In this paper, we develop an extremely simple method to establish the sharpened Adams-type inequalities on higher-order Sobolev spaces $W^{m,\frac {n}{m}}(\mathbb {R}^n)$ in the entire space $\mathbb {R}^n$ , which can be stated as follows: Given $\Phi \left ( t\right ) =e^{t}-\underset {j=0}{\overset {n-2}{\sum }} \frac {t^{j}}{j!}$ and the Adams sharp constant $\beta _{n,m}$ . Then,

$$ \begin{align*}\sup_{\|\nabla^mu\|_{\frac{n}{m}}^{\frac{n}{m}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}}\leq1}\int_{\mathbb{R}^n}\Phi\Big(\beta_{n,m} (1+\alpha \|u\|_{\frac{n}{m}}^{\frac{n}{m}} )^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\Big)dx<\infty, \end{align*} $$

$0<\alpha <1$

$\alpha $

$\alpha \ge 1$

Our argument avoids applying the complicated blow-up analysis often used in the literature to deal with such sharpened inequalities.

In this paper we obtain some improved $L^p$ -Hardy and $L^p$ -Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.

In this paper, we will use optimal mass transport combining with suitable transforms to study the sharp constants and optimizers for a class of the Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities. Moreover, we will investigate these inequalities with and without the monomial weights $x_{1}^{A_{1}} \cdots x_{N}^{A_{N}}$ on ℝN.

We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [−1, 1], and is the r-maximal operator, then for any weight w and any function f on [0, 1) we have

for some constant Cr depending only on r. We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md. The proof rests on the Bellman function method; using this technique we establish related weighted Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.

Let f:ℝ→ℝ be a locally integrable function of bounded lower oscillation. The paper contains the proofs of sharp strong-type, weak-type and exponential estimates for the mean oscillation of f. In particular, this yields the precise value of the norm of the embedding BLO⊂BMOp, 1≤p<∞. Higher-dimensional analogues for anisotropic BLO spaces are also established.