Page 14 - Textos de Matemática Vol. 34

4 DAVID E. EDMUNDS
When p ∈ (1, ∞), the extremal functions f (ones for which ∥H0 f | Lp ((0, 1))∥ = γp ∥f | Lp((0, 1))∥ , f ̸= 0) have been determined (see, for example, [2]). To explain what they are, consider the function Fp : [0, 1] → R defined by
x dt
inverse of Fp by sinp, so that
sinp (x) =
(1 − tp)1/p . Initially sinp is defined on the interval [0,πp/2], where
(1 − tp)1/p . −1 x dt
Fp(x) =
Evidently Fp is strictly increasing and F2(x) = sin−1 x; by analogy we denote the
0
πp 1 dt
0
2 =
on this interval it is strictly increasing while sinp(0) = 0 and sinp(πp/2) = 1. We
extend it to [0, πp] by defining sinp x = sinp(πp −x) for x ∈ [πp/2, πp], to [−πp, πp] by
oddness, and finally to all of R by 2πp−periodicity. Now define cosp x := d sinp x: dx
this is an even, 2πp−periodic function which is odd about πp/2. When x ∈ [0, πp/2] we have, with y = sinp x,
cospx=(1−yp)1/p =(1−(sinpx)p)1/p,
from which it is apparent that cosp is strictly decreasing on [0, πp /2], cosp 0 = 1,
0
(1−tp)1/p;
cosp(πp/2) = 0 and
|cosp x|p + |sinp x|p = 1
This last equality holds for all x ∈ R because of the symmetry and periodicity of the functions. We may now define tanp(x) = sinp(x)/cosp(x). Of course, when p = 2 these functions are simply the standard trignometric functions. Further details of the properties of sinp and cosp (defined slightly differently) can be found in [16]. Note in particular that while when p = 2 we have the familiar addition formulae, such as
sin(x + y) = sin x cos y + cos x sin y,
at present no analogues of these are known when p ̸= 2. For the trignometric func- tions these properties follow directly from the property of the exponential function that
ex+y = exey, but there is nothing like this in the case p ̸= 2.
(2.7) πp = p
As for the constant πp, the change of variable t = s1/p shows that 21 2 2π
0
(1−s)−1/ps1/p−1ds= pB(1−1/p,1/p)= psin(π/p).