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We prove hypoellipticity of second order linear operators on $$\mathbb {R}^{n+m}$$Rn+m of the form $$L(x,y,D_x,D_y) = L_1(x,D_x) + g(x) L_2(y,D_y)$$L(x,y,Dx,Dy)=L1(x,Dx)+g(x)L2(y,Dy), where $$L_j$$Lj, $$j=1,2$$j=1,2, satisfy Morimoto’s super-logarithmic estimates $$||\log \!\left^2 \hat{u}(\xi… Click to show full abstract
We prove hypoellipticity of second order linear operators on $$\mathbb {R}^{n+m}$$Rn+m of the form $$L(x,y,D_x,D_y) = L_1(x,D_x) + g(x) L_2(y,D_y)$$L(x,y,Dx,Dy)=L1(x,Dx)+g(x)L2(y,Dy), where $$L_j$$Lj, $$j=1,2$$j=1,2, satisfy Morimoto’s super-logarithmic estimates $$||\log \!\left<\xi \right>^2 \hat{u}(\xi )||^2 \le \varepsilon (L_j u,u) + C_{\varepsilon ,K} ||u||^2$$||logξ2u^(ξ)||2≤ε(Lju,u)+Cε,K||u||2, and g is smooth, nonnegative, and vanishes only at the origin in $$\mathbb {R}^n$$Rn to any arbitrary order. We also show examples in which our hypotheses are necessary for hypoellipticity.
Journal Title: Journal of Pseudo-Differential Operators and Applications
Year Published: 2019
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