Calderón commutators and the Cauchy integral on Lipschitz curves revisited II: The Cauchy integral and its generalizations
This is the second paper of three ones, whose aim is to give new proofs to the well-known theorems of Calderón, Coifman, McIntosh and Meyer, concerning \(L^p\) estimates for the Calderón's commutators and the Cauchy integral on Lipschitz curves. This one describes the case of the Cauchy integral on Lipschitz curves and its generalizations. More precisely, the author treats the Cauchy integral \(C_\Gamma(f)\) defined by NEWLINE\[NEWLINEC_\Gamma(f)(x) =\mathrm{p.v.}\,\int_{\mathbb R}\frac{f(y)}{(x-y)+i(A(x)-A(y))}\,dy,NEWLINE\]NEWLINE where \(A'=a\in L^\infty(\mathbb R)\). As is well-known, standard arguments reduce the \(L^p\) boundedness of this operator to the problem of proving polynomial bounds for the associated \(d\)-th Calderón commutators defined by NEWLINE\[NEWLINEC_d(f)(x) =\mathrm{p.v.}\,\int_{\mathbb R}\frac{(A(x)-A(y))^d}{(x-y)^{d+1}}f(y)\,dy.NEWLINE\]NEWLINE As is known, this can be seen as a \(d+1\)-linear Fourier multiplier with singular symbol. \(C_d\) has \(d+1\) natural adjoints \(C_d^{*i}\) \((i=1,\dots,d+1)\), i.e. NEWLINE\[NEWLINE\int_{\mathbb R}C_d(f_1,\dots,f_{d+1})(x)f_{d+2}(x)\,dx=\int_{\mathbb R} C_d^{*i}(f_1,\dots,f_{i-1},f_{i+1},\dots,f_{d+2})(x)f_{i}(x)\,dx.NEWLINE\]NEWLINE Let \(C_d^{*i}=C_d\). In this paper, the author proves that for \(1\leq i\leq d+2\), NEWLINE\[NEWLINE(*)\quad \|C_d^{*i}(f_1,\dots,f_{d+1})\|_{p}\leq C(d)C(\ell )C(p_1)\cdots C(p_{d+1}) \|f_1\|_{p_1}\cdots\|f_{d+1}\|_{p_{d+1}},NEWLINE\]NEWLINE where \(1\leq p_1,\dots,p_{d+1}\leq \infty\), \(1\leq p<\infty\), \(1/p=1/p_1+\cdots+1/p_{d+1}\), \(\ell\) is the number of indices for which \(pI\neq\infty\), \(C(d)\) grows at most polynomially in \(d\) and \(C_{p_i}=1\) if \(p_i=\infty\).NEWLINENEWLINEHe proves (\(*\)) by developing the ideas for treating multilinear Fourier multiplier theory developed by him, Lacey, Pipher, Tao, Thiele, i.e. by decomposing and discretizing \(C_d^{*i}\) and reducing it to a finite and well localized model operator. He also discuss several generalizations of the above ideas.NEWLINENEWLINENEWLINEEditorial remark: For part I see Zbl 1306.42026.
scientific article; zbMATH DE number 6353618