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Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. (English) Zbl 1030.35024
Consider the equations $i\partial_{t}u + \Delta u - {a\over |x|^{2}}u = 0, \qquad u(0,x) = f(x), \tag{1}$
$\partial_{t}^{2}u + \Delta u - {a\over |x|^{2}}u = 0, \qquad u(0,x) = f(x),\;\partial_{t}u(0,x) = g(x),\tag{2}$ where $$\Delta$$ denotes the $$n$$-dimensional Laplacian and $$a$$ is a real number. Let $$n\geq 2$$, $$\lambda(n) = {n-2\over 2}$$ and for $$a\geq -\lambda(n)^{2}$$ set $$\nu_{d}= \sqrt{(\lambda(n) +d)^{2} + a}$$.
In the case that $$u$$ solves (1) the authors prove the following estimate for $$f\in L^{2}(\mathbb R^{n})$$ $\|u\|_{L^{p}_{t}(L^{q}_{x})} \leq C\|f\|_{L^{2}}$ where $$p\geq 2$$ and $$q$$ such that $${2\over p} + {n\over q} = {n\over 2}$$ with $$(n,p) \neq (2,2)$$. The proof is based on a weighted $$L^{p}$$-estimate for the operator $$P_{a} = -\Delta + {a\over |x|^{2}}$$.
If $$u$$ solves the wave equation (2) it is proven that for Cauchy data $$(f,g) \in {\dot H}^{1/2} \times {\dot H}^{-1/2}$$ $\|(-\Delta)^{\sigma/2}u\|_{L_{t}^{p}(L_{x}^{q})} \leq C(\|f\|_{{\dot H}^{1/2}} + \|g\|_{{\dot H}^{-1/2}})$ holds, where $$p\geq 2$$ and $$q$$ such that $${2\over p} + {n-1\over q} = {n-1 \over 2}$$ and $$a+\lambda^{2}>0$$, $$\sigma = {1\over p} +{n\over q} - {n-1\over 2}$$. This result is based on a generalized Morawetz estimate again for $$P_{a}$$.
The last section contains estimates of $$(-\Delta)^{s/2}u$$ in suitable norms. This is applied to show that the nonlinear initial value problem for $$\partial_{t}^{2}u + P_{a}u = \pm |u|^{\kappa}$$, for $$\kappa \geq {n+3 \over n-1}$$ with appropriate Cauchy data has a unique global solution.

MSC:
 35G10 Initial value problems for linear higher-order PDEs 35J10 Schrödinger operator, Schrödinger equation 35L05 Wave equation 35L15 Initial value problems for second-order hyperbolic equations
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