The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the re...

6 Lie Derivatives and the Commutator Revisited 21 6.1 Integral Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2 Congruence of Curves . . . . . . . . . . . . . . . . . . . . . . . 21 6.3 The Commutator Revisited: A Geometric Interpretation . . . 22 6.4 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.5 Lie Derivatives of a Function . . . . . . . . . . . ....

Let I denote the commutator ideal in the free associative algebra on m variables over an arbitrary field. In this article we prove there are exactly m! finite Gröbner bases for I , and uncountably many infinite Gröbner bases for I with respect to total division orderings. In addition, for m = 3 we give a complete description of its universal Gröbner basis. Let A be a finite set and let K be a f...

We prove the regularity of weak 1/2−harmonic maps from the real line into a sphere. The key point in our result is first a formulation of the 1/2−harmonic map equation in the form of a non-local linear Schrödinger type equation with a 3-terms commutators in the right-hand-side . We then establish a sharp estimate for these 3-commutators.

Let K be the kernel of an epimorphism G→ Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3 or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group G, then any homomorphism from K onto the symmetric group S2 (resp. Z3) lifts to a homomorphism onto S3 (resp. alternating group A4).

For a given prime p, what is the smallest integer n such that there exists a group of order p in which the set of commutators does not form a subgroup? In this paper we show that n = 6 for any odd prime and n = 7 for p = 2.

The fact that symbols in the modulation space M1,1 generate pseudo-differential operators of the trace class was first mentioned by Feichtinger and the proof was given by Gröchenig [12]. In this paper, we show that the same is true if we replace M1,1 by more general α-modulation spaces which include modulation spaces (α = 0) and Besov spaces (α = 1) as special cases. The result with α = 0 corre...

We show that if an operator T is bounded on weighted Lebesgue space L(w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator [b, T ] with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that the kth-order commutator T k b = [b, T k−1 b ] will obey a bound that is a power (k + 1) of the A2 consta...