Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function ss="section_image" src="/cms/asset/96f31c91-6ef2-4487-a60e-3e958a47a478/mana201700376-math-0001.png"> on the upper half‐spaces. Using the decomposition, we study the boundedness of ss="section_image" src="/cms/asset/943d9cea-1e55-4c0d-a66d-edbbe6510b3b/mana201700376-math-0002.png">. We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.     

Generalized Hirano inverses in rings

We introduce and study a new class of generalized inverses in rings. An element a in a ring R has generalized Hirano inverse if there exists <span class="NLM_disp-formula-image inline-formula">script>src="//:0" data-src='{"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0001.gif"}' />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0001.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula inline-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />scroll" altimg="eq-00001.gif">b∈Rspan> such that <span class="NLM_disp-formula-image inline-formula">script>src="//:0" data-src='{"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0002.gif"}' />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0002.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula inline-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />scroll" altimg="eq-00002.gif">bab=b,b∈comsup>m2sup>stretchy="false">(astretchy="false">),sup>a2sup>?ab∈sup>Rqnilsup>.span> We prove that the generalized Hirano inverse of an element is its generalized Drazin inverse. An element <span class="NLM_disp-formula-image inline-formula">script>src="//:0" data-src='{"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0003.gif"}' />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0003.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula inline-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />scroll" altimg="eq-00003.gif">a∈Rspan> has generalized Hirano inverse if and only if there exists <span class="NLM_disp-formula-image inline-formula">script>src="//:0" data-src='{"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0004.gif"}' />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0004.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula inline-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />scroll" altimg="eq-00004.gif">p=sup>p2sup>∈comsup>m2sup>stretchy="false">(astretchy="false">)span> such that <span class="NLM_disp-formula-image inline-formula">script>src="//:0" data-src='{"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0005.gif"}' />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/lagb20/2019/lagb20.v047.i07/00927872.2018.1546391/20190531/images/lagb_a_1546391_ilm0005.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula inline-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />scroll" altimg="eq-00005.gif">sup>a2sup>?p∈sup>Rqnilsup>span>. We then completely determine when a 2?×?2 matrix over projective-free rings has generalized Hirano inverse. Cline’s formula and additive properties for generalized Hirano inverses are thereby obtained.     

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let e<sub>?sub>, for ? = 1,2,3, be orthogonal unit vectors in ss="section_image" src="/cms/asset/52ceedd5-5bf8-481d-b7b9-be17e2e8e474/mma5466-math-0001.png"> and let ss="section_image" src="/cms/asset/cb6de83e-4f6c-428e-85cc-1060941f0562/mma5466-math-0002.png"> be a bounded open set with smooth boundary ?Ω. Denoting by ss="section_image" src="/cms/asset/b0a4af4d-5c02-479a-8ca2-eced460c7fe7/mma5466-math-0003.png"> a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: ss="" src="/cms/asset/d1fde17e-5c5b-4abd-ac68-eeb59bae2057/mma5466-math-0004.png" alt="urn:x-wiley:mma:media:mma5466:mma5466-math-0004" title="urn:x-wiley:mma:media:mma5466:mma5466-math-0004"> into the conservation of energy law, here a, b, ss="section_image" src="/cms/asset/b744c327-5ea3-467d-8147-418995b7e95f/mma5466-math-0005.png"> are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, ss="section_image" src="/cms/asset/11890952-8eb9-44ec-9d6e-f3397e17e3b0/mma5466-math-0006.png"> the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.     

Existence of strong minimizers for the Griffith static fracture model in dimension two

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate s="true">s="true">s="true">Ds="true">s="true">psup> functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy s="true">s="true">s="true">Ds="true">s="true">psup> functions by s="true">s="true">s="true">Vs="true">s="true">psup> ones.     

