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- W2950240870 abstract "We consider degenerate differential operators $A = displaystyle{sum_{k,j=1}^d partial_k (a_{kj} partial_j)}$ on $L^2(mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given a function $chi in C_b^infty(mathbb{R}^d)$ (respectively, $Omega$ a bounded Lipschitz domain) and suppose that $(a_{kj}) ge mu > 0$ a.e. on $ supp chi$ (resp., a.e. on $Omega$). We prove a spectral multiplier type result: if $Fcolon [0, infty) to mathbb{C}$ is such that $sup_{t > 0} | varphi(.) F(t .) |_{C^s} < infty$ for some non-trivial function $varphi in C_c^infty(0,infty)$ and some $s > d/2$ then $M_chi F(I+A) M_chi$ is weak type $(1,1)$ (resp. $P_Omega F(I+A) P_Omega$ is weak type $(1,1)$). We also prove boundedness on $L^p$ for all $p in (1,2]$ of the partial Riesz transforms $M_chi nabla (I + A)^{-1/2}M_ chi$. The proofs are based on a criterion for a singular integral operator to be weak type $(1,1)$." @default.
- W2950240870 created "2019-06-27" @default.
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- W2950240870 date "2012-02-09" @default.
- W2950240870 modified "2023-09-29" @default.
- W2950240870 title "Partial spectral multipliers and partial Riesz transforms for degenerate operators" @default.
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- W2950240870 doi "https://doi.org/10.48550/arxiv.1202.2136" @default.
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