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For n > 1, there are additional conditions, i.e. Sp(2n, F) is then a proper subgroup of SL(2n, F). Typically, the field F is the field of real numbers R or complex numbers C. In these cases Sp(2n, F) is a real or complex Lie group of real or complex dimension n(2n + 1), respectively. These groups are connected but non-compact.
Property (c) says the operator M is bounded on L p (R n); it is clearly true when p = ∞, since we cannot take an average of a bounded function and obtain a value larger than the largest value of the function. Property (c) for all other values of p can then be deduced from these two facts by an interpolation argument.
An alternative version uses the fact that the Poisson distribution converges to a normal distribution by the Central Limit Theorem. [5]Since the Poisson distribution with parameter converges to a normal distribution with mean and variance , their density functions will be approximately the same:
This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from L p (R d) to itself for p > 1. That is, if f ∈ L p (R d) then the maximal function Mf is weak L 1-bounded and Mf ∈ L p (R d). Before stating the theorem more precisely, for simplicity, let {f > t} denote the set {x | f(x) > t}. Now we have:
f has degree at most p − 2 (since the leading terms cancel), and modulo p also has the p − 1 roots 1, 2, ..., p − 1. But Lagrange's theorem says it cannot have more than p − 2 roots. Therefore, f must be identically zero (mod p), so its constant term is (p − 1)! + 1 ≡ 0 (mod p). This is Wilson's theorem.
If the right-hand side is finite, equality holds if and only if f(x) = 0 almost everywhere. Hardy's inequality was first published and proved (at least the discrete version with a worse constant) in 1920 in a note by Hardy. [1] The original formulation was in an integral form slightly different from the above.
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Thus, n p = [G : N G (P)], and it follows that this number is a divisor of [G : P] = | G |/q. Now let P act on Ω by conjugation, and again let Ω 0 denote the set of fixed points of this action. Let Q ∈ Ω 0 and observe that then Q = xQx −1 for all x ∈ P so that P ≤ N G (Q).