Compact half step approximation in exponential form for the system of 2D second-order quasi-linear elliptic partial differential equations

This paper deals with devising a novel exponential scheme, implicit in nature, using half step discretization of order four for computing the numerical solution of quasi-linear elliptic partial differential equations of the type <span class="NLM_disp-formula-image inline-formula">script>src="/na101/home/literatum/publisher/tandf/journals/content/gdea20/2019/gdea20.v025.i05/10236198.2019.1624737/20190703/images/gdea_a_1624737_ilm0001.gif" alt="" />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gdea20/2019/gdea20.v025.i05/10236198.2019.1624737/20190703/images/gdea_a_1624737_ilm0001.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula inline-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />span>, <span class="NLM_disp-formula-image inline-formula">script>src="/na101/home/literatum/publisher/tandf/journals/content/gdea20/2019/gdea20.v025.i05/10236198.2019.1624737/20190703/images/gdea_a_1624737_ilm0002.gif" alt="" />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gdea20/2019/gdea20.v025.i05/10236198.2019.1624737/20190703/images/gdea_a_1624737_ilm0002.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula inline-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />span>. Nine mesh points in a single compact cell are being used in this method. An elaborate convergence analysis of the proposed method has been worked out. The technique for a scalar equation is eventually extended to solve the systems of quasi-linear elliptic equations. The technique is then applied to noteworthy problems and computational outcomes corroborate the theoretical rate of convergence. Linear and non-linear singular problems are tackled separately ensuring the usage of nine spatial grid points of a single computing cell.     

Non-trivial 1d and 2d Segal–Bargmann transforms

Special generating functions and Mehler's formula for the univariate complex Hermite polynomials are obtained and next employed to introduce and study some one- and two-dimensional integral transforms of Segal–Bargmann type in the framework of some specific functional Hilbert spaces; including the so-called generalized Bargmann–Fock spaces that are realized as <span class="NLM_disp-formula-image inline-formula">script>src="//:0" data-src='{"type":"image","src":"/na101/home/literatum/publisher/tandf/journals/content/gitr20/2019/gitr20.v030.i07/10652469.2019.1593407/20190428/images/gitr_a_1593407_ilm0001.gif"}' />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gitr20/2019/gitr20.v030.i07/10652469.2019.1593407/20190428/images/gitr_a_1593407_ilm0001.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula inline-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />span>-eigenspaces of a special magnetic Schrödinger operator.     

Heinz type inequalities for mappings satisfying Poisson’s equation

In this paper, we study the Heinz type inequalities for mappings satisfying Poisson’s equation. Some results generalize the ones obtained by Partyka and Sakan.     

Existence of homoclinic solutions for nonlinear second-order coupled systems

This work presents sufficient conditions for the existence of homoclinic solutions for second order coupled discontinuous systems of differential equations on the real line without the usual growth condition in the literature.The arguments apply the fixed point theory, Green's functions technique, s="true">s="true">s="true">Ls="true">s="true">1sup>-Carathéodory functions, lower and upper solutions and Schauder's fixed point theorem.     

A Near-Infrared-Controllable Artificial Metalloprotease Used for Degrading Amyloid-β Monomers and Aggregates

Proteolysis of amyloid-β (Aβ) is a promising approach against Alzheimer's disease. However, it is not feasible to employ natural hydrolases directly because of their cumbersome preparation and purification, poor stability, and hazardous immunogenicity. Therefore, artificial enzymes have been developed as potential alternatives to natural hydrolases. Since specific cleavage sites of Aβ are usually embedded inside the β-sheet structures that restrict access by artificial enzymes, this strongly hinders their efficiency for practical applications. Herein, we construct a NIR (near-IR) controllable artificial metalloprotease (MoS<sub>2sub>-Co) using a molybdenum disulfide nanosheet (MoS<sub>2sub>) and a cobalt complex of 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid (Codota). Evidenced by detailed experimental and theoretical studies, the NIR-enhanced MoS<sub>2sub>-Co can circumvent the restriction by simultaneously inhibition of β-sheet formation and destroying β-sheet structures of the preformed Aβ aggregates in living cell. Furthermore, our designed MoS<sub>2sub>-Co is an easy to graft Aβ-target agent that prevents misdirected or undesirable hydrolysis reactions, and has been demonstrated to cross the blood brain barrier. This method can be adapted for hydrolysis of other kinds of amyloids.