Singular oscillatory integrals on {mathbb{R}^n} |
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| Authors: | M. Papadimitrakis I. R. Parissis |
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| Affiliation: | 1. Department of Mathematics, University of Crete, Knossos Avenue, 71409, Heraklion, Crete, Greece
2. Institutionen f?r Matematik, Kungliga Tekniska H?gskolan, 100 44, Stockholm, Sweden
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| Abstract: | Let ${mathcal{P}_{d,n}}Let Pd,n{mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on mathbbRn{mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? Pd,1{Pin mathcal{P}_{d,1}} . Using this estimate, we prove that | supP ? Pd,n| p.v.òmathbbRneiP(x)fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),mathop{rm sup}limits_ {P in mathcal{P}_{d,n}}left| p.v.int_{mathbb{R}^{n}}{e^{iP(x)}}{frac{Omega(x/|x|)}{|x|^n}dx}right | leq c,{rm log},d,(||Omega||_L log L(S^{n-1})+1), |
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