Canonical Sobolev projections of weak type \((1,1)\) (Q2709430)

The authors investigate boundedness properties of Fourier multipliers associated with Sobolev projections that are said to be anisotropic because only a subset of weak derivatives up to a fixed order are required to belong to \(L^{p}\). Typically such projections are \(L^{p}\)-bounded for \(1<p<\infty \). But when \(p=1\) some conditions are required for boundedness from \(L^{1}\) to weak-\(L^{1}\). The authors consider the Euclidean case as well as the periodic case and, importantly, connections between them. This review will only discuss the Euclidean result. By a \textit{smoothness} \(S\) one means a set of multiindices or \(n\)-tuples of nonnegative integers that is convex in the sense that if \(\alpha =(\alpha _{1},\dots ,\alpha _{n})\in S\) then \(\beta =(\beta _{1},\dots ,\beta _{n})\in S\) whenever \(\beta _{i}\leq \alpha _{i}\) for all \(i=1,\dots ,n\). For \(1\leq p<\infty \) the anisotropic Sobolev space \(L_{S}^{p}(R^{n})\) is defined as the completion of the Schwartz class in the norm \(\|f\|_{S,p}=( \sum_{\alpha \in S}\|\partial ^{\alpha }f\|_{p}^{p}) ^{1/p}\). One then defines the canonical embedding \(J_{S}\) from \(L_{S}^{p}(R^{n})\) into the direct sum of the cardinality of \(S\) copies of \(L^{p}(R^{n})\) by setting \(J_{S}(f)=(\partial ^{\alpha }f)_{\alpha \in S}\). The canonical projection \(P_{S}\) is the orthogonal projection of \(\oplus _{S}L^{2}(R^{n})\) onto \(J_{S}L_{S}^{2}(R^{n})\). Setting \(\widehat{f}(\xi)=\int f(x)e^{-2\pi ix\cdot \xi } dx\) one can define the canonical projection through its Fourier multiplier matrix with entries \(\varphi _{\alpha ,\beta }(2\pi \xi)\) where \(\varphi _{\alpha ,\beta }(\xi)=i^{|\alpha |-|\beta |} \frac{\xi ^{\alpha +\beta }}{Q_{S}(\xi)}\), where \(Q_{S}=\sum_{\alpha \in S}\xi ^{2\alpha }\) is called the fundamental polynomial of \(S\). Thus \(L^{p}\)-boundedness of the canonical projection follows from that of the multiplier matrix, as Pelczynski and Senator proved in the mid-80s. Weak-\(L^{1}\) boundedness requires extra properties of \(S\). Set \(S^{\#}\) to be the maximal multiindices in \(S\) and define \(Q_{S^{\#}}=\sum_{\alpha \in S^{\#}}\xi ^{2\alpha }\) to be the generating polynomial of \(S\). The order of \(S\) is half the degree of \(Q_{S}\) or, equivalently, the maximum of the orders \(|\alpha |=\sum \alpha _{i}\) of all \(\alpha \in S\). One says that \(S\) is homogeneous if \(Q_{S^{\#}}\) is a homogeneous polynomial. \(S\) is nondegenerate if it contains all multiindices \(e_{i}=(\dots ,0,1,0,\dots) \) with \(1\) in the \(i\)th coordinate, and is irreducible if no coordinate function divides \(Q_{S^{\#}}\). Finally, one says that the canonical projection \(P_{S}\) is weak-\(L^{1}\) bounded provided each of the entries of its multiplier matrix is bounded from \(L^{1}(R^{n})\) to weak-\( L^{1}(R^{n})\). The main theorem applies to non-degenerate second-order smoothnesses on \(R^{n}\). The authors prove that, for such a smoothness \(S\), \( P_{S}\) is weak-\(L^{1}\) bounded if and only if one of the following three sets of conditions hold: (i) \(S\) is reducible, (ii) \(S\) is irreducible, homogeneous, and for each pair \((j,k)\) of integers such that \(1\leq j<k\leq n \), at least one of \(e_{j}+e_{k},2e_{j}\) or \(2e_{k}\) belongs to \(S\) or (iii) \(S \) is non-homogeneous and for each pair \((j,k)\) of integers such that \(1\leq j<k\leq n\), if \(e_{j}+e_{k}\in S\) then both \(2e_{j}\in S\) and \(2e_{k}\in S\). Note that, as a corollary \(P_{S}\) is weak-\(L^{1}\) bounded for every nondegenerate two-dimensional smoothness, but the authors show that, in fact, \(P_{S}\) is weak-\(L^{1}\) bounded for every second order smoothness in two-dimensions. The authors show that \(P_{S}\) is not weak-\(L^{1}\) bounded for the smoothness \(S=\{(1,0,\dots ,0),(0,1,1,\dots ,1)\}\) when \(n\geq 3\). The sufficiency part of the homogeneous case relies on a modified Hörmander-Mikhlin theorem. The case of condition (iii) relies on the fact that individual multipliers must meet a certain ellipticity condition. The necessity part of (ii) relies on a combinatorial argument that yields reduction to the case of a specific unbounded rational multiplier.