On general local \(Tb\) theorems (Q2841372)

To state the main result requires the following concepts: upper doubling measures, Calderón-Zygmund operators with standard kernel, and \(L^\infty\) and \(L^2\) accretive systems. A Borel measure \(\mu\) with associated \(\lambda:\mathbb{R}^n\times (0,\infty)\to (0,\infty)\) such that \(r\mapsto \lambda(x,r)\) is nondecreasing for each \(x\) and such that \(\lambda(x,2r)\leq C\lambda(x,r)\) is said to be \(\lambda\)-upper doubling if \(\mu(B(x,r))\leq \lambda(x,r)\). One assumes that \(\lambda\) also satisfies \(\lambda(x,r)\leq C\lambda(y,r)\) if \(|x-y|<r\). Then a standard Calderón-Zygmund kernel is one that satisfies NEWLINE\[NEWLINEK(x,y)\leq {C\over \lambda(x,|x-y|)},\quad x\neq y, NEWLINE\]NEWLINE NEWLINE\[NEWLINE|K(x,y)-K(x,y')|\leq C{|y-y'|^\alpha\over |x-y|^\alpha \lambda(x,|x-y|)}, \quad |x-y|\geq 2|y-y'| NEWLINE\]NEWLINE with a corresponding inequality with roles of \(x\) (and \(x'\)) replaced by \(y\) (and \(y'\)). A Calderón-Zygmund operator with standard kernel is then an operator \(T\) from \(L^2(\mu)\) to itself such that \(Tf(x)=\int K(x,y)\, f(y)\, d\mu(y)\) (\(x\notin \text{supp}\, f\)).NEWLINENEWLINEAn \(L^\infty\) system of accretive functions is a pair \(\{b_Q^1\}\), \(\{b_Q^2\}\) indexed by cubes in \(\mathbb{R}^n\) such that \(b_Q^i\) is supported in \(Q\), \(\|b_Q^{i}\|_{L^\infty(\mu)}\leq C\), \(\|Tb_Q^{i}\|_{L^\infty(\mu)}\leq C\) and similarly for \(T^\ast\), and \(\int_Q b_Q^{i}\, d\mu =\mu(Q)\) (\(i=1,2\)) for all cubes \(Q\). An \(L^2\)-accretive system is defined similarly but with \(\|b_Q^i\|_{L^\infty(\mu)}\leq C\) replaced by \(\int_Q |b_Q^i|^2\, d\mu\leq C\mu(Q)\) and \(\|Tb_Q^i\|_{L^\infty(\mu)}\leq C\) replaced by \(\int_Q |T b_Q^i|^s\, d\mu\leq C\mu(Q)\) for some fixed \(s>2\) (and similarly for \(T^\ast\)).NEWLINENEWLINEWith these definitions, the main theorem is stated as follows: Let \(\mu\) be a \(\lambda\)-upper doubling measure and \(T: L^2(\mu)\to L^2(\mu)\) a Calderón-Zygmund operator with a standard kernel \(K\). Assuming the existence of accretive \(L^\infty\) systems \(\{b_Q^{i}\}\), \(i=1,2\), we have \(\|T\|\leq C\) where \(C\) depends on the dimension \(n\) and on the explicit constants in the definitions of the quantities defined above. In case \(\mu\) is a doubling measure, the same conclusion holds assuming only the existence of accretive \(L^2\)-systems \(\{b_Q^i\}\).NEWLINENEWLINEMuch of the analytic approach is guided by the work of \textit{F. Nazarov, S. Treil} and \textit{A. Volberg} [Duke Math. J. 113, No. 2, 259--312 (2002; Zbl 1055.47027)] and a good deal of preliminary discussion addresses points of departure from that work. In particular, martingale difference operators \(\Delta_Q f(x) \) are defined by the difference between \(\langle f\rangle_{R} b_R^1\) and \(\langle f\rangle_{Q} b_Q^1\) when \(x\in R\), one of the \(2^n\) children subcubes of \(Q\). It is observed that desired martingale square function estimates are not true in general for \(L^\infty\) accretive systems. Because the local operators \(\Delta_Q\) are not pairwise orthogonal one has to approach the decomposition of \(f\) into {\textit{good}} and {\textit{bad}} parts differently from global \(Tb\) arguments in order to make a telescoping argument in a paraproduct estimate work. Averaging over random dyadic grids is needed to obtain \(L^2\) estimates for the pairing \(\langle Tf,\, g\rangle\).