Abstract
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.
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