Bound for mixed partial derivatives

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#25

From: Enrico Talinucci
Date: Fri, 28 Jan 2000 00:00

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Hi everybody!

I'm in trouble with the following problem.

f: R^2 -> R is an infinitely differentiable function. I know an upper
bound for the modulus of its pure partial derivatives, i.e.
|d^n f/dx^n| and |d^n f/dy^n|, for every (x,y) \in R^2. I need to find a
bound on its mixed p.d., d^(k+h) f /(dx^k dy^h).


Now I'll detail the question.
The data is:

   |d^k f/dx^k|  < A_x ^k ||(x,y)||
   |d^k f/dy^k|  < A_y ^k ||(x,y)||

for every (x,y) and every k, where A_x, A_y are known constants
and ||..|| is a norm on R^2, let's say the Euclidean norm. The
inequalities holds for k=0 too.
The target is finding a bound B(k, h, A_x, A_y) so that:

   |d^(k+h) f /(dx^k dy^h)| < B(k, h, A_x, A_y) ||(x,y)||

for every (x,y), k, h.


An hypotesis could be:

  B = A_x^k  A_y^h

I couldn't prove it neither find a counterexample. It's just to give an
idea of what I'm looking for.


Does such a bound exist? If so, can you help me to find it? If not,
can you give me a counterexample, i.e. a function who satisfies the
given bounds, but for which at least one m.p.d. is unbounded with
respect to ||(x,y)||?


Thanks in advance for your help.

Enrico Talinucci

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From: "Enrico Talinucci" <talinucci@tin.it>
Subject: Bound for mixed partial derivatives
Date: 2000/01/28
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