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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
$d\ge 4$
.
Let
$M=(\begin {smallmatrix}\rho ^{-1} & 0 \\0 & \rho ^{-1} \\\end {smallmatrix})$
be an expanding real matrix with
$0<\rho <1$
, and let
${\mathcal D}_n=\{(\begin {smallmatrix} 0\\ 0 \end {smallmatrix}),(\begin {smallmatrix} \sigma _n\\ 0 \end {smallmatrix}),(\begin {smallmatrix} 0\\ \gamma _n \end {smallmatrix})\}$
be digit sets with
$\sigma _n,\gamma _n\in \{-1,1\}$
for each
$n\ge 1$
. Then the infinite convolution
is called a Moran–Sierpinski measure. We give a necessary and sufficient condition for
$L^2(\,\mu _{M,\{{\mathcal D}_n\}})$
to admit an infinite orthogonal set of exponential functions. Furthermore, we give the exact cardinality of orthogonal exponential functions in
$L^2(\,\mu _{M,\{{\mathcal D}_n\}})$
when
$L^2(\,\mu _{M,\{{\mathcal D}_n\}})$
does not admit any infinite orthogonal set of exponential functions based on whether
$\rho $
is a trinomial number or not.
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
This condition was introduced by Riveros and de la Torre [33] as a one-sided counterpart of the $C_{p}$ condition studied first by Muckenhoupt and Sawyer [30, 34]. In this paper we show that given $1< p< q<\infty$ if $w\in C_{q}^+$ then
and conversely if such an inequality holds, then $w\in C_{p}^+$. This result is the one-sided counterpart of Yabuta's main result in [37]. Combining this estimate with known pointwise estimates for $M^{\sharp,+}$ in the literature we recover and extend the result for maximal one-sided singular integrals due to Riveros and de la Torre [33] obtaining counterparts a number of operators.
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
from the Sobolev space based on $\mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$ to the space $\mathrm {BMO}(\mathbb {R}^n)$, and from the Sobolev space based on the Hardy space $H^1(\mathbb {R}^n)$ to $H^1(\mathbb {R}^n)$. Moreover, for any $\lambda \in (0,\,\infty )$, we also obtain the unique solvability of the non-local elliptic equation $Lu-\lambda u=f$ in $\mathbb {R}^n$, with $f\in \mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$ or $H^1(\mathbb {R}^n)$, in the Sobolev space based on $\mathrm {BMO}(\mathbb {R}^n)$ or $H^1(\mathbb {R}^n)$. The boundedness and unique solvability results given in this paper are further devolvement for the corresponding results in the scale of the Lebesgue space $L^p(\mathbb {R}^n)$ with $p\in (1,\,\infty )$, established by H. Dong and D. Kim [J. Funct. Anal. 262 (2012), 1166–1199], in the endpoint cases of $p=1$ and $p=\infty$.
In this paper, we investigate the $H^{p}(G) \rightarrow L^{p}(G)$, $0< p \leq 1$, boundedness of multiplier operators defined via group Fourier transform on a graded Lie group $G$, where $H^{p}(G)$ is the Hardy space on $G$. Our main result extends those obtained in [Colloq. Math. 165 (2021), 1–30], where the $L^{1}(G)\rightarrow L^{1,\infty }(G)$ and $L^{p}(G) \rightarrow L^{p}(G)$, $1< p <\infty$, boundedness of such Fourier multiplier operators were proved.
In this paper, we study divergence properties of the Fourier series on Cantor-type fractal measure, also called the mock Fourier series. We give a sufficient condition under which the mock Fourier series for doubling spectral measure is divergent on a set of strictly positive measure. In particular, there exists an example of the quarter Cantor measure whose mock Fourier sums are not almost everywhere convergent.
A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.
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 Linéaire26 (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
$p\in (2-{1}/{n}, 2+{1}/{(n-1)})$
.
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.