Robert Kesler

We prove in Chapter 1 that for every integer n$\geq$3 the n -sublinear map nˆfj ([xi]j) e2 [pi]ix[xi]j d[xi]n C $\alpha$:(f1,...,fn) $\Rightarrow$ sup M [xi]$\cdot$$\alpha$>0,[xi]n<M j=1 [-]1 satisfies no Lp estimates provided $\alpha$∈ Rn satisfies $\alpha$j=$\alpha$+qj for some q ∈ Qn with distinct entries and $\alpha$∈ R with qj = [-]$\alpha$ for all 1 $\leq$ j $\leq$ n. Furthermore, if [-]1 2 n $\geq$ 5 and $\alpha$∈Rn satisfies $\alpha$j = qj + $\alpha$qj for some q ∈ Qn with distinct, non-zero entires such that qj$\alpha$= [-]1 for all 1 $\leq$ j $\leq$ n, it is shown that there is a symbol m:Rn $\Rightarrow$ C adapted to the hyperplane $\Gamma$ a=supported in [xi]: dist([xi], $\Gamma$ $\alpha$) [xi] ∈ Rn:n j=1 [xi]j $\cdot$ aj = 0 and 1 for which the associated n-linear multiplier also satisfies no Lp estimates. Next, we construct a HOrmander-Marcinkiewicz symbol o [PI] : R2$\Rightarrow$R, which is a paraproduct of ([phi], [psi]) type, such that the trilinear operator Tm whose symbol m is given by m([xi]1 , [xi]2 , [xi]3 ) = sgn([xi]1 + [xi]2 )[PI]([xi]2 , [xi]3 ) satisfies no Lp estimates. Finally, we establish 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 (d, n) s.t. n 2 + 3 2 $\leq$ d<n we construct an explicit collection C of uncountably many d-dimensional non-degenerate subspaces of Rn such that for each $\Gamma$∈C there is an associated symbol m$\Gamma$ adapted to $\Gamma$ in the Mikhlin-HOrmander sense and o supported in [xi] : dist([xi], $\Gamma$ 1 for which the associated multilinear multiplier Tm$\Gamma$ is unbounded. In Chapter 2, we consider for each 2<p $\leq$ $\infty$ the space Wp (R) = ˆ f ∈ Lp (R) : f ∈ Lp (R) ˆ with norm ||f2 ||Wp2 (R) = ||f2 ||Lp2 (R) . Letting $\Gamma$={[xi]1 + [xi]2 = 0} $\subseteq$ R2 and a1 , a2 : R2 $\Rightarrow$ C satisfy ∂ [alpha] aj [xi] 1 $\alpha$dist([xi],$\Gamma$|$\alpha$| for each j ∈ {1, 2} and sufficiently many multi-indices [alpha] ∈ (N {0})2 , we construct a time-frequency framework to show that the degenerate trilinear simplex multiplier defined for any (f1 , f2 , f3 ) ∈ S(R)3 by the formula 3 B[a1, a2] : (f1, f2, f3)$\Rightarrow$p1 ˆ fj ([xi]j )e2[pi]ix[xi]j d[xi]1 d[xi]2 d[xi]3 a1 ([xi]1 , [xi]2 )a2 ([xi]2 , [xi]3 ) R3 p3 maps L (R) x Wp2 (R) x L (R) into L 1 < p1 , p3 < $\infty$, 2 < p2 < $\infty$, j=1 1 1 1 1 p1 + p2 + p3 (R) provided 1 1 1 1 + < 1, + <1. p1 p2 p2 p3 Unlike non-degenerate simplex multipliers, B[a1, a2] cannot be written as a global paracomposition modulo harmless error terms. Nonetheless, B[a1 , a2 ] can be written as a local paracomposition, and we show this is enough to obtain generalized restricted type mixed estimates. Mixed Marcinkiewicz interpolation then finishes the argument. Lastly, in Chapter 3, we turn our attention to proving a wide range of Lp estimates for two so-called semi-degenerate simplex multipliers defined on tuples of Schwartz functions by the following maps:ˆˆˆ f1 ([xi]1 )f2 ([xi]2 )f3 ([xi]3 )e2[pi]ix([xi]1 +[xi]2 [-]2[xi]3 ) d[xi] C 1,1,[-]2 : (f1 , f2 , f3 )$\Rightarrow$R3 ˆ ˆ ˆ ˆ f1 ([xi]1 )f2 ([xi]2 )f3 ([xi]3 )f4 ([xi]4 )e2[pi]ix([xi]1 +[xi]2 +[xi]3 [-]2[xi]4 ) d[xi]. C 1,1,1,[-]2 : (f1 , f2 , f3 , f4 )$\Rightarrow$R4 We also construct elementary counterexamples to show that our target Lp ranges for both C 1,1,[-]2 and C 1,1,1,[-]2 are sharp. A crucial ingredient in our analysis of the semi-degenerate setting is a novel l1 -based energy estimate.