We prove in Chapter 1 that for every integer n ≥ 3 the n-sublinear map (n/a) satisfies no Lp estimates provided (n/a) satisfies (n/a) for some (n/a) with distinct entries and α ∈ R with qj ≠ –α for all 1 ≤ j ≤ n. Furthermore, if n ≥ 5 and (n/a) satisfies (n/a) for some (n/a) with distinct, non-zero entries such that qjα ≠ –1 or all 1 ≤ j ≤ n, it is shown that there is a symbol (n/a) adapted to the hyperplane (n/a) and supported in (n/a) for which the associated n-linear multiplier also satisfies no Lp estimates. Next, we construct a Hörmander-Marcinkiewicz symbol (n/a), which is a paraproduct of (&phis;,ψ) type, such that the trilinear operator Tm whose symbol m is (n/a) satisfies no Lp estimates. Finally, we state a converse to a theorem of C. Muscalu, T. Tao, and C. Thiele concerning estimates for multipliers with subspace singularities of dimension at least half of the total space dimension using Riesz kernels in the spirit of C. Muscalu's recent work. Specifically, for every pair of integers (n/a) we construct an explicit collection C of uncountably many d-dimensional non-degenerate subspaces of Rn such that for each (n/a) there is an associated symbol mΓ adapted to Γ in the Mikhlin-Hörmander sense and supported in (n/a) for which the associated multilinear multiplier TmΓ is unbounded.
|School Location:||United States -- New York|
|Source:||DAI-B 78/04(E), Dissertation Abstracts International|
|Keywords:||Fourier analysis, Multiplier estimates, Simplex operators, Unbounded multipliers|
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