arXiv · 0705.4124
This is an overview of merging the techniques of Riesz space theory and convex geometry.
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S. S. Kutateladze. 2007-05-29. Interaction of Order and Convexity. https://arxiv.org/abs/0705.4124
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We introduce the mixed Bourgain-Morrey spaces and obtain their preduals. The boundedness of Hardy-Littlewood maximal operator, iterated maximal operator, fractional integral operator, singular integral operator on these spaces is proved. In addition, we give a description of the dual of mixed Bourgain-Morrey spaces and conclude the reflexivity of these spaces.
A central problem in harmonic analysis is to understand how frequency resolution constrains stability under nonlinear changes of coordinates. Motivated by Mallat's wavelet scattering theory, we establish a sharp resolution--robustness principle for nonlinear representations built from two-dimensional wave-packet systems, showing that the transverse geometry of the frequency channels determines the critical Sobolev regularity. Whenever the channels are thinner than in the wavelet regime, arbitrarily small smooth deformations can move high-frequency mass between adjacent channels, producing instability already at the level of the linear filter bank and the first scattering layer. We match this obstruction with scale-sharp commutator estimates propagated through the full scattering cascade, obtaining Lipschitz stability at the critical regularity threshold. Together, the results provide the first intrinsic deformation-stability theory for non-wavelet Euclidean scattering transforms and identify a microlocal transport mechanism governing robustness across nonlinear multiscale representations.
We study the positive Helson form $h_0[a,b]=\sum_{m,n\ge1}α(mn)a_n\overline{b_m}$ on the finitely supported sequences $c_{00}\subset\ell^2(\mathbb{N})$, where $α(n)=\int_{\mathbb{R}}n^{-s}dμ(s)<\infty$ for a finite positive Borel measure $μ$. Under the Carleman condition $\sum_{k\ge1}α(4^k)^{-1/(2k)}=\infty$, closability of $h_0$ is equivalent to $\sqrt{n}α(n)\to0$, and hence to $μ((-\infty,1/2])=0$. For measures concentrated on $(1/2,\infty)$, we identify the closure of the operator $N_0a=\sum_{n\ge1}a_nn^{-s}$, defined on $c_{00}$, with its maximal realization from $\ell^2(\mathbb{N})$ to $L^2(μ)$. We further prove that the sequences $a\in c_{00}$ satisfying $\sum_{n\ge1}a_n/\sqrt{n}=0$ form an operator core for the positive self-adjoint operator associated with the closure of $h_0$.