The Sobolev space 
,
for an open subset 
with 
, 
, and nonnegative integer 
, is defined by
 |
(1)
|
where 
is a vector of nonnegative
integers, 
,
and the derivatives 
are taken in a weak sense.
When endowed with the norm
 |
(2)
|
is a Banach
space.
In the special case 
,
is denoted by 
. This space is a Hilbert
space for the inner product
 |
(3)
|
Sobolev spaces play an important role in the theory of partial differential equations.
On 
, noninteger orders can be defined
using the Bessel potential space convention.
In particular, with 
,
the norm of 
is
 |
(4)
|
This fractional regularity is used in the local smoothing conjecture and the Schrödinger pointwise convergence problem.
Sum spaces of Sobolev spaces can behave differently from their gradient-constrained analogs. The sum norm is 
.
Zhang (2026) constructed smooth real-valued mean-zero functions 
on the 
-torus, for every 
and integer 
, such that 
but 
, with positive constants
independent of 
and 
.
The spaces in these norms are over the same 
-torus. This disproves the proposed
Bourgain-Brezis sum-space estimate controlling the latter norm by the former. Zhang
(2026) reports AI assistance with exploratory calculations and checking the proof.
