Abstrakt
Given a bounded measurable function sigma on R-n, we let T-sigma be the operator obtained by multiplication on the Fourier transform by sigma. Let 0 Pi(n)(i=1)(I - partial derivative(2)(i))(si/2) [Pi(n)(i=1)(psi) over cap(xi(i))sigma(2(j1) xi(1), ... , 2(jn) xi(n))] belongs to it uniformly in j(1), ... , j(n) is an element of Z, then T-sigma is bounded on L-p(R-n) when vertical bar 1/p - 1/2 vertical bar < s(1) and 1 < p < infinity.
In the case where s(i) not equal s(i+1) for all i, it was proved in [12] that the Lorentz space L-1/s1,L-1(R-n) is the function space sought. Here we address the significantly more difficult general case when for certain indices i we might have s(i) = s(i+1).
We obtain a version of the Marcinkiewicz multiplier theorem in which the space L-1/s1,L-1 is replaced by an appropriate Lorentz space associated with a certain concave function related to the number of terms among s(2), ... , s(n) that equal s(1). Our result is optimal up to an arbitrarily small power of the logarithm in the defining concave function of the Lorentz space. (c) 2021 Elsevier Inc.
All rights reserved.