\(Tb\) theorem for the generalized singular integral operator on product Lipschitz spaces with para-accretive functions
Spaces \(\mathrm{Lip}_b(\alpha_1,\alpha_2)(\mathbb{R}^n\times\mathbb{R}^m)\) are defined for product \(b=b_1b_2\) para-accretive functions and a \(Tb\) theorem for generalized singular integral operators on these spaces is established. A function \(b\) defined on \(\mathbb{R}^n\) is para-accretive if there are \(C>0\) and \(\delta>0\) such that for any cube \(Q\subset\mathbb{R}^n\) there is a subcube \(Q^\prime\) such that \(\delta|Q|\leq |Q^\prime|\) and \(\frac{1}{|Q|}\Bigl| \int_{Q^\prime} b(x)\, dx\Bigr|\geq C\). The space \(\mathrm{Lip}(\alpha_1,\alpha_2)(\mathbb{R}^n\times\mathbb{R}^m)\) consists of those \(f\) such that \[|f(x-u,y-v)-f(x,y-v)-f(x-u,y)+f(x,y)|\leq C|u|^{\alpha_1}|v|^{\alpha_2} \] (with norm defined as the smallest \(C\) such that the inequality holds for all \((x,y)\in (\mathbb{R}^n\times\mathbb{R}^m\)). One defines \(\mathrm{Lip}_b(\alpha_1,\alpha_2)=\{f: fb\in\mathrm{Lip}(\alpha_1,\alpha_2)\}\) where \(b(x,y)=b_1(x)b_2(y)\). One defines \(C_0^\eta(\mathbb{R}^n)\) to be those compactly supported \(f\) such that \(\sup_{x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\eta}<\infty\). One defines a corresponding product space with product norms in terms of the differences defined above. One defines a product Calderón-Zygmund operator in terms of a product kernel \(K(x_1,y_1;x_2,y_2)\) satisfying standard CZO estimates in each parameter along with additional joint decay estimates on cross-parameter differences. One defines a general singular integral operator \(T\) as a continuous linear operator from \(b C_0^\eta(\mathbb{R}^n\times\mathbb{R}^m)\) into \((b C_0^\eta(\mathbb{R}^n\times\mathbb{R}^m))^\prime\) associated with a product CZO kernel such that the formal integrals \(\langle M_b T M_b f_1\otimes f_2, \, g_1\otimes g_2\rangle\) are defined by product integration against the kernel for Hölder continuous test functions \(f_i\), \(g_i\) having disjoint supports. One also defines test functions of type \((\beta_1,\beta_2,\gamma_1,\gamma_2)\) centered at \((x_0,y_0)\in \mathbb{R}^n\times\mathbb{R}^m\) satisfying standard molecular-type estimates, again, with additional decay on cross-parameter differences, and satisfying vanishing mean values in each parameter weighted against the para-accretive functions \(b_i\), e.g., \(\int_{\mathbb{R}^n} f(x,y) b_1(x)\, dx=0\) for all \(y\). One defines a class \(\mathcal{M}(x_0,y_0;d_1,d_2;\beta_1,\beta_2;\gamma_1,\gamma_2)\) satisfying all the desired inequalities and moment conditions. The first theorem establishes that \(\mathrm{Lip}(\alpha_1,\alpha_2)\) and \(\mathrm{Lip}_b(\alpha_1,\alpha_2)\) consist of elements of the classes \(\mathcal{M}(\beta_1,\beta_2;\gamma_1,\gamma_2)\) satisfying uniform bounds with respect to the semi-norms defining these classes. The main result is the following \(Tb\) theorem: let \(b(x,y)=b_1(x)b_2(y)\) where \(b_1\) and \(b_2\) are para-accretive in \(\mathbb{R}^n\) and \(\mathbb{R}^m\) respectively. Let \(T\) be a generalized singular integral operator on \(L^2(\mathbb{R}^{n+m})\). Then \(T\) is bounded from \(\mathrm{Lip}_b(\alpha_1,\alpha_2)(\mathbb{R}^n\times\mathbb{R}^m)\) to \(\mathrm{Lip}(\alpha_1,\alpha_2)(\mathbb{R}^n\times\mathbb{R}^m)\) for \(\alpha_i\in (0,\epsilon)\) (where \(\epsilon\) is the usual parameter that appears in the CZO kernel estimates) if and only if \(T_1b_1=T_2b_2=0\). A product approximate identity allows for Littlewood-Paley type estimates to be formulated. A further Calderón-type reproducing formula is also used, and a new class of (homogeneous) product Besov spaces is introduced.
scientific article; zbMATH DE number 7258098
scientific article