arXiv subjects
Publications and source records attributed to Cristian Rios.
We prove a smooth Alpert testing characterization of convolution type for the Fourier extension conjecture on paraboloids, that further extends the characterizations in arXiv:2506.23376v4.
We show that the Fourier extension conjecture on the paraboloid in three dimensions is equivalent to a local single scale smooth Alpert trilinear inequality, which is an improvement of an analogous multiscale trilinear inequality in arXiv:2506.03992.
We prove that a local trilinear extension inequality on the paraboloid in three dimensions is equivalent to the Fourier restriction conjecture, and then we prove a variant involving smooth Alpert wavelets that represents the weakest such inequality the authors could find that characterizes the Fourier extension conjecture.
We prove the equivalence of two Kakeya conjectures: 1.The Kakeya maximal operator conjecture 2.The disjoint trilinear dual form of the Kakeya maximal operator conjecture
We show that if $\mathbb{R}^{n}$ is equipped with certain non-doubling metric and an Orlicz-Sobolev inequality holds for a special family of Young functions $Φ$, then weak solutions to quasilinear infinitely degenerate elliptic divergence equations of the form $$\mathrm{div}\mathcal{A}\left( x,u\right) \nabla u=ϕ_{0}-\mathrm{div}_{A} \vecϕ_{1}$$ are locally bounded. Furthermore, we establish a maximum principle for solutions whenever a global Orlicz-Soblev estimate is available. We obtain these results via the implementation of a Moser iteration method, what constitutes the first instance of such technique applied to infinite degenerate equations. These results partially extend previously known estimates for solutions of these equations but for which the right hand side did not have a drift term. We also obtain bounds for small negative powers of nonnegative solutions; these will be applied to obtain continuity of solutions in a subsequent paper.
We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form equations, and the local boundedness turns out to be sharp in more than two dimensions, answering the `Moser gap' problem left open in arXiv:1506.09203v5. Finally we obtain a maximum principle for weak solutions under the same condition on the degeneracy.
We obtain a generalization of the DeGiorgi Lemma to the infinitely degenerate regime and apply it to obtain continuity of weak solutions to certain infinitely degenerate equations. This reproduces the continuity result obtained in arXiv:1506.09203 via Moser iteration, but only for homogeneous equations. However, the proofs are much less technical and more transparent.
We prove that second order linear operators on $\mathbb{R}^{n+m}$ of the form $L(x,y,D_x,D_y) = L_1(x,D_x) + g(x) L_2(y,D_y)$, where $L_1$ and $L_2$ satisfy Morimoto's super-logarithmic estimates and $g$ is smooth, nonnegative, and vanishes only at the origin in $\mathbb{R}^n$ (but to any arbitrary order) are hypoelliptic without loss of derivarives. We also show examples in which our hypotheses are necessary for hypoellipticity.
We study the Kato problem for degenerate divergence form operators. This was begun by Cruz-Uribe and Rios who proved that given an operator $L_w=-w^{-1}{\rm div}(A\nabla)$, where $w\in A_2$ and $A$ is a $w$-degenerate elliptic measure (i.e, $A=w\,B$ with $B$ an $n\times n$ bounded, complex-valued, uniformly elliptic matrix), then $L_w$ satisfies the weighted estimate $\|\sqrt{L_w}f\|_{L^2(w)}\approx\|\nabla f\|_{L^2(w)}$. Here we solve the $L^2$-Kato problem: under some additional conditions on the weight $w$, the following unweighted $L^2$-Kato estimates hold $$ \|L_w^{1/2}f\|_{L^2(\mathbb{R}^n)}\approx\|\nabla f\|_{L^2(\mathbb{R}^n)}. $$ This extends the celebrated solution to the Kato conjecture by Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, allowing the differential operator to have some degeneracy in its ellipticity. For example, we consider the family of operators $L_γ=-|x|^γ{\rm div}(|x|^{-γ}B(x)\nabla)$, where $B$ is any bounded, complex-valued, uniformly elliptic matrix. We prove that there exists $ε>0$, depending only on dimension and the ellipticity constants, such that $$ \|L_γ^{1/2}f\|_{L^2(\mathbb{R}^n)}\approx\|\nabla f\|_{L^2(\mathbb{R}^n)}, \qquad -ε<γ<\frac{2\,n}{n+2}. $$ This gives a range of $γ$'s for which the classical Kato square root $γ=0$ is an interior point. Our main results are obtained as a consequence of a rich Calderón-Zygmund theory developed for some operators associated with $L_w$. These results, which are of independent interest, establish estimates on $L^p(w)$, and also on $L^p(v\,dw)$ with $v\in A_\infty(w)$, for the associated semigroup, its gradient, the functional calculus, the Riesz transform, and square functions. As an application, we solve some unweighted $L^2$-Dirichlet, Regularity and Neumann boundary value problems for degenerate elliptic operators.
We obtain a new general extension theorem in Banach spaces for operators which are not required to be symmetric, and apply it to obtain Harnack estimates and a priori regularity for solutions of fractional powers of several second order differential operators. These include weighted elliptic and subellitptic operators in divergence form (nonnecessarily self-adjoint), and nondivergence form operators with rough coefficients. We utilize the reflection extension technique introduced by Caffarelli and Silvestre.
We develop subrepresentation inequalities for infinitely degenerate metrics, and obtain corresponding Poincare and Sobolev inequalities. We then derive conditions on the degenerate metric under which weak solutions to associated infinitely degenerate equations with rough coefficients are locally bounded, satisfy a maximum principle, or are continuous. As an application we obtain W-hypoellipticity of certain infinitely degenerate quasilinear equations with smooth coefficients having mild nonlinearities and degeneracies.
In various analytical contexts, it is proved that a weak Sobolev inequality implies a doubling property for the underlying measure.
The main result of the paper is on the continuity of weak solutions of infinitely degenerate quasilinear second order equations. Namely, we show that every weak solution to a certain class of degenerate quasilinear equations is continuous. More precisely, we show that it is Hölder continuous with respect to a certain metric associated to the operator. One of the essential features of this metric is that the metric balls are non doubling with respect to Lebesgue measure. The proof of the continuity together with a recent result by Rios et al. completes the result on hypoellipticity of a class of second order quasilinear infinitely degenerate elliptic operators.
We prove that every weak solution to a certain class of infinitely degenerate quasilinear equations is continuous. An essential feature of the operators we consider is that their Fefferman-Phong associated metric may be non doubling with respect to Lebesgue measure.
In this paper we establish a hypoellipticity result for second order linear operators comprised by a linear combination, with infinite vanishing coefficients, of subelliptic operators in separate spaces. This generalizes previous known results.
In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds just in terms of the supremum norm for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness and interior regularity of solutions for the Dirichlet problem with continuous boundary data.
We characterize when radial weak solutions to Monge-Ampere equations are smooth. This paper extends previous partial results and also covers Generalized Monge-Ampere equations and infinitely vanishing right hand side.
Given two elliptic operators L and M in nondivergence form, with coefficients A_L(x), A_M(x) and drift terms b_L(x), b_M(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in R^{n+1}, then their harmonic measures are mutually absolutely continuous. As an application of this, a new approximation argument and known results we obtain necessary and sufficient conditions for a single operator L (in divergence or nondivergence form) to have regular harmonic measure with respect to Lebesgue measure. The results are sharp in all cases.