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Remarks on Strichartz estimates for null forms

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12 Pages
English

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Niveau: Supérieur, Doctorat, Bac+8
Remarks on Strichartz estimates for null forms Fabrice Planchon Laboratoire d'Analyse Numerique, URA CNRS 189, Universite Pierre et Marie Curie, 4 place Jussieu BP 187, 75 252 Paris Cedex Abstract We prove some improved Strichartz estimates for null forms, some of which were conjectured recently in [7]. The results follow from combining the usual Strichartz estimates with div-curl lemma techniques rather than space-time Fourier transform. Introduction Consider the bilinear form (1) Q ij (f; g) = @ i f@ j g @ i g@ j f; where f and g are functions of x 2 R n . Such forms appear in various instances connected with geometric PDEs, either elliptic ([10] and references therein) or hyperbolic ([17] and references therein). In the later context, they are called null forms (along with other bilinear forms which we will introduce later). These forms have cancellation properties. For the simple product Holder yields (2) rf;rg 2 L 2 =) @f@g 2 L 1 ; while for the null form one has (3) rf;rg 2 L 2 =) Q ij (f; g) 2 H 1 ; where H 1 denotes the Hardy space: h 2 H 1 if and only if h 2 L

  • summing over dyadic

  • strichartz estimates

  • over

  • space cancellation

  • understood when considering

  • rather than

  • when studying

  • besov space

  • time fourier


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4