Publications (8)6.97 Total impact
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ABSTRACT: We establish endpoint Lebesgue space bounds for convolution and restricted Xray transforms along curves satisfying fairly minimal differentiability hypotheses, with affine and Euclidean arclengths. We also explore the behavior of certain natural interpolants and extrapolants of the affine and Euclidean versions of these operators.Journal of Functional Analysis 02/2015; 268(3). DOI:10.1016/j.jfa.2014.10.012 · 1.15 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We establish linear profile decompositions for the fourth order Schr\"odinger equation and for certain fourth order perturbations of the Schr\"odinger equation, in dimensions greater than or equal to two. We apply these results to prove dichotomy results on the existence of extremizers for the associated SteinTomas/Strichartz inequalities; along the way, we also obtain lower bounds for the norms of these operators.  [Show abstract] [Hide abstract]
ABSTRACT: We analyze the blowup behaviour of solutions to the focusing nonlinear KleinGordon equation in spatial dimensions $d\geq 2$. We obtain upper bounds on the blowup rate, both globally in space and in light cones. The results are sharp in the conformal and subconformal cases. The argument relies on Lyapunov functionals derived from the dilation identity. We also prove that the critical Sobolev norm diverges near the blowup time.Mathematische Annalen 03/2012; 358(12). DOI:10.1007/s002080130960z · 1.20 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We establish nearoptimal mixednorm estimates for the Xray transform restricted to polynomial curves with a weight that is a power of the affine arclength. The bounds that we establish depend only on the spatial dimension and the degree of the polynomial. Some of our results are new even in the wellcurved case.Journal of Functional Analysis 11/2011; 262(12). DOI:10.1016/j.jfa.2012.03.020 · 1.15 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We establish strongtype endpoint Lp(Rd)→Lq(Rd) bounds for the operator given by convolution with affine arclength measure on polynomial curves for d⩾4. The bounds established depend only on the dimension d and the degree of the polynomial.Journal of Functional Analysis 12/2010; 259(12):32053229. DOI:10.1016/j.jfa.2010.08.008 · 1.15 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider both the defocusing and focusing cubic nonlinear KleinGordon equations $$ u_{tt}  \Delta u + u \pm u^3 =0 $$ in two space dimensions for realvalued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the defocusing case, solutions are global and have finite global $L^4_{t,x}$ spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state. These results rely on analogous statements for the twodimensional cubic nonlinear Schr\"odinger equation, which are known in the defocusing case and for sphericallysymmetric initial data in the focusing case. Thus, our results are mostly unconditional. It was previously shown by Nakanishi that spacetime bounds for KleinGordon equations imply the same for nonlinear Schr\"odinger equations.Transactions of the American Mathematical Society 08/2010; 364. DOI:10.1090/S000299472011055364 · 1.10 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We prove that convolution with affine arclength measure on the curve parametrized by $h(t) := (t,t^2,...,t^n)$ is a bounded operator from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for the full conjectured range of exponents, improving on a result due to M. Christ. We also obtain nearly sharp Lorentz space bounds. Comment: 24 pages, The final version of this article will appear in the Journal of the London Math. SocJournal of the London Mathematical Society 05/2009; 80(2). DOI:10.1112/jlms/jdp033 · 0.88 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: If T is the operator given by convolution with surface measure on the sphere, (E, F) is a quasiextremal pair of sets for T if 〈TχE, χF〉≳Ed/(d+1)Fd/(d+1). In this article, we explicitly define a family $\mathcal{F}$ of quasiextremal pairs of sets for T. We prove that $\mathcal{F}$ is fundamental in the sense that every quasiextremal pair (E, F) is comparable (in a rather strong sense) to a pair from $\mathcal{F}$ . This extends work carried out by Christ for convolution with surface measure on the paraboloid.Illinois journal of mathematics 01/2009; 53(2009). · 0.34 Impact Factor
Publication Stats
42  Citations  
6.97  Total Impact Points  
Top Journals
Institutions

2012–2015

University of Wisconsin–Madison
Madison, Wisconsin, United States


2010–2011

University of California, Los Angeles
 Department of Mathematics
Los Angeles, California, United States


2009

University of California, Berkeley
 Department of Mathematics
Berkeley, California, United States
