Using the technique of Gabor analysis, we characterize the boundedness of $e^{i\Delta }: W^{p_1,q_1}_m\rightarrow W^{p_2,q_2}$ with modulation and translation operators, where and m is a v-moderate weight. The sharp exponents for the boundedness are also characterized in the case of power weight.

We show that the fractional integral operator $I_{\alpha }$, $0<\alpha <n$, and the fractional maximal operator $M_{\alpha }$, $0\le \alpha <n$, are bounded on weak Choquet spaces with respect to Hausdorff content. We also investigate these operators on Choquet–Morrey spaces. The results for the fractional maximal operator $M_\alpha $ are extensions of the work of Tang [‘Choquet integrals, weighted Hausdorff content and maximal operators’, Georgian Math. J. 18(3) (2011), 587–596] and earlier work of Adams and Orobitg and Verdera. The results for the fractional integral operator $I_{\alpha }$ are essentially new.

We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.

We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables $\left\{X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions

\begin{equation*}\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}\end{equation*}

\begin{equation*}\Omega_{\text{rand},\text{lac}} := \left\{\frac{\pi X_k}{2^k} : k\geq 1 \right\}.\end{equation*}

We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p({\mathbb{R}}^2)$ for any $1 \lt p \lt \infty$.

We discuss how countable subadditivity of operators can be derived from subadditivity under mild forms of continuity, and provide examples manifesting such circumstances.

Let $\mathcal {M}$ be an Ahlfors $n$-regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : \mathcal {M} \to \mathbb {R}$ by a Carleson measure condition of its $\sigma$-harmonic extension $U:\mathcal {M}\times \mathbb {R}_+ \to \mathbb {R}$. This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO & Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.

In this paper, we study the behaviours of the commutators $[\vec b,\,T]$ generated by multilinear Calderón–Zygmund operators $T$ with $\vec b=(b_1,\,\ldots,\,b_m)\in L_{\rm loc}(\mathbb {R}^n)$ on weighted Hardy spaces. We show that for some $p_i\in (0,\,1]$ with $1/p=1/p_1+\cdots +1/p_m$, $\omega \in A_\infty$ and $b_i\in \mathcal {BMO}_{\omega,p_i}$ ($1\le i\le m$), which are a class of non-trivial subspaces of ${\rm BMO}$, the commutators $[\vec b,\,T]$ are bounded from $H^{p_1}(\omega )\times \cdots \times H^{p_m}(\omega )$ to $L^p(\omega )$. Meanwhile, we also establish the corresponding results for a class of maximal truncated multilinear commutators $T_{\vec b}^*$.

We prove that the uncentered Hardy–Littlewood maximal operator is discontinuous on ${BMO}(\mathbb {R}^n)$ and maps ${VMO}(\mathbb {R}^n)$ to itself. A counterexample to the boundedness of the strong and directional maximal operators on ${BMO}(\mathbb {R}^n)$ is given, and properties of slices of ${BMO}(\mathbb {R}^n)$ functions are discussed.

Let $\sigma \in (0,\,2)$, $\chi ^{(\sigma )}(y):={\mathbf 1}_{\sigma \in (1,2)}+{\mathbf 1}_{\sigma =1} {\mathbf 1}_{y\in B(\mathbf {0},\,1)}$, where $\mathbf {0}$ denotes the origin of $\mathbb {R}^n$, and $a$ be a non-negative and bounded measurable function on $\mathbb {R}^n$. In this paper, we obtain the boundedness of the non-local elliptic operator

\[ Lu(x):=\int_{\mathbb{R}^n}\left[u(x+y)-u(x)-\chi^{(\sigma)}(y)y\cdot\nabla u(x)\right]a(y)\,\frac{{\rm d}y}{|y|^{n+\sigma}} \]

$\mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$

$\mathrm {BMO}(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$\lambda \in (0,\,\infty )$

$Lu-\lambda u=f$

$\mathbb {R}^n$

$f\in \mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$

$H^1(\mathbb {R}^n)$

$\mathrm {BMO}(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$L^p(\mathbb {R}^n)$

$p\in (1,\,\infty )$

$p=1$

$p=\infty$

The aim of this paper is to establish the boundedness of fractional type Marcinkiewicz integral $\mathcal {M}_{\iota ,\rho ,m}$ and its commutator $\mathcal {M}_{\iota ,\rho ,m,b}$ on generalized Morrey spaces and on Morrey spaces over nonhomogeneous metric measure spaces which satisfy the upper doubling and geometrically doubling conditions. Under the assumption that the dominating function $\lambda $ satisfies $\epsilon $ -weak reverse doubling condition, the author proves that $\mathcal {M}_{\iota ,\rho ,m}$ is bounded on generalized Morrey space $L^{p,\phi }(\mu )$ and on Morrey space $M^{p}_{q}(\mu )$ . Furthermore, the boundedness of the commutator $\mathcal {M}_{\iota ,\rho ,m,b}$ generated by $\mathcal {M}_{\iota ,\rho ,m}$ and regularized $\mathrm {BMO}$ space with discrete coefficient (= $\widetilde {\mathrm {RBMO}}(\mu )$ ) on space $L^{p,\phi }(\mu )$ and on space $M^{p}_{q}(\mu )$ is also obtained.

In this paper we study boundedness and compactness characterizations of the commutators of Cauchy type integrals on bounded strongly pseudoconvex domains D in $\mathbb C^{n}$ with boundaries $bD$ satisfying the minimum regularity condition $C^{2}$ , based on the recent results of Lanzani–Stein and Duong et al. We point out that in this setting the Cauchy type integral is the sum of the essential part which is a Calderón–Zygmund operator and a remainder which is no longer a Calderón–Zygmund operator. We show that the commutator is bounded on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the BMO space on $bD$ . Moreover, the commutator is compact on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the VMO space on $bD$ .

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.

This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.

We discuss the dyadic John–Nirenberg space that is a generalization of functions of bounded mean oscillation. A John–Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John–Nirenberg space and provide a method to construct nontrivial functions in the dyadic John–Nirenberg space. Moreover, we prove that the John–Nirenberg space is complete. Several open problems are also discussed.

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

We establish the global regularity of multilinear Fourier integral operators that are associated to nonlinear wave equations on products of $L^p$ spaces by proving endpoint boundedness on suitable product spaces containing combinations of the local Hardy space, the local BMO and the $L^2$ spaces.

On a compact Lie group $G$ of dimension $n$, we study the Bochner–Riesz mean $S_{R}^{\unicode[STIX]{x1D6FC}}(f)$ of the Fourier series for a function $f$. At the critical index $\unicode[STIX]{x1D6FC}=(n-1)/2$, we obtain the convergence rate for $S_{R}^{(n-1)/2}(f)$ when $f$ is a function in the block-Sobolev space. The main theorems extend some known results on the $m$-torus $\mathbb{T}^{m}$.

Block decomposition of $L^{p}$ spaces with weighted Hausdorff content is established for $0<p\leqslant 1$ and the Fefferman–Stein type inequalities are shown for fractional integral operators and some variants of maximal operators.

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a metric measure space satisfying the geometrically doubling condition and the upper doubling condition. In this paper, the authors establish the John-Nirenberg inequality for the regularized BLO space $\widetilde{\operatorname{RBLO}}(\unicode[STIX]{x1D707})$.