In the field of mathematical analysis, an interpolation inequality is an inequality of the form
\|u0\|0\leqC\|u1
\alpha1 | |
\| | |
1 |
\|u2
\alpha2 | |
\| | |
2 |
...\|un
\alphan | |
\| | |
n |
, n\geq2,
where for
0\leqk\leqn
uk
Xk
\| ⋅ \|k
\alphak
C
u0,..,un
u0=u1= … =un
u0,..,un
The main applications of interpolation inequalities lie in fields of study, such as partial differential equations, where various function spaces are used. An important example are the Sobolev spaces, consisting of functions whose weak derivatives up to some (not necessarily integer) order lie in Lp spaces for some p. There interpolation inequalities are used, roughly speaking, to bound derivatives of some order with a combination of derivatives of other orders. They can also be used to bound products, convolutions, and other combinations of functions, often with some flexibility in the choice of function space. Interpolation inequalities are fundamental to the notion of an interpolation space, such as the space
Ws,p
sth
Lp
s | |
B | |
p,q |
(\Omega)
A simple example of an interpolation inequality — one in which all the are the same, but the norms are different — is Ladyzhenskaya's inequality for functions
u:R2\rarrR
\int | |
R2 |
|u(x)|4dx\leq2
\int | |
R2 |
|u(x)|2dx
\int | |
R2 |
|\nablau(x)|2dx,
i.e.
\|u
\| | |
L4 |
\leq\sqrt[4]{2}\|u
1/2 | |
\| | |
L2 |
\|\nablau
1/2 | |
\| | |
L2 |
.
A slightly weaker form of Ladyzhenskaya's inequality applies in dimension 3, and Ladyzhenskaya's inequality is actually a special case of a general result that subsumes many of the interpolation inequalities involving Sobolev spaces, the Gagliardo-Nirenberg interpolation inequality.[3]
The following example, this one allowing interpolation of non-integer Sobolev spaces, is also a special case of the Gagliardo-Nirenberg interpolation inequality.[4] Denoting the
L2
Hk=Wk,2
u\inHm
\|u\| | |
H\ell |
\leq
| ||||
\|u\| | ||||
Hk |
| ||||
\|u\| | ||||
Hm |
.
The elementary interpolation inequality for Lebesgue spaces, which is a direct consequence of the Hölder's inequality reads: for exponents
1\leqp\ler\leq\leinfty
f\inLp(X,\mu)\capLq(X,\mu)
Lr(X,\mu),
\|f\| | |
Lr |
\leq
t | |
\|f\| | |
Lp |
1-t | |
\|f\| | |
Lq |
,
where, in the case of
p<q<infty,
r
r=tp+(1-t)q
t:= | q-r |
q-p |
1-t= | r-p |
q-p |
p<q=infty
r
r= | pt |
t:= | pr |
1-t= | r-p |
r. |
An example of an interpolation inequality where the elements differ is Young's inequality for convolutions.[5] Given exponents
1\leqp,q,r\leqinfty
\tfrac{1}{p}+\tfrac{1}{q}=1+\tfrac{1}{r}
f\inLp, g\inLq
Lr
\|f*
g\| | |
Lr |
\leq
\|f\| | |
Lp |
\|g\| | |
Lq |
.