We present several continuous embeddings of the critical Besov space $B_p^{n/p,\rho }\left(ℝⁿ\right)$. We first establish a Gagliardo-Nirenberg type estimate ${\left|\right|u\left|\right|}_{{Ḃ}_{q,w_r}^{0,\nu }}\le Cₙ{(1/(n-r))}^{1/q+1/\nu -1/\rho }{(q/r)}^{1/\nu -1/\rho }{\left|\right|u\left|\right|}_{{Ḃ}_p^{0,\rho }}^{(n-r)p/nq}{\left|\right|u\left|\right|}_{{Ḃ}_p^{n/p,\rho }}^{1-(n-r)p/nq}$, for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function $w_r\left(x\right)=1/{\left(\right|x|}^r)$ with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding $B_p^{n/p,\rho }\left(ℝⁿ\right)\hookrightarrow B_{\Phi ₀,w_r}^{0,\nu }\left(ℝⁿ\right)$, where the function Φ₀ of the weighted Besov-Orlicz space $B_{\Phi ₀,w_r}^{0,\nu }\left(ℝⁿ\right)$ is a Young function of the exponential type. Another point of interest is to embed $B_p^{n/p,\rho }\left(ℝⁿ\right)$ into the weighted Besov space $B_{p,wₙ}^{0,\rho }\left(ℝⁿ\right)$ with...

We consider the half-linear differential equation of the form $$\left(p\left(t\right)\right|x^{\text{'}}{|}^{\alpha }\mathrm{sgn}{x}^{\text{'}}{)}^{\text{'}}+q\left(t\right){\left|x\right|}^{\alpha }\mathrm{sgn}x=0,t\ge t_0,$$ under the assumption that $p{\left(t\right)}^{-1/\alpha }$ is integrable on $[t_0,\infty )$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t\to \infty$.

Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by $\sigma \left(A\right)={\int }_B{\chi }_A(x,\phi \left(x\right))dx$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^p\left(ℝ³\right)$ to $L^q(\Sigma ,d\sigma )$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

Let $x\left(t\right)=(t,\frac{1}{2}{t}^2,\frac{1}{6}{t}^3)$ a certain curve in ${\mathbf{R}}^3.$ We investigate inequalities of the type $\{\int |\widehat{f}\left(x\right(t\left)\right){|}^{b}dt{\}}^{1/b}\le C\parallel f{\parallel }_a$ for $f\in 𝒮(\mathbf{R}$ 3). Our results improve improve an earlier restriction theorem of Prestini. Various generalizations are also discussed.

We prove that under the Gaussian measure, half-spaces are uniquely the most noise stable sets. We also prove a quantitative version of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This extends a theorem of Borell, who proved the same result but without uniqueness, and it also answers a question of Ledoux, who asked whether it was possible to prove Borell’s theorem using a direct semigroup argument. Our quantitative uniqueness result has various...