We establish local-in-time Strichartz estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains $\Omega \subset \mathbb {R}^ 3$ with smooth boundary $\partial \Omega \neq \emptyset $ . The key ingredients to prove Strichartz estimates are dispersive estimates, energy estimates, interpolation and $TT^*$ arguments. Strichartz estimates for waves inside an arbitrary domain $\Omega $ have been proved by Blair, Smith and Sogge [‘Strichartz estimates for the wave equation on manifolds with boundary’, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1817–1829]. We provide a detailed proof of the usual Strichartz estimates from dispersive estimates inside cylindrical convex domains for a certain range of the wave admissibility.

Let $\Omega \subset \mathbb {R}^{n+1}$ , $n\ge 2$ , be a $1$ -sided nontangentially accessible domain, that is, a set which is quantitatively open and path-connected. Assume also that $\Omega $ satisfies the capacity density condition. Let $L_0 u=-\mathop {\operatorname {div}}\nolimits (A_0 \nabla u)$ , $Lu=-\mathop {\operatorname {div}}\nolimits (A\nabla u)$ be two real (not necessarily symmetric) uniformly elliptic operators in $\Omega $ , and write $\omega _{L_0}, \omega _L$ for the respective associated elliptic measures. We establish the equivalence between the following properties: (i) $\omega _L \in A_{\infty }(\omega _{L_0})$ , (ii) L is $L^p(\omega _{L_0})$ -solvable for some $p\in (1,\infty )$ , (iii) bounded null solutions of L satisfy Carleson measure estimates with respect to $\omega _{L_0}$ , (iv) $\mathcal {S}<\mathcal {N}$ (i.e., the conical square function is controlled by the nontangential maximal function) in $L^q(\omega _{L_0})$ for some (or for all) $q\in (0,\infty )$ for any null solution of L, and (v) L is $\mathrm {BMO}(\omega _{L_0})$ -solvable. Moreover, in each of the properties (ii)-(v) it is enough to consider the class of solutions given by characteristic functions of Borel sets (i.e, $u(X)=\omega _L^X(S)$ for an arbitrary Borel set $S\subset \partial \Omega $ ).

Also, we obtain a qualitative analog of the previous equivalences. Namely, we characterize the absolute continuity of $\omega _{L_0}$ with respect to $\omega _L$ in terms of some qualitative local $L^2(\omega _{L_0})$ estimates for the truncated conical square function for any bounded null solution of L. This is also equivalent to the finiteness $\omega _{L_0}$ -almost everywhere of the truncated conical square function for any bounded null solution of L. As applications, we show that $\omega _{L_0}$ is absolutely continuous with respect to $\omega _L$ if the disagreement of the coefficients satisfies some qualitative quadratic estimate in truncated cones for $\omega _{L_0}$ -almost everywhere vertex. Finally, when $L_0$ is either the transpose of L or its symmetric part, we obtain the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for $\omega _{L_0}$ -almost every vertex.

This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions.

More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions.

These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the $\phi ^4$ problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.

Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson, and Wei), we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here, we extend this work to all dimensions, which turns out to have many surprising difficulties due to the fact that $d+1$, not $2^d$, grids is the optimal number in an adjacent dyadic system in $\mathbb {R}^d$. As a byproduct, we show that a collection of $d+1$ dyadic systems in $\mathbb {R}^d$ is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on $\mathbb {R}$. The underlying geometric structures that arise in this higher-dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and n-adic, for any n) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific exa.

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.

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of $L^{p}$ norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it.

In this paper, we study global-in-time, weighted Strichartz estimates for the Dirac equation on warped product spaces in dimension $n\geq 3$ . In particular, we prove estimates for the dynamics restricted to eigenspaces of the Dirac operator on the compact spin manifolds defining the ambient manifold under some explicit sufficient condition on the metric and estimates with loss of angular derivatives for general initial data in the setting of spherically symmetric and asymptotically flat manifolds.

We consider the $L^{p}$ -regularity of the Szegö projection on the symmetrised polydisc $\mathbb {G}_{n}$ . In the setting of the Hardy space corresponding to the distinguished boundary of the symmetrised polydisc, it is shown that this operator is $L^{p}$ -bounded for $p\in (2-{1}/{n}, 2+{1}/{(n-1)})$ .

We show that the system of equations

$$ \begin{align*} \sum_{i=1}^{s} (x_{i}^{\,j}-y_{i}^{\,j}) = a_{j} \quad (1 \leqslant j \leqslant k) \end{align*} $$

has appreciably fewer solutions in the subcritical range $s < \tfrac 12k(k+1)$ than its homogeneous counterpart, provided that $a_{\ell } \neq 0$ for some $\ell \leqslant k-1$ . Our methods use Vinogradov’s mean value theorem in combination with a shifting argument.

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 present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely $T$ is bounded on $L^{p}(v)$ for every $v$ in an strictly smaller class of weights than the Muckenhoupt class $A_p$. Important examples will be the Bochner–Riesz operators $BR_\lambda$ with $0<\lambda <{(n-1)}/2$, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$, $1 < r < \infty$.

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.

In this paper, we consider an equivalence relation on the space $AP(\mathbb {R},X)$ of almost periodic functions with values in a prefixed Banach space X. In this context, it is known that the normality or Bochner-type property, which characterizes these functions, is based on the relative compactness of the family of translates. Now, we prove that every equivalence class is sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class, i.e., the condition of almost periodicity of a function $f\in AP(\mathbb {R},X)$ yields that every sequence of translates of f has a subsequence that converges to a function equivalent to f. This extends previous work by the same authors on the case of numerical almost periodic functions.

We obtain a nontrivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to the growth of conjugates classes, estimates of exponential sums, and restriction phenomenon.

Our aim in this paper is to establish Trudinger’s exponential integrability for Riesz potentials in weighted Morrey spaces on the half space. As an application, we obtain Trudinger’s inequality for Riesz potentials in the framework of double phase functionals.

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.

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.