# American Institute of Mathematical Sciences

June  2017, 37(6): 3387-3410. doi: 10.3934/dcds.2017143

## Generalized inhomogeneous Strichartz estimates

 Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

Received  September 2016 Revised  December 2016 Published  February 2017

We prove new inhomogeneous generalized Strichartz estimates, which do not follow from the homogeneous generalized estimates by virtue of the Christ-Kiselev lemma. Instead, we make use of the bilinear interpolation argument worked out by Keel and Tao and refined by Foschi presented in a unified framework. Finally, we give a sample application.

Citation: Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143
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,Figure 2,p.5]. The axes refer to the spatial integrability coefficients. The rectangle $ABCD$ corresponds to estimates found from factorization and the application of the Christ-Kiselev lemma up to endpoints. The origin relates to the dispersive estimate; one finds local estimates to hold in the wedge $AOCD$ by virtue of interpolation, restrictions on global estimates cut off estimates with too large spatial integrability coefficients">Figure 1.  This pictorial representation generalizes [7,Figure 2,p.5]. The axes refer to the spatial integrability coefficients. The rectangle $ABCD$ corresponds to estimates found from factorization and the application of the Christ-Kiselev lemma up to endpoints. The origin relates to the dispersive estimate; one finds local estimates to hold in the wedge $AOCD$ by virtue of interpolation, restrictions on global estimates cut off estimates with too large spatial integrability coefficients
,Figure 1,p.1907]. In the setting of [3,Corollary 1.,p.1907] we find $A=(\frac{n-a}{2n},\frac{1}{2}), \, B= (\frac{n}{n+a}-\frac{n}{2(n+1)},\frac{n}{n+a}-\frac{n}{2(n+1)}), \, C=(\frac{n}{2(n+1)},\frac{n}{2(n+1)}), \, D=(\frac{1}{2},0)$, where the open line $\overline{BC}$ corresponds to the range from [3,Theorem 1.2.,p.1908] and the closed line $\overline{AD}$ corresponds to estimates found from factorization and application of the Christ-Kiselev lemma">Figure 2.  We give a pictorial representation similar to [3,Figure 1,p.1907]. In the setting of [3,Corollary 1.,p.1907] we find $A=(\frac{n-a}{2n},\frac{1}{2}), \, B= (\frac{n}{n+a}-\frac{n}{2(n+1)},\frac{n}{n+a}-\frac{n}{2(n+1)}), \, C=(\frac{n}{2(n+1)},\frac{n}{2(n+1)}), \, D=(\frac{1}{2},0)$, where the open line $\overline{BC}$ corresponds to the range from [3,Theorem 1.2.,p.1908] and the closed line $\overline{AD}$ corresponds to estimates found from factorization and application of the Christ-Kiselev lemma
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