Ü Bounds for Spectral Multipliers on Nilpotent Groups
نویسنده
چکیده
A criterion is given for the Lp boundedness of a class of spectral multiplier operators associated to left-invariant, homogeneous subelliptic second-order differential operators on nilpotent Lie groups, generalizing a theorem of Hörmander for radial Fourier multipliers on Euclidean space. The order of differentiability required is half the homogeneous dimension of the group, improving previous results in the same direction. The Hörmander multiplier theorem on the group Rn gives a sufficient condition for a Fourier multiplier operator ff(Ç) = m(Ç)f(Ç), with m e L°°(R"), to extend to an operator bounded on Lp(Rn) for p e (1, oo) and of weak type (1, 1). Write R+ = (0, oo) and fix any auxiliary function n e C0°°(R+), not identically zero. For ß > 0 let L2 denote the L2 Sobolev space of order ß . Then the condition is that sup||z/(|-|)zn(i-)||L2(Rn) zz/2 [H]. It is actually independent of the choice of n. Specialized to radial multipliers m(|^|), the hypothesis becomes (1) sup||z/(-)m(i-)||L2(E) n/2. This order of differentiability is essentially optimal, even in the radial case. For if m(s) = sn for 5 e M+, where tel, then m(x) = cn x\x\~n~n where \cn T| ~ |t|"/2 as |t| -> oo [S2, pp. 51-52]. Thus the multiplier operator is readily seen to be of weak type (1,1) with a bound which grows at least as fast as |t|"' , which is comparable to the bound in (1) with a = zz/2 . Received by the editors April 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 42B15, 43A22; Secondary 35P99.
منابع مشابه
Bounds for the dimension of the $c$-nilpotent multiplier of a pair of Lie algebras
In this paper, we study the Neumann boundary value problem of a class of nonlinear divergence type diffusion equations. By a priori estimates, difference and variation techniques, we establish the existence and uniqueness of weak solutions of this problem.
متن کاملSharp Bounds on the PI Spectral Radius
In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.
متن کاملOn a conjecture of a bound for the exponent of the Schur multiplier of a finite $p$-group
Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $exp(G)$ and let $m=lfloorlog_pk floor$. We show that $exp(M^{(c)}(G))$ divides $exp(G)p^{m(k-1)}$, for all $cgeq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that $exp( M(G))$ divides $exp(G)$, for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result is an improvement...
متن کاملScaled Relators and Dehn Functions for Nilpotent Groups
A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphism. In this paper, we present combinatorial methods for finding such bounds. U...
متن کامل