Abstract
Abstract We prove mixed L p ( L q ) -estimates, with p , q ∈ ( 1 , ∞ ) , for higher-order elliptic and parabolic equations on the half space R + d + 1 with general boundary conditions which satisfy the Lopatinskii–Shapiro condition. We assume that the elliptic operators A have leading coefficients which are in the class of vanishing mean oscillations both in the time variable and the space variable. In the proof, we apply and extend the techniques developed by Krylov [24] as well as Dong and Kim in [13] to produce mean oscillation estimates for equations on the half space with general boundary conditions.
Highlights
In this paper, we study the Weyl product acting on weighted modulation spaces with Lebesgue parameters in
We work out conditions on the weights and the Lebesgue parameters that are sufficient for continuity of the Weyl product, and we prove necessary conditions
The Weyl product or twisted product is the product of symbols in the Weyl calculus of pseudodifferential operators corresponding to operator composition
Summary
We study the Weyl product acting on weighted modulation spaces with Lebesgue parameters in We work out conditions on the weights and the Lebesgue parameters that are sufficient for continuity of the Weyl product, and we prove necessary conditions. In [7] conditions on the Lebesgue parameters were found that are both necessary and sufficient for continuity of the Weyl product, characterizing the Weyl product acting on Banach modulation spaces. Our technique to prove the sufficient conditions consists of a discretization of the Weyl product by means of a Gabor frame. In Appendix, we show a Fubini type result for Gelfand–Shilov distributions that is needed in the definition of the short-time Fourier transform (STFT) of a Gelfand– Shilov distribution
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