In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations, the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) in terms of these weights under an additional mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned (two-)weighted boundedness of the parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón–Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality, which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.

In this article, the authors consider the boundedness on Triebel–Lizorkin spaces for the d-dimensional Calderón commutator defined by

$$ \begin{align*}T_{\Omega,a}f(x)=\mathrm{p.\,v.}\int_{\mathbb{R}^d}\frac{\Omega(x-y)}{|x-y|^{d+1}}\big(a(x)-a(y)\big)f(y){d}y,\end{align*} $$

$\Omega $

$S^{d-1}$

$\mathbb {R}^d$

$$ \begin{align*}\sup_{\zeta\in S^{d-1}}\int_{S^{d-1}}|\Omega(\theta)|\log ^{\beta} \big(\frac{1}{|\theta\cdot\zeta|}\big)d\theta<\infty\end{align*} $$

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

$T_{\Omega ,a}$

$\dot {F}_{p}^{0,q}(\mathbb {R}^d)$

$1+\frac {1}{2\beta -1}<p,q<2\beta $

We establish hyperweak boundedness of area functions, square functions, maximal operators, and Calderón–Zygmund operators on products of two stratified Lie groups.

We enrich the class of power-constructible functions, introduced in [CCRS23], to a class $\mathcal {C}^{\mathcal {M,F}}$ of algebras of functions which contains all complex powers of subanalytic functions and their parametric Mellin and Fourier transforms, and which is stable under parametric integration. By describing a set of generators of a special prepared form, we deduce information on the asymptotics and on the loci of integrability of the functions of $\mathcal {C}^{\mathcal {M,F}}$. We furthermore identify a subclass $\mathcal {C}^{\mathbb {C},\mathcal {F}}$ of $\mathcal {C}^{\mathcal {M,F}}$, which is the smallest class containing all power-constructible functions and stable under parametric Fourier transforms and right-composition with subanalytic maps. This class is also stable under parametric integration, under taking pointwise and $\text {L}^p$-limits and under parametric Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in the power-logarithmic scale, uniformly in the parameters, for functions in $\mathcal {C}^{\mathbb {C},\mathcal {F}}$.

Let $L=-\Delta +V$ be a Schrödinger operator in ${\mathbb R}^n$ with $n\geq 3$, where $\Delta $ is the Laplace operator denoted by $\Delta =\sum ^{n}_{i=1}({\partial ^{2}}/{\partial x_{i}^{2}})$ and the nonnegative potential V belongs to the reverse Hölder class $(RH)_{q}$ with $q>n/2$. For $\alpha \in (0,1)$, we define the operator

$$ \begin{align*} T_N^{L^{\alpha}} f(x) =\sum_{j=N_1}^{N_2} v_j(e^{-a_{j+1}L^\alpha} f(x)-e^{-a_{j}L^\alpha} f(x)) \quad \mbox{for all }x\in \mathbb R^n, \end{align*} $$

where $\{e^{-tL^\alpha } \}_{t>0}$ is the fractional heat semigroup of the operator L, $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequence and $\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence.

We investigate the boundedness of the operator $T_N^{L^{\alpha }}$ and the related maximal operator $T^*_{L^{\alpha }}f(x):=\sup _N \vert T_N^{L^{\alpha }} f(x)\vert $ on the spaces $L^{p}(\mathbb {R}^{n})$ and $BMO_{L}(\mathbb {R}^{n})$, respectively. As extensions of $L^{p}(\mathbb {R}^{n})$, the boundedness of the operators $T_N^{L^{\alpha }}$ and $T^*_{L^{\alpha }}$ on the Morrey space $L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ and the weak Morrey space $WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$ has also been proved.

We present a new proof of the compactness of bilinear paraproducts with CMO symbols. By drawing an analogy to compact linear operators, we first explore further properties of compact bilinear operators on Banach spaces and present examples. We then prove compactness of bilinear paraproducts with CMO symbols by combining one of the properties of compact bilinear operators thus obtained with vanishing Carleson measure estimates and interpolation of bilinear compactness.

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, k-point configuration sets given by

$$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} $$

$E\subset \mathbb {R}^d$

$k\ge 3$

$\Delta ^{\mathrm{diag}}(E)$

(0.1)

$$ \begin{align} \dim(E)> \begin{cases} \frac{2d+1}3, & k=3, \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{align} $$

$C^\omega $

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty }$ converging to zero and initial data $f\in H^s(\mathbb {R}^N)$, $N\geq 2$, we consider the almost everywhere convergence problem for sequences of Schrödinger means ${\rm e}^{it_{n}\Delta }f$, which was proposed by Sjölin, and was open until recently. In this paper, we prove that if $\{t_n\}_{n=1}^{\infty }$ belongs to Lorentz space ${\ell }^{r,\infty }(\mathbb {N})$, then the a.e. convergence results hold for $s>\min \{\frac {r}{\frac {N+1}{N}r+1},\,\frac {N}{2(N+1)}\}$. Inspired by the work of Lucà-Rogers, we construct a counterexample to show that our a.e. convergence results are sharp (up to endpoints). Our results imply that when $0< r<\frac {N}{N+1}$, there is a gain over the a.e. convergence result from Du-Guth-Li and Du-Zhang, but not when $r\geq \frac {N}{N+1}$, even though we are in the discrete case. Our approach can also be applied to get the a.e. convergence results for the fractional Schrödinger means and nonelliptic Schrödinger means.

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 study $L^p$-Sobolev regularity estimates for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for the restricted X-ray transforms in $\mathbb {R}^{d+1}$, $d\ge 3$. This extends the result due to Pramanik and Seeger in $\mathbb {R}^3$.

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 $T_{\vec b}^*$.

Let $\mu$ be a Radon measure on the nth Heisenberg group ${\mathbb{H}}^n$. In this note we prove that if the $(2n+1)$-dimensional (Heisenberg) Riesz transform on ${\mathbb{H}}^n$ is $L^2(\mu)$-bounded, and if $\mu(F)=0$ for all Borel sets with ${\text{dim}}_H(F)\leq 2$, then $\mu$ must have $(2n+1)$-polynomial growth. This is the Heisenberg counterpart of a result of Guy David from [Dav91].

We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma $ in $\mathbb R^d$, $d\ge 3$. Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every $d\ge 3$. As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $\gamma $ for $d\ge 4$.

We recall that $w\in C_{p}^+$ if there exist $\varepsilon >0$ and $C>0$ such that for any $a< b< c$ with $c-b< b-a$ and any measurable set $E\subset (a,b)$, the following holds

\[ \int_{E}w\leq C\left(\frac{|E|}{(c-b)}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^+\chi_{(a,c)}\right)^{p}w<\infty. \]

$C_{p}$

$1< p< q<\infty$

$w\in C_{q}^+$

\[ \|M^+f\|_{L^{p}(w)}\lesssim\|M^{\sharp,+}f\|_{L^{p}(w)} \]

$w\in C_{p}^+$

$M^{\sharp,+}$

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 $bD$.

We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$, where $x\in {\mathbb R}^{n}$ and Q is a cube in ${\mathbb R}^{n}$. When applied to operators, this result recovers our recent works [37, 39]. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalised Poincaré–Sobolev inequalities, tent spaces and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localisable in the sense of [39], as we will demonstrate in an application to vector-valued square functions.

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.