Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harmonics, an explicit gauge transformation and a change of variables identify each radial operator with a shifted Jacobi operator. This yields a complete eigenbasis indexed by $(\ell,k)\in\mathbb{N}_0^2$, where $\ell$ is the angular degree and $k$ is the radial mode number.
We study $(2+1)$-dimensional oscillatory integral operators of the form \[ T_\lambda f(x,y)=\int_{\mathbb{R}}e^{i\lambda P(x,y)t^k}\psi(x,y,t)f(t)dt,\qquad k\geq 1, \] where the phase $P$ is a real-analytic function with a critical point at the origin. We establish the sharp $L^2\to L^2$ decay rate of $\frac12\min\{1/h_{P}, 1/k\}$, where $h_{P}$ denotes Varchenko's Newton height of $P$. The two terms in the minimum reflect a natural competition between the spatial degeneracy of $P$ and the temporal degeneracy of $t^k$; their optimality is confirmed by a Knapp-type and a focusing example, respectively. A $TT^{*}$ reduction transforms the $L^2$ estimate into a scalar oscillatory integral, allowing Varchenko's theorem to apply directly. Building on this foundation, complex interpolation yields the sharp $L^2\to L^{2k+2}$ bound. Finally, in the regime $h_{P}\geq k$, we obtain sharp $L^2\to L^p$ decay estimates for all $p$.
Let $X$ be a vector field on an open set of $\mathbb{R}^n$ with $X(p)=0$ and Jacobian $J=DX(p)$ of rank $n-1$ having $0$ as an eigenvalue of algebraic multiplicity two. Let $q_0$ span $\ker J$ and let $e_k(A)$ denote the sum of the principal $k\times k$ minors of $A$. Under the usual hyperbolicity assumption on the transverse block, we prove that the quadratic coefficients $a,b$ of the Bogdanov--Takens normal form on the centre manifold are $a=-\frac{1}{2}\frac{D_{q_0}e_n}{e_{n-2}}$ and $b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}D_{q_0}e_n}{e_{n-2}^2}$. The underlying spectral identity requires only invertibility of the transverse block and remains valid without hyperbolicity. Both coefficients arise as the lowest-order terms of a generating identity for the first-order spectral jet of $DX$ along $\ker J$; all higher coefficients depend on the transverse block. We prove that this dichotomy is sharp. The planar formulas $a=-\frac{1}{2}D_{q_0}\det$ and $b=D_{q_0}\operatorname{tr}$ are recovered when $n=2$. We also obtain a coordinate-free nondegeneracy test and a geometric interpretation in terms of the transversality of the kernel line to two invariant hypersurfaces. A self-contained Wolfram Language notebook accompanies the paper and verifies the results symbolically.
Let $\mathcal H=-\Delta+|x|^2$ be the Hermite operator on $\mathbb R^2$, and let $\Pi_\lambda$ denote the spectral projection corresponding to $\lambda=2N+2$. We prove the sharp log-free endpoint estimate $||\Pi_\lambda||_{L^2(\mathbb R^2)\to L^{10/3}(\mathbb R^2)}\lesssim\lambda^{-1/10}$. The proof uses a spectral decomposition in polar coordinates and combines Koch-Tataru localized spectral projection bounds with a Liouville-Green representation, van der Corput estimates for exponential sums, and a weighted $TT^*$ argument across radial scales.
Let $H_0$ be the standard discrete Laplacian on $\mathbb Z^d$, $d\geq4$, let $R_0(z)=(H_0-z)^{-1}$, and let $p'$ denote the H\"older conjugate of $p$, with $p'=\infty$ when $p=1$. We establish uniform diagonal resolvent estimates from $\ell^p(\mathbb Z^d)$ to $\ell^{p'}(\mathbb Z^d)$ by proving local Fourier-decay bounds for surface measure on the Fermi surfaces of the lattice dispersion relation. At a regular point, the decay is governed by the number of coordinate directions in which the quadratic term vanishes: after a local change of variables, these directions produce cubic one-dimensional phases, while the remaining directions are quadratic. For $d\geq5$, we prove the sharp global range \[ 1\leq p\leq\frac{2(d+2)}{d+5}. \] Away from the threshold energies, this range remains sharp in odd dimensions, whereas for even $d\geq6$ the sharp range improves to \[ 1\leq p\leq\frac{2(2d+5)}{2d+11}. \] In dimension four, we obtain uniform estimates for $1\leq p<4/3$ and an $\ell^{4/3}$-to-$\ell^4$ endpoint estimate with a square-root logarithmic loss. Matching anisotropic Knapp-type constructions give the corresponding necessary conditions. We also derive Birman--Schwinger and Kato smoothing bounds, complex-eigenvalue estimates, and improved thin spectral projection estimates.
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $f\in L^{(p+1)/p}_g(\Omega)$. Under certain integrability condition on $g$, the existence of solution is proven for every $\lambda\in \mathbb{R}$. Moreover, the regularity of solution is also obtained.
Rubén Fiñana, A. Molino· Nonlinear Analysis· 0 citations
We prove the Harnack inequality (with tails) and local H\"older regularity for the fractional logarithmic $p$-Laplace operator, which is derived by differentiating the fractional $p$-Laplace operator with respect to its order. To be more precise, for a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-\Delta_p)^{s+\log} u:= \frac{{\rm d}}{{\rm d}t}(-\Delta_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order $s\in (0, 1).$ The kernel of this operator involves a logarithmic factor. As a consequence, it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional $p$-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in this setting. We also construct an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear setup $p=2$.
