Weighted Norm Inequalities for Multilinear Fourier Multipliers with Mixed Norm

In this paper, weighted norm inequalities for multilinear Fourier multipliers satisfying Sobolev regularity with mixed norm are discussed. Our result can be understood as a generalization of the result by Fujita and Tomita by using the L r -based Sobolev space, 1 < r ≤ 2 with mixed norm.

Note on an elementary inequality and its application to the regularity of p-harmonic functions

We study the Sobolev regularity of \(p\)-harmonic functions. We show that \(|Du|^{\frac{p-2+s}{2}}Du\) belongs to the Sobolev space \(W^{1,2}_{\operatorname{loc}}\), \(s>-1-\frac{p-1}{n-1}\), for any \(p\)-harmonic function \(u\). The proof is based on an elementary inequality.

Sobolev Regularity of Multilinear Fractional Maximal Operators on Infinite Connected Graphs

Let G be an infinite connected graph. We introduce two kinds of multilinear fractional maximal operators on G. By assuming that the graph G satisfies certain geometric conditions, we establish the bounds for the above operators on the endpoint Sobolev spaces and Hajłasz–Sobolev spaces on G.

Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production

<p style='text-indent:20px;'>This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega, t&gt;0,\\ { }{ v_t = \Delta v- v +w},\quad x\in \Omega, t&gt;0,\\ { }{\tau w_t+\delta w = u},\quad x\in \Omega, t&gt;0\\ \end{array}\right. \end{align} (*)$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^N(N\geq1) $\end{document}</tex-math></inline-formula> under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> as well as <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> are positive. Based on an <b>new</b> energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \mu&gt;\left\{ \begin{array}{ll} {0, \; \; \; {\rm{if}}\; \; N\leq4},\\ {\frac{(N-4)_{+}}{N-2}\max\{1,\lambda_{0}\},\; \; \; {\rm{if}}\; \; N\geq5}\\ \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the initial data <inline-formula><tex-math id="M5">\begin{document}$ (u_0,v_0,w_0) $\end{document}</tex-math></inline-formula> are sufficiently regular. Here <inline-formula><tex-math id="M6">\begin{document}$ \lambda_0 $\end{document}</tex-math></inline-formula> is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors.</p>

Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

<p style='text-indent:20px;'>We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" <i>Analysis &amp; PDE</i>, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript.</p>

Smoothing theorems for Radon transforms over hypersurfaces and related operators

We extend the theorems of [M. Greenblatt, L^{p} Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron, J. Funct. Anal. 276 2019, 5, 1510–1527] on {L^{p}} to {L^{p}_{s}} Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving {L^{p}} to {L^{q}_{s}} boundedness results for such operators. Here {q\geq p} but s can be positive, negative, or zero. For many such operators we will have a triangle {Z\subset(0,1)\times(0,1)\times{\mathbb{R}}} such that one has {L^{p}} to {L^{q}_{s}} boundedness for {({1\over p},{1\over q},s)} beneath Z, and in the case of Radon transforms one does not have {L^{p}} to {L^{q}_{s}} boundedness for {({1\over p},{1\over q},s)} above the plane containing Z, thereby providing a Sobolev space improvement result which is sharp up to endpoints for {({1\over p},{1\over q})} below Z. This triangle Z intersects the plane {\{(x_{1},x_{2},x_{3}):x_{3}=0\}}, and therefore we also have an {L^{p}} to {L^{q}} improvement result that is also sharp up to endpoints for certain ranges of p and q.