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...

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.

Let $B_w\left({\ell }^p\right)$ denote the space of infinite matrices $A$ for which $A\left(x\right)\in {\ell }^p$ for all $x={\left\{x_k\right\}}_{k=1}^{\infty }\in {\ell }^p$ with $|x_k|\searrow 0$. We characterize the upper triangular positive matrices from $B_w\left({\ell }^p\right)$, $1<p<\infty$, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.