Related papers: Sharp form for improved Moser-Trudinger inequality

Let $d \ge 1$, $p \ge d$, and let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $\Omega$ such that…

Given $\alpha >0$, we establish the following two supercritical Moser-Trudinger inequalities \[ \sup\limits_{u \in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( (\alpha_n + |x|^\alpha) |u|^{\frac{n}{n-1}} \big)…

In the paper, the authors discover the best constants $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$, and $\beta_{2}$ for the double inequalities $$ \alpha_{1}\bar{C}(a,b)+(1-\alpha_{1}) A(a,b)< T(a,b) <\beta_{1} \bar{C}(a,b)+(1-\beta_{1})A(a,b)…

In this paper, we examine the boundary $L^2$ term of the sharp Sobolev trace inequality $\|u\|_{L^{q}(\pa M)}^2\leq S \|\nabla_g u\|_{L^2(M)}^2 +A(M,g)\|u\|^2_{L^2(\pa M)}$ on Riemannian manifolds $(M,g)$ with boundaries $\pa M$, where…

In this note we deal with some inequalities for the tangent function that are valid for $x$ in $(-\pi/2,\pi/2)$. These inequalities are optimal in the sense that the best values of the exponents involved are obtained.

We show that optimal $L^2$-convergence in the finite element method on quasi-uniform meshes can be achieved if, for some $s_0 > 1/2$, the boundary value problem has the mapping property $H^{-1+s} \rightarrow H^{1+s}$ for $s \in [0,s_0]$.…

New Strichartz estimates for the modulated cubic nonlinear Schr\"{o}dinger equation are proved. These Strichartz estimates allow us to show that this equation is pathwise locally well-posed. We also show that improved Strichartz estimates…

In this paper, we combine the argument of [12] and [27] to prove the maximal estimates for fractional Schr\"odinger equations $(i\partial_t+\mathcal{L}_{\mathbf{A}}^{\frac\alpha 2})u=0$ in the purely magnetic fields which includes the…

We give a short linear--algebraic proof of the inequality \[ \|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{p}}{2}\,\|x\|_2^2, \] valid for every \(x\in\mathbb{R}^p\). This inequality relates three fundamental norms on finite-dimensional spaces…

We prove Aubin's "Hypothese fondamentale" concerning the existence of Moser-Trudinger type inequalities on any integral compact K\"ahler manifold X. In the case of the anti-canonical class on a Fano manifold the constants in the…

We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a…

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$\|\varphi\|_{L^p(\mathbb{R}^d)}\le C\|\varphi\|_{\dot H^{s}(\mathbb{R}^d)}^{\beta} \left(\iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{|\varphi (x)|^q\,|\varphi…

Among all metrics on $\mathbb S^d$ with $d>4$ that are conformal to the standard metric and have positive scalar curvature, the total $\sigma_2$-curvature, normalized by the volume, is uniquely (up to M\"obius transformations) minimized by…

We obtain the Strichartz inequalities $$ \| u \|_{L^q_t L^r_x([0,1] \times M)} \leq C \| u(0) \|_{L^2(M)}$$ for any smooth $n$-dimensional Riemannian manifold $M$ which is asymptotically conic at infinity (with either short-range or…

We obtain all extreme and exposed points of the closed unit ball of the space of bilinear forms $T:\ell_{\infty}^{2}\times\ell_{\infty}^{2}\rightarrow \mathbb{R}.$ We also show that any (norm one) bilinear form $T:\ell_{\infty…

We establish an improved version of the Moser-Trudinger inequality in the hyperbolic space $\mathbb H^n$, $n\geq 2$. Namely, we prove the following result: for any $0 \leq \lambda < \left(\frac{n-1}n\right)^n$, then we have $$…

In this paper, on a compact Riemann surface $(\Sigma, g)$ with smooth boundary $\partial\Sigma$, we concern a Trudinger-Moser inequality with mean value zero. To be exact, let $\lambda_1(\Sigma)$ denotes the first eigenvalue of the…

In this note we prove the following good-$\lambda$ inequality, for $r>2$, all $\lambda > 0$, $\delta \in \big(0, \frac{1}{2} \big)$ \[ \nu\big\{ V_r(f) > 3 \lambda ; \mathcal{M}(f) \leq \delta \lambda\big\} \leq 4 \nu\{s(f) > \delta…

This paper deals with the construction of an optimal quadrature formula for the approximation of Fourier integrals in the Sobolev space $L_2^{(1)}[a,b]$ of non-periodic, complex valued functions which are square integrable with first order…

We study convergence properties of the full truncation Euler scheme for the Cox-Ingersoll-Ross process in the regime where the boundary point zero is inaccessible. Under some conditions on the model parameters (precisely, when the Feller…