_1,...,f_N) can be decomposed into a sum of a solenoidal field and a gradient. When testing with such decomposition, the solenoidal field is calceled out, and we remain with the scalar product of two gradients. Any opinions? Thanks in advance. joe Thanks. Joe

I was flipping through a book and saw something along the lines of

sup( \int_\Omega grad u grad v ) >= c || u ||

where the norm is the W_0^{1,p}(\Omega) one and we are supping over v in W_0^{1,q} with norm of v <=1.

Here q is conjugate of p. They said result followed

from some Zygmund - Calderon theorem but I never figure out which one.

If you have this figure out I would really like to know some of the details or where to look.

I am in some open , bounded subset Omega of R^n .

thanks

craig
Stephen Montgomery-Smith1
science forum Guru

Joined: 01 May 2005
Posts: 487

Posted: Tue Jul 11, 2006 6:02 am    Post subject: Re: characterization of W^{1,p} norms

don wrote:
 Quote: Hi, I have a question concerning W^{1,p} norms: is it true that the W^{1,p} norm of a function $v$ can be characterized as \int u v + \nabla u \nabla v with $u$ ranging in W^{1,q}? (q is the dual exponent of p) A possible proof is the followinig: a standard representation theorem for the dual of W^{1,p} in N dimensions states that any dual F can be represented through a N+1 ple of L^q functions f_i, so that F,v>=\int f_0 u + (f_1,...f_N) \nabla u So, if I want to compute the norm of u, I must test its gradient against all vector fields. On the other hand the vector field (f_1,...,f_N) can be decomposed into a sum of a solenoidal field and a gradient. When testing with such decomposition, the solenoidal field is calceled out, and we remain with the scalar product of two gradients. Any opinions? Thanks in advance. joe Thanks. Joe I was flipping through a book and saw something along the lines of sup( \int_\Omega grad u grad v ) >= c || u || where the norm is the W_0^{1,p}(\Omega) one and we are supping over v in W_0^{1,q} with norm of v <=1. Here q is conjugate of p. They said result followed from some Zygmund - Calderon theorem but I never figure out which one. If you have this figure out I would really like to know some of the details or where to look. I am in some open , bounded subset Omega of R^n . thanks craig

Here are ideas.

What you need to show is that if u is a vector field on Omega, and you
decompose it using the Hodge decomposition as nabla p + v, where div v
= 0, then ||v||_p <= C ||u||. You work out what the operator is - on
R^n it would be some kind of composition of Riesz projections, but on
Omega it would be something different, but still you can probably
calculate that it satisfies the hypothesis of the Calderon-Zygmund
Theorem. (Incidently I have a feeling that the smoothness of the
boundary of Omega will play a role, as I recall seeing somewhere that
it isn't necessarily true for all p in (1,infty) if the boundary of
Omega only satisfies Lipschitz boundary conditions.)

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