An analogue of the Littlewood-Paley theorem for orthoprojectors onto wavelet subspaces (Q2821886)

The standard wavelet-based Littlewood-Paley characterization of the \(L^p(\mathbb{R}^n)\) norm involves inequalities NEWLINE\[NEWLINEc_1\|f\|_{L^p(\mathbb{R}^d)}\leq \Bigl\|\Bigl(\sum_k |Q_kf(x)|^2 \Bigr)^{1/2}\Bigr\|_{L^p(\mathbb{R}^d)} \leq c_2\|f\|_{L^p(\mathbb{R}^d)} NEWLINE\]NEWLINE where \(Qf\) is the multiresolution projection onto the span of wavelets at scale \(j\). In the tensor wavelet case, \(Q_j f=2^j \sum_{\epsilon\in E}\sum_{k\in\mathbb{Z}^d} \langle f,\, \psi_\epsilon(2^j\cdot-k)\rangle\, \psi_\epsilon(2^j\cdot-k)\) where \(\epsilon\) ranges over \(2^{d}-1\) generators. In the present work, the author considers nonisotropic wavelet type projections in which different scales apply in different spatial coordinates. He defines projections \(\mathcal{E}_k\), \(k\in \mathbb{Z}^d_{+}\) (nonnegative integers in each coordinate) by taking advantage of the product structure of \(\mathbb{R}^d\), starting with projections on \(L^2(\mathbb{R})\). Set \((E_\kappa f)(x)=\sum_{\nu\in\mathbb{Z}} 2^\kappa \langle f,\, \widetilde\varphi (2^\kappa\cdot -\nu)\rangle \varphi(2^\kappa x-\nu)\). It is assumed that \(\langle \varphi,\, \widetilde\varphi(\cdot-\nu)\rangle=\delta_\nu\) and that \(\{\varphi(\cdot-\nu):\, \nu\in\mathbb{Z}\}\) forms a Riesz basis for its \(L^2(\mathbb{R})\)-closed linear span. Set \(\mathcal{E}_\kappa =E_\kappa-E_{\kappa-1}\), \(\kappa\in\mathbb{Z}_+\), (\(E_{-1}=0\)). Next, for \(j\in \{1,\dots, d\}\) let \(V_j^X\) denote the operation that extends an operator \(T\) defined on a function space \(X\) on \(\mathbb{R}\) to \(\mathcal{T}_j\) defined on the \(d\)-fold tensor product of \(X\), regarded as a function space on \(\mathbb{R}^d\), by NEWLINE\[NEWLINE(\mathcal{T}_j f)(x_1,\dots, x_{j-1},\cdot,x_{j+1},\dots, x_d)=T(f(x_1,\dots, x_{j-1},\cdot,x_{j+1},\dots, x_d))(\cdot).NEWLINE\]NEWLINE Let NEWLINE\[NEWLINE\mathcal{E}_k=\sum_{\epsilon\in\{0,1\}^d, \,\epsilon\leq k} (\pm 1) E_{k-\epsilon}, \;\;E_k=\prod_{j=1}^d V_j^{L^p} E_{k_j}NEWLINE\]NEWLINE where, in each coordinate, \(E_{k_j}\) depends on dual functions \(\varphi\) and \(\widetilde\varphi\). The coordinatewise condition \( \epsilon\leq k\) and the ``\(\pm\)'' signify that, unless \(k_j=0\), the univariate operator \(\mathcal{E}_{k_j}=E_{k_j}-E_{k_{j-1}}\) will appear as the corresponding coordinate operator in the sum when \(\epsilon_j=0\) and \(E_{k_j}\) will appear when \(\epsilon_j=0\).NEWLINENEWLINEGiven these technicalities, the main result, Theorem 2.3, states that there are constants \(c=c(d,\varphi,\widetilde\varphi,d)\) and \(C=C(d,\varphi,\widetilde\varphi,d)\) such that for all \(f\in L^p(\mathbb{R}^d)\) one has NEWLINE\[NEWLINEc\|f\|_{L^p(\mathbb{R}^d)}\leq \Bigl\|\Bigl(\sum_{k\in\mathbb{Z}_{d}^+} |(\mathcal{E}_k f)(\cdot)|^2 \Bigr)^{1/2}\Bigr\|_{L^p(\mathbb{R}^d)} \leq C\|f\|_{L^p(\mathbb{R}^d)}\, . NEWLINE\]NEWLINE The functions \(\varphi,\widetilde\varphi\) are tensor products: \(\varphi(x)=\varphi_1(x_1)\,\varphi_2(x_2)\cdots\varphi_d(x_d)\) and similarly for \(\widetilde\varphi\). Besides the biorthogonality and Riesz properties, one also assumes that each coordinate \(\varphi_j\) satisfies a nondegeneracy condition in the form of \(|\varphi(t)|\) being pointwise bounded by the convolution of the indicator of the interval of length one centered at the origin and a nonnegative \(L^1\) function \(\Phi\) that is also convexly decreasing (the technical hypotheses are stated in Lemma 2.1 and Proposition 1.2). A fair amount of harmonic analysis (e.g., weak-type estimates) is needed to prove the desired integral bounds under hypotheses of this generality. In particular, Lemma 2.1, which gives the univariate bound \(\|E_\kappa f\|_{L^p(\mathbb{R})}\leq c\|f\|_{L^p(\mathbb{R})}\), requires a ten-page proof